problem solving involving ratio and proportion

5.4 Ratios and Proportions

Learning objectives.

After completing this section, you should be able to:

• Construct ratios to express comparison of two quantities.
• Use and apply proportional relationships to solve problems.
• Determine and apply a constant of proportionality.
• Use proportions to solve scaling problems.

Ratios and proportions are used in a wide variety of situations to make comparisons. For example, using the information from Figure 5.15 , we can see that the number of Facebook users compared to the number of Twitter users is 2,006 M to 328 M. Note that the "M" stands for million, so 2,006 million is actually 2,006,000,000 and 328 million is 328,000,000. Similarly, the number of Qzone users compared to the number of Pinterest users is in a ratio of 632 million to 175 million. These types of comparisons are ratios.

Constructing Ratios to Express Comparison of Two Quantities

Note there are three different ways to write a ratio , which is a comparison of two numbers that can be written as: a a to b b OR a : b a : b OR the fraction a / b a / b . Which method you use often depends upon the situation. For the most part, we will want to write our ratios using the fraction notation. Note that, while all ratios are fractions, not all fractions are ratios. Ratios make part to part, part to whole, and whole to part comparisons. Fractions make part to whole comparisons only.

Example 5.28

Expressing the relationship between two currencies as a ratio.

The Euro (€) is the most common currency used in Europe. Twenty-two nations, including Italy, France, Germany, Spain, Portugal, and the Netherlands use it. On June 9, 2021, 1 U.S. dollar was worth 0.82 Euros. Write this comparison as a ratio.

Using the definition of ratio, let a = 1 a = 1 U.S. dollar and let b = 0.82 b = 0.82 Euros. Then the ratio can be written as either 1 to 0.82; or 1:0.82; or 1 0.82 . 1 0.82 .

Your Turn 5.28

Example 5.29, expressing the relationship between two weights as a ratio.

The gravitational pull on various planetary bodies in our solar system varies. Because weight is the force of gravity acting upon a mass, the weights of objects is different on various planetary bodies than they are on Earth. For example, a person who weighs 200 pounds on Earth would weigh only 33 pounds on the moon! Write this comparison as a ratio.

Using the definition of ratio, let a = 200 a = 200 pounds on Earth and let b = 33 b = 33 pounds on the moon. Then the ratio can be written as either 200 to 33; or 200:33; or 200 33 . 200 33 .

Your Turn 5.29

Using and applying proportional relationships to solve problems.

Using proportions to solve problems is a very useful method. It is usually used when you know three parts of the proportion, and one part is unknown. Proportions are often solved by setting up like ratios. If a b a b and c d c d are two ratios such that a b = c d , a b = c d , then the fractions are said to be proportional . Also, two fractions a b a b and c d c d are proportional ( a b = c d ) ( a b = c d ) if and only if a × d = b × c a × d = b × c .

Example 5.30

Solving a proportion involving two currencies.

You are going to take a trip to France. You have $520 U.S. dollars that you wish to convert to Euros. You know that 1 U.S. dollar is worth 0.82 Euros. How much money in Euros can you get in exchange for $520?

Step 1: Set up the two ratios into a proportion; let x x be the variable that represents the unknown. Notice that U.S. dollar amounts are in both numerators and Euro amounts are in both denominators.

Step 2: Cross multiply, since the ratios a b a b and c d c d are proportional, then a × d = b × c a × d = b × c .

You should receive 426.4 426.4 Euros ( 426.4 € ) ( 426.4 € ) .

Your Turn 5.30

Example 5.31, solving a proportion involving weights on different planets.

A person who weighs 170 pounds on Earth would weigh 64 pounds on Mars. How much would a typical racehorse (1,000 pounds) weigh on Mars? Round your answer to the nearest tenth.

Step 1: Set up the two ratios into a proportion. Notice the Earth weights are both in the numerator and the Mars weights are both in the denominator.

Step 2: Cross multiply, and then divide to solve.

So the 1,000-pound horse would weigh about 376.5 pounds on Mars.

Your Turn 5.31

Example 5.32, solving a proportion involving baking.

A cookie recipe needs 2 1 4 2 1 4 cups of flour to make 60 cookies. Jackie is baking cookies for a large fundraiser; she is told she needs to bake 1,020 cookies! How many cups of flour will she need?

Step 1: Set up the two ratios into a proportion. Notice that the cups of flour are both in the numerator and the amounts of cookies are both in the denominator. To make the calculations more efficient, the cups of flour ( 2 1 4 ) ( 2 1 4 ) is converted to a decimal number (2.25).

Step 2: Cross multiply, and then simplify to solve.

Jackie will need 38.25, or 38 1 4 38 1 4 , cups of flour to bake 1,020 cookies.

Your Turn 5.32

Part of the definition of proportion states that two fractions a b a b and c d c d are proportional if a × d = b × c a × d = b × c . This is the "cross multiplication" rule that students often use (and unfortunately, often use incorrectly). The only time cross multiplication can be used is if you have two ratios (and only two ratios) set up in a proportion. For example, you cannot use cross multiplication to solve for x x in an equation such as 2 5 = x 8 + 3 x 2 5 = x 8 + 3 x because you do not have just the two ratios. Of course, you could use the rules of algebra to change it to be just two ratios and then you could use cross multiplication, but in its present form, cross multiplication cannot be used.

People in Mathematics

Eudoxus was born around 408 BCE in Cnidus (now known as Knidos) in modern-day Turkey. As a young man, he traveled to Italy to study under Archytas, one of the followers of Pythagoras. He also traveled to Athens to hear lectures by Plato and to Egypt to study astronomy. He eventually founded a school and had many students.

Eudoxus made many contributions to the field of mathematics. In mathematics, he is probably best known for his work with the idea of proportions. He created a definition of proportions that allowed for the comparison of any numbers, even irrational ones. His definition concerning the equality of ratios was similar to the idea of cross multiplying that is used today. From his work on proportions, he devised what could be described as a method of integration, roughly 2000 years before calculus (which includes integration) would be fully developed by Isaac Newton and Gottfried Leibniz. Through this technique, Eudoxus became the first person to rigorously prove various theorems involving the volumes of certain objects. He also developed a planetary theory, made a sundial still usable today, and wrote a seven volume book on geography called Tour of the Earth , in which he wrote about all the civilizations on the Earth, and their political systems, that were known at the time. While this book has been lost to history, over 100 references to it by different ancient writers attest to its usefulness and popularity.

Determining and Applying a Constant of Proportionality

In the last example, we were given that 2 1 4 2 1 4 cups of flour could make 60 cookies; we then calculated that 38 1 4 38 1 4 cups of flour would make 1,020 cookies, and 720 cookies could be made from 27 cups of flour. Each of those three ratios is written as a fraction below (with the fractions converted to decimals). What happens if you divide the numerator by the denominator in each?

The quotients in each are exactly the same! This number, determined from the ratio of cups of flour to cookies, is called the constant of proportionality . If the values a a and b b are related by the equality a b = k , a b = k , then k k is the constant of proportionality between a a and b b . Note since a b = k , a b = k , then b = a k . b = a k . and b = a k . b = a k .

One piece of information that we can derive from the constant of proportionality is a unit rate. In our example (cups of flour divided by cookies), the constant of proportionality is telling us that it takes 0.0375 cups of flour to make one cookie. What if we had performed the calculation the other way (cookies divided by cups of flour)?

In this case, the constant of proportionality ( 26.66666 … = 26 2 3 ) ( 26.66666 … = 26 2 3 ) is telling us that 26 2 3 26 2 3 cookies can be made with one cup of flour. Notice in both cases, the "one" unit is associated with the denominator. The constant of proportionality is also useful in calculations if you only know one part of the ratio and wish to find the other.

Example 5.33

Finding a constant of proportionality.

Isabelle has a part-time job. She kept track of her pay and the number of hours she worked on four different days, and recorded it in the table below. What is the constant of proportionality, or pay divided by hours? What does the constant of proportionality tell you in this situation?

To find the constant of proportionality, divide the pay by hours using the information from any of the four columns. For example, 87.5 7 = 12.5 87.5 7 = 12.5 . The constant of proportionality is 12.5, or $12.50. This tells you Isabelle's hourly pay: For every hour she works, she gets paid $12.50.

Your Turn 5.33

Example 5.34, applying a constant of proportionality: running mph.

Zac runs at a constant speed: 4 miles per hour (mph). One day, Zac left his house at exactly noon (12:00 PM) to begin running; when he returned, his clock said 4:30 PM. How many miles did he run?

The constant of proportionality in this problem is 4 miles per hour (or 4 miles in 1 hour). Since a b = k , a b = k , where k k is the constant of proportionality, we have

a miles b hours = k a miles b hours = k

a 4 .5 = 4 a 4 .5 = 4 (30 minutes is ½ ½ , or 0.5 0.5 , hours)

a = 4 ( 4.5 ) a = 4 ( 4.5 ) , since from the definition we know a = k b a = k b

a = 18 a = 18

Zac ran 18 miles.

Your Turn 5.34

Example 5.35, applying a constant of proportionality: filling buckets.

Joe had a job where every time he filled a bucket with dirt, he was paid $2.50. One day Joe was paid $337.50. How many buckets did he fill that day?

The constant of proportionality in this situation is $2.50 per bucket (or $2.50 for 1 bucket). Since a b = k , a b = k , where k k is the constant of proportionality, we have

a dollars b buckets = k 337.50 b = 2.50 a dollars b buckets = k 337.50 b = 2.50

Since we are solving for b b , and we know from the definition that b = a k : b = a k :

b = 337.50 2.50 b = 135 b = 337.50 2.50 b = 135

Joe filled 135 buckets.

Your Turn 5.35

Example 5.36, applying a constant of proportionality: miles vs. kilometers.

While driving in Canada, Mabel quickly noticed the distances on the road signs were in kilometers, not miles. She knew the constant of proportionality for converting kilometers to miles was about 0.62—that is, there are about 0.62 miles in 1 kilometer. If the last road sign she saw stated that Montreal is 104 kilometers away, about how many more miles does Mabel have to drive? Round your answer to the nearest tenth.

The constant of proportionality in this situation is 0.62 miles per 1 kilometer. Since a b = k , a b = k , where k k is the constant of proportionality, we have

a miles b kilometers = k a 104 = 0.62 a = 0.62 ( 104 ) a = 64.48 a miles b kilometers = k a 104 = 0.62 a = 0.62 ( 104 ) a = 64.48

Rounding the answer to the nearest tenth, Mabel has to drive 64.5 miles.

Your Turn 5.36

Using proportions to solve scaling problems.

Ratio and proportions are used to solve problems involving scale . One common place you see a scale is on a map (as represented in Figure 5.16 ). In this image, 1 inch is equal to 200 miles. This is the scale. This means that 1 inch on the map corresponds to 200 miles on the surface of Earth. Another place where scales are used is with models: model cars, trucks, airplanes, trains, and so on. A common ratio given for model cars is 1:24—that means that 1 inch in length on the model car is equal to 24 inches (2 feet) on an actual automobile. Although these are two common places that scale is used, it is used in a variety of other ways as well.

Example 5.37

Solving a scaling problem involving maps.

Figure 5.17 is an outline map of the state of Colorado and its counties. If the distance of the southern border is 380 miles, determine the scale (i.e., 1 inch = how many miles). Then use that scale to determine the approximate lengths of the other borders of the state of Colorado.

When the southern border is measured with a ruler, the length is 4 inches. Since the length of the border in real life is 380 miles, our scale is 1 inch = 95 = 95 miles.

The eastern and western borders both measure 3 inches, so their lengths are about 285 miles. The northern border measures the same as the southern border, so it has a length of 380 miles.

Your Turn 5.37

Example 5.38, solving a scaling problem involving model cars.

Die-cast NASCAR model cars are said to be built on a scale of 1:24 when compared to the actual car. If a model car is 9 inches long, how long is a real NASCAR automobile? Write your answer in feet.

The scale tells us that 1 inch of the model car is equal to 24 inches (2 feet) on the real automobile. So set up the two ratios into a proportion. Notice that the model lengths are both in the numerator and the NASCAR automobile lengths are both in the denominator.

This amount (216) is in inches. To convert to feet, divide by 12, because there are 12 inches in a foot (this conversion from inches to feet is really another proportion!). The final answer is:

The NASCAR automobile is 18 feet long.

Your Turn 5.38

Check your understanding, section 5.4 exercises.

As an Amazon Associate we earn from qualifying purchases.

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Access for free at https://openstax.org/books/contemporary-mathematics/pages/1-introduction

• Authors: Donna Kirk
• Publisher/website: OpenStax
• Book title: Contemporary Mathematics
• Publication date: Mar 22, 2023
• Location: Houston, Texas
• Book URL: https://openstax.org/books/contemporary-mathematics/pages/1-introduction
• Section URL: https://openstax.org/books/contemporary-mathematics/pages/5-4-ratios-and-proportions

© Dec 21, 2023 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.

WORD PROBLEMS ON RATIO AND PROPORTION

Problem 1 :

The average age of three boys is 25 years and their ages are in the proportion 3 : 5 : 7. Find the age of the youngest boy.

From the ratio 3 : 5 : 7, the ages of three boys are 3x, 5x and 7x.

Average age of three boys = 25

(3x + 5x + 7x)/3 = 25

Age of the first boy = 3x

Age of the first boy = 5x

Age of the first boy = 7x

So, the age of the youngest boy is 15 years.

Problem 2 :

John weighs 56.7 kilograms. If he is going to reduce his weight in the ratio 7 : 6, find his new weight.

Given :  Original weight of John = 56.7 kg. He is going to reduce his weight in the ratio 7:6.

We can use the following hint to find his new weight, after it is reduced in the ratio 7 : 6.

His new weight is

= (6  ⋅  56.7)/7

So, John's new weight is 48.6 kg.

Problem 3 :

The ratio of the no. of boys to the no. of girls in a school of 720 students is 3 : 5. If 18 new girls are admitted in the school, find how many new boys may be admitted so that the ratio of the no. of boys to the no. of girls may change to 2 : 3.

Sum of the terms in the given ratio is

So, no. of boys in the school is

= 720  ⋅  (3/8)

= 270 

No. of girls in the school is

= 720  ⋅  (5/8) 

Given : Number of new girls admitted in the school is 18.

Let x be the no. of new boys admitted in the school.

After the above new admissions,

No. of boys in the school = 270 + x

No. of girls in the school = 450 + 18 = 468

Given : The ratio after the new admission is 2 : 3.

Then, we have

(270 + x) : 468 = 2 : 3

Use cross product rule.

3(270 + x) = 468  ⋅  2

810 + 3x = 936

So, the number of new boys admitted in the school is 42.

Problem 4 :

The monthly incomes of two persons are in the ratio 4 : 5 and their monthly expenditures are in the ratio 7 : 9. If each saves $50 per month, find the monthly income of the second person.

From the given ratio of incomes ( 4 : 5 ),

Income of the 1st person = 4x

Income of the 2nd person = 5x

(Expenditure  =  Income - Savings)

Then, expenditure of the 1st person = 4x - 50

Expenditure of the 2nd person = 5x - 50

Expenditure ratio = 7 : 9 (given)

So, we have

(4x - 50) : (5x - 50) = 7 : 9

9(4x - 50) = 7(5x - 50)

36x - 450 = 35x - 350

Then, the income  of the second person is

So, income of the second person is $500.

Problem 5 :

The ratio of the prices of two houses was 16 : 23. Two years later when the price of the first has increased by 10% and that of the second by $477, the ratio of the prices becomes 11 : 20. Find the original price of the first house.

From the given ratio 16 : 23,

Original price of the 1st house = 16x

Original price of the 2nd house = 23x

After increment in prices,

Price of the 1st house = 16x + 10% of 16x 

= 16x + 1.6x

Price of the 2nd house = 23x + 477

After increment in prices, the ratio of prices becomes 11:20.

17.6x : (23x + 477) = 11 : 20

20(17.6x) = 11(23x + 477)

352x = 253x + 5247

Then, original price of the first house is

So, original price of the first house is $848.

Problem 6 :

Two numbers are respectively 20% and 50% are more than a third number, Find the ratio of the two numbers.

Let x be the third number.

Then, the first number is

= (100 + 20)% of x

= 120% of x

The second number is

= (100 + 50)% of x

= 150% of x

The ratio between the first number and second number is

= 1.2x : 1.5x

= 1.2 : 1.5

So, the ratio of two numbers is 4 : 5.

Problem 7 :

The milk and water in two vessels A and B are in the ratio 4:3 and 2:3 respectively. In what ratio, the liquids in both the vessels be mixed to obtain a new mixture in vessel C consisting half milk and half water ?    

[4 : 3 ----> 4 + 3 = 7, M ----> 4/7, W ----> 3/7]

Let x be the quantity of mixture taken from vessel A to obtain a new mixture in vessel C.

Quantity of milk in x = (4/7)x = 4x/7

Quantity of water in x = (3/7)x = 3x/7

[2 : 3 ----> 2 + 3 = 5, M ---> 2/5, W ----> 3/5]

Let y be the quantity of mixture taken from vessel B to obtain a new mixture in vessel C.

Quantity of milk in y = (2/5)y = 2y/5

Quantity of water in y = (3/5)y = 3y/5

Vessel A and B :

Quantity of milk from A and B is

= 4x/7 + 2y/5

= (20x + 14y)/35

Quantity of water from A and B is

= (3x/7) + (3y/5)

= (15x + 21y)/35

According to the question, vessel C must consist half of the milk and half of the water.

That is, in vessel C, quantity of milk and water must be same.

There fore,

quantity of milk in (A + B) = quantity of water in (A + B)

(20x + 14y)/35 = (15x + 21y)/35

20x + 14y = 15x + 21y

x : y = 7 : 5

So, the required ratio is 7 : 5.

Problem 8 :

A vessel contains 20 liters of a mixture of milk and water in the ratio 3:2. From the vessel, 10 liters of the mixture is removed  and replaced with an equal quantity of pure milk. Find the ratio of milk and water in the final mixture obtained.    

[3 : 2 ----> 3 + 2 = 5, M ----> 3/5, W ----> 2/5]

In 20 liters of mixture,

no. of liters of milk = 20  ⋅  3/5 = 12

no. of liters of water = 20  ⋅  2/5 = 8

Now, 10 liters of mixture removed.

In this 10 liters of mixture, milk and water will be in the ratio 3 : 2.

no. of liters of milk in this 10 liters = 10  ⋅  3/5 = 6

no. of liters of water in this 10 liters = 10  ⋅  2/5 = 4

After removing 10 liters (1st time),

no. of liters of milk in the vessel = 12 - 6 = 6

no. of liters of water in the vessel = 8 - 4 = 4

Now,  we add 10 liters of pure milk in the vessel,

After adding 10 liters of pure milk in the vessel,

no. of liters of milk in the vessel = 6 + 10 = 16

no. of liters of water in the vessel = 4 + 0 = 4

After removing 10 liters of mixture and adding 10 liters of pure milk, the ratio of milk and water is

So, the required ratio is 4 : 1.

Kindly mail your feedback to   [email protected]

We always appreciate your feedback.

© All rights reserved. onlinemath4all.com

• Sat Math Practice
• SAT Math Worksheets
• PEMDAS Rule
• BODMAS rule
• GEMDAS Order of Operations
• Math Calculators
• Transformations of Functions
• Order of rotational symmetry
• Lines of symmetry
• Compound Angles
• Quantitative Aptitude Tricks
• Trigonometric ratio table
• Word Problems
• Times Table Shortcuts
• 10th CBSE solution
• PSAT Math Preparation
• Privacy Policy
• Laws of Exponents

Recent Articles

Multi Step Algebra Word Problems

Apr 23, 24 09:10 PM

Solving Multi Step Word Problems Worksheet

Apr 23, 24 12:32 PM

Solving Multi Step Word Problems

Apr 23, 24 12:07 PM

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons
• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

1.4: Proportions

• Last updated
• Save as PDF
• Page ID 130921

In the previous section, we learned that a ratio is a comparison of two quantities. However, many of the problems we solved involved comparing multiple ratios, and often required finding an equality between two ratios. These sorts of problems are most easily solved using proportions , which are the subject of this section.

In this section, you will learn to:

• Recognize and set up proportion problems
• Apply the Cross Multiplication and Division undoes Multiplication methods to solve proportion problems
• Use the Compare to the Whole method to solve problems involving proportions

Proportions: Definition and Basic Methods

Let's recall our starting example from the previous section, in which we had a recipe that called for \(2\) cups of flour and \(1\) cup of sugar. We asked the question: if we wanted to make a larger version of the same recipe using \(3\) cups of sugar, how much flour should we use?

We know now that ratios can be expressed as fractions, and whenever ratios are equivalent, the fractional representations of those ratios are equal. So, let's call the number of cups of flour \(x\) and set up the following equality:

\[\frac{2 \text{ cups of flour}}{1 \text{ cup of sugar}} = \frac{x \text{ cups of flour}}{3 \text{ cups of sugar}}\]

We need to figure out the value of \(x\) that makes this equation true. That is, what number can be substituted in place of \(x\) so that these fractions truly are equal? In this case, the easiest way is to guess-and-check, which is a perfectly valid solution method with small numbers. If you just try some small numbers, you can find that \(x = 6\) is the solution, because \(\frac{6}{3}\) reduces to \(\frac{2}{1}\), and thus \(\frac{6}{3}\) and \(\frac{2}{1}\)are equivalent ratios.

In this chapter, we'll learn how to solve problems like this generally, including those that can't be solved using guess-and-check. Our primary tool will be a proportion.

Definition: Proportion

A proportion is an equality between ratios.

This means that, in a strictly mathematical sense, a proportion is an equation. For example,

\[\frac{5}{2} = \frac{40}{16}\]

is a proportion, because it contains two ratios that are equal to one another. You may have heard the word proportion used in other ways, and that's because the word "proportion" has a different mathematical meaning than its typical English usage. That doesn't mean either usage is wrong; rather, it is dependent upon context. In this book, we will use the word proportion to mean any equation that looks like this:

\[\frac{a}{b} = \frac{c}{d}\]

where \(a, b, c\), and \(d\) will usually be numbers or variables.

The reasons we care about proportions is that they give us a way to find an unknown part of one of the ratios involved . Recall the following example, which we saw in the Ratios section:

Example \(\PageIndex{1}\)

On the Western Oregon University website , the total enrollment is listed as \(3752\) students, and the student-faculty ratio is listed as \(13 \colon 1\). You want to know how many faculty there are at WOU. How might you find this out, and how do you explain your answer?

In solving this problem before, we set up two ratios

\[3752 \colon x \quad \text{and} \quad 13 \colon 1 \]

Why did we do this? Well, it turns out that all proportion problems can be solved using a method from algebra known as cross multiplication . While this text mostly stays away from algebra, this procedure is essential. The good news is that it works the same way every time, and it's not very complicated.

Cross Multiplication

If you have a proportion of the form:

then "cross multiplication" refers to rewriting the equation in the following equivalent way:

\[c \times b= a \times d\]

In other words, we are "crossing" from \(a\) to \(d\) in the \(\searrow\) direction and from \(c\) to \(b\) in the \(\swarrow\) direction.

A quick mathematical note: what we're really doing is multiplying both sides by \(b\) and \(d\), and then canceling common factors -- but calling it cross multiplication seems to make it easier for students to understand and remember.

Let's practice cross multiplying in our example. We had the proportion:

\[\frac{3752}{x} = \frac{13}{1}\]

Cross multiplying this gives us the following equation:

\[13 \times x = 3752 \times 1\]

What good did that do? Well, note that we can simplify a little bit. It's typical to omit the \(\times\) when a variable is multiplied by a number, so we can rewrite \(13x\) for \(13 \times x\). We also know that \(3752 \times 1 = 3752\). So our equation becomes

\[13x = 3752\]

Now what? We've eliminated the fractions, but we can't yet say what \(x\) is. In order to find \(x\), we need one more algebra procedure, which we will call Division undoes Multiplication.

Division undoes Multiplication

Given an equation of the form \[Ax = B\] where \(A\) and \(B\) are numbers, we can find the value of \(x\) by dividing both sides by \(A\). That is, \[x = \frac{B}{A} = B \div A\]

Once again, we are using properties of fractions here: mathematically, we are dividing both sides by \(A\) and then reducing:

\[\begin{align*} Ax & = B\\ \frac{Ax}{A} & = \frac{B}{A} \\ \frac{\cancel{A}x}{\cancel{A}} & = \frac{B}{A} \\ x & = \frac{B}{A} \\ \end{align*} \]

But this is another procedure used so frequently that it's worth giving it a name.

Back to our example: we had the equation \[13x = 3752\]

We now have a tool to find \(x\) -- the fact that Division undoes Multiplication ! Using this procedure, we have \[x = \frac{3752}{13} = 3752 \div 13 \approx 288.6\]

This is the same answer we found before, but we used a slightly different method. And keep in mind that, just as the previous section, we would need to round this answer to \(289\) faculty to make sense in context. That said; the main point is now we now have a fool-proof way to solve this type of equation! 

While this may seem more complicated at first, you'll find that the following sequence of steps will always work to solve proportions:

Solving Proportions

• Set up the proportion with exactly one unknown value, called \(x\).
• Apply the Cross Multiplication.
• Apply Division undoes Multiplication.

We will get lots of practice with this procedure in the exercises for this section. Once you practice with the procedures above, you'll find that it's not too bad. The hardest part is often the first step — setting up the proportion correctly. That's the part that depends on reading the question very carefully! In general, the way to set up a proportion involves keeping track of units. Let's see an example to understand.

Example \(\PageIndex{2}\)

In an office supply store, \(8\) markers cost a total of \(\$12.00\). Assuming all markers are equally priced, how much would 6 markers cost?

This is a problem that is suitable to be solved using proportions because the markers are all equally priced, meaning that the ratio of total cost : number of markers purchased will be the same, no matter how many markers are purchased. That means we can set up the following proportion:

\[\frac{12 \text{ dollars}}{8 \text{ markers}} = \frac{x \text{ dollars}}{6 \text{ markers}}\]

Notice how, in the equation above, we are labeling the units of all quantities involved. Moreover, the units on each side match: dollars are on top, markers are on bottom, and the corresponding quantities are grouped on each side of the equation -- \(\$12\) for \(8\) markers, and \(\$x\) for 6 markers. Labeling your units in this way will help you avoid mistakes with units!

Now that we've gotten our proportion set up correctly, we can rewrite it without labels: \[\frac{12}{8} = \frac{x}{6}\]

From here, we'll follow the last two steps: cross multiply, and then use division to find \(x\). Using Cross Multiplication, we have \[x \times 8 = 12 \times 6 \]

On the left, we can rewrite \(x \times 8\) as \(8 x\), since multiplication can always switch orders. Then we can simplify to get \[8x = 72\]

Now we can use Division undoes Multiplication to get

\[x = \frac{72}{8} = 72 \div 8 = 9\]

Therefore, \(x = 9\). Now, we want to make sure our answer actually means something. What are the units on \(x\)? Well, if we look back at our original proportion,

we see that \(x\) is a number of dollars. Thus, we can say that \(x = \$9\), which means that 6 markers will cost \(\$9\).

You may be thinking: there is a much faster way to do that! And that may be true for you. Once again, the point is not to mimic a particular method for problem solving here — these notes will show some good ways of solving a problem, but they cannot cover every good solution. They are intended to highlight themes and strategies that will work for many types of situations. Other ways you may have solved the problem above include:

• Calculate the cost per marker to be \(\$1.50\), and multiply that number by 6 markers to get \(\$9\).
• Calculate that \(6\) is \(\frac{3}{4}\) of \(8\), so the cost of \(6\) markers would be \(\frac{3}{4}\) the cost of \(8\) markers, and \(\frac{3}{4}\) of \(\$12\) is \(\$9\).
• Set up a different initial proportion, such as \(\frac{12 \text{ dollars}}{x \text{ dollars}} = \frac{8 \text{ markers}}{6 \text{ markers}}\) or \(\frac{8 \text{ markers}}{12 \text{ dollars}} = \frac{6 \text{ markers}}{x \text{ dollars}}\) and then solved that proportion.

What's amazing about the last point above is that both of those proportions — which were different than the method used in the solution above — still give the same answer! This shows that there are many  different ways of approaching the same problem. All you need to do is find the one that works for you, and be able to explain your work.

Comparing to the Whole

Sometimes a problem involving proportions will be less straightforward. For example, consider the following:

Example \(\PageIndex{3}\)

In a rainforest in Panama, the ratio of two-toed sloths to three-toed sloths is \(10 \colon 3\). There are \(741\) total sloths in the rainforest. How many of them are two-toed?

In the problem above, we are given one ratio that compares the quantity of two-toed versus three-toed sloths. However, we are not given any information about the actual numbers of either two- or three-toed sloths. We simply know the comparison between them. Instead, we are just given the total number of sloths, but no actual breakdown into how many fall into each category. How are we supposed to find the number of two-toed sloths from just this information? We can't readily write down a proportion like we were able to in the previous example, because the units would be wrong; we need to compare like quantities. This situation calls for one more procedure.

Compare to the Whole

Assume there are two quantities, \(x\) and \(y\), neither of which you know. However, you know two things about them

• The total \(x + y\) (the total number of both quantities)
• The ratio of quantity \(x\) to quantity \(y\) is \(a \colon b\)

Then you can use the Compare to the Whole method. This says that, to find quantity \(x\), you use the proportion \[\frac{a}{a+b} = \frac{x}{x+y}\] and then find \(x\). Note: you already know \(x+y\), since it is the total number of both quantities.

Let's see how this procedure can be applied to the sloth example.

Example \(\PageIndex{3}\) Revisited

In this question, our two quantities \(x\) and \(y\) are the number of two- and three-toed sloths, respectively. We are asked to find \(x\), the number of two-toed sloths. Our known ratio is \(10 \colon 3\), so using the notation of the Compare to the Whole method, we have \(a = 10\) and \(b = 3\), and \(a +b = 13\). We also know that the total number of sloths is 741, so \(x + y = 741\). So we'll set up the following proportion -- pay close attention to the labels!

\[\frac{10 \text{ two-toed sloths}}{13 \text{ total sloths}} = \frac{x \text{ two-toed sloths}}{741 \text{ total sloths}}\]

On the righthand side of the proportion above, the ratio \(\frac{x}{741}\) represents the actual number of sloths, in which there are \(x\) two-toed sloths out of a total of \(741\) total sloths.

On the lefthand side, the ratio \(\frac{10}{13}\) represents an imaginary "smaller but proportional rainforest," in which there are only \(10\) two-toed and \(3\) three-toed sloths, for a total of \(13\) sloths in our imaginary smaller rainforest.

Proportionality says that these proportions must be equal, but since we don't know the breakdown of the total number of sloths, we must compare to the whole , which means we must compare the total number of sloths on each side. We get a total number of \(13\) on the left by computing \(10 + 3\), and on the right, we know the total to be \(741\).

Once we have that proportion, we can simply solve it using our processes from the previous section. From the proportion \[\frac{10}{13} = \frac{x}{741}\]

we use Cross Multiplication to obtain \[13x = 7410\]

and then use Division undoes Multiplication to get \[x = \frac{7410}{13} = 570\]

Looking back, we see that \(x\) represents the number of two-toed sloths. Therefore, there are \(570\) two-toed sloths in the rainforest.

That's the best way to think about the Compare to the Whole method -- the ratio you are given represents a "smaller version" of the situation described, and to find the total quantity in the smaller version, you simply add the two parts together. Then compare that to the actual total quantity using a proportion. If the problem statement contains words like "total," "whole," or "all together," it's likely that you'll need to use the Compare to the Whole method . However, as always, the most important thing is to read the problem and think critically about what it's asking!

P.S. for this section: You may notice that some of the algebra is becoming less explicit as we see more and more examples. If you are confused about why an algebra or arithmetic step is true, try looking for a similar problem earlier in this book — there is likely an explanation there. If you can't find one, or are still confused, you should ask your instructor or email the author of this book at [email protected] .

When you are completing these exercises, make sure to show supporting work. 

• You can walk 2 miles in 36 minutes. How long will it take you to walk 5 miles? Give you answer as a number of hours plus a number of minutes (that is, you would express 70 minutes as "1 hour and 10 minutes"). Remember that there are 60 minutes in an hour!
• You can mow 1/3 of an acre of lawn in 90 minutes. How long would it take you to mow 2 acres of lawn? Give your answer as a number of hours.
• When brewing an amber ale (a type of beer), recipes typically call for an 8:2 ratio of pale malts to crystal malts (these are types of grain in the beer). If you are brewing a 10 gallon batch of amber ale, you need a total of 22 pounds of malt. How many pounds of each type of malt (pale and crystal) should you buy? Make sure to indicate both answers clearly, and do not round them -- decimals are fine. [Hint: they should add up to 22 pounds!]
• How long of a shadow does a 6 foot tall person cast?
• If a shadow of a tree is 20 feet long, how tall is the tree?
• The ratio of registered Democrats to registered Republicans is 47 : 52 in Polk County. There is a total of 8920 registered Democrat and Republican voters. How many of them are Democrats?

"More than 200,000 sea turtles nest on or near Raine, a tiny 80-acre curl of sand along the northern edge of the  Great Barrier Reef , the portion hardest hit by warming waters. The other portion of that sea turtle population nests further from the equator, near Brisbane, where temperature increases have not been as dramatic.

What Allen and Jensen discovered was significant. Older turtles that had emerged from their eggs 30 or 40 years earlier were also mostly female, but only by a 6 to 1 ratio. But younger turtles for at least the last 20 years had been  more than 99 percent female . And as evidence that rising temperatures were responsible, female turtles from the cooler sands near Brisbane currently still only outnumber males 2 to 1.

Six weeks after Allen and Jensen published their results,  another study  from Florida looking at loggerheads revealed that temperature is just one factor. If sands are moist and cool, they produce more males. If sands are hot and dry, hatchlings are more female.

But new research in the last year also offered rays of hope." 

• What ratios can you find above? Write them down, stating explicitly what they are comparing.
• What two factors does this article assert affect the sex of sea turtles? List them.
• Given the information in the article, if a randomly selected group of 120 turtles from Brisbane have their sex examined, how many do you expect to be female? Show your work.
• Write a 2-4 sentence reaction to the article excerpt above, and make sure to answer the following question: do you feel that that ratios in the article are presented in a way that makes sense? If not, how else could you present this same information?

[*Note: you can access the article for free if you enter your email when prompted; however, you do NOT need to access the article answer this question.] 

Direct & Inverse Proportions/Variations

In these lessons, we will learn how to solve direct proportions (variations) and inverse proportions (inverse variations) problems. (Note: Some texts may refer to inverse proportions/variations as indirect proportions/variations.)

Related Pages: Direct Variations Proportion Word Problems More Algebra Lessons

The following diagram gives the steps to solve ratios and direct proportion word problems. Scroll down the page for examples and step-by-step solutions.

Direct Proportions/Variations

Knowing that the ratio does not change allows you to form an equation to find the value of an unknown variable.

Example : If two pencils cost $1.50, how many pencils can you buy with $9.00?

How To Solve Directly Proportional Questions?

Example 1: F is directly proportional to x. When F is 6, x is 4. Find the value of F when x is 5. Example 2: A is directly proportional to the square of B. When A is 10, B is 2. Find the value of A when B is 3.

How To Use Direct Proportion?

How To Solve Word Problems Using Proportions?

This video shows how to solve word problems by writing a proportion and solving 1. A recipe uses 5 cups of flour for every 2 cups of sugar. If I want to make a recipe using 8 cups of flour, how much sugar do I use? 2. A syrup is made by dissolving 2 cups of sugar in 2/3 cups of boiling water. How many cups of sugar should be used for 2 cups of boiling water? 3. A school buys 8 gallons of juice for 100 kids. how many gallons do they need for 175 kids?

Solving More Word Problems Using Proportions

1. On a map, two cities are 2 5/8 inches apart. If 3/8 inches on the map represents 25 miles, how far apart are the cities (in miles)? 2. Solve for the sides of similar triangles using proportions

Inverse Proportions/Variations Or Indirect Proportions

Two values x and y are inversely proportional to each other when their product xy is a constant (always remains the same). This means that when x increases y will decrease, and vice versa, by an amount such that xy remains the same.

Knowing that the product does not change also allows you to form an equation to find the value of an unknown variable

Example : It takes 4 men 6 hours to repair a road. How long will it take 8 men to do the job if they work at the same rate?

Solution : The number of men is inversely proportional to the time taken to do the job. Let t be the time taken for the 8 men to finish the job. 4 × 6 = 8 × t 24 = 8t t = 3 hours

Usually, you will be able to decide from the question whether the values are directly proportional or inversely proportional.

How To Solve Inverse Proportion Questions?

This video shows how to solve inverse proportion questions. It goes through a couple of examples and ends with some practice questions Example 1: A is inversely proportional to B. When A is 10, B is 2. Find the value of A when B is 8 Example 2: F is inversely proportional to the square of x. When A is 20, B is 3. Find the value of F when x is 5.

How To Use Inverse Proportion To Work Out Problems?

How to use a more advanced form of inverse proportion where the use of square numbers is involved.

More examples to explain direct proportions / variations and inverse proportions / variations

How to solve Inverse Proportion Math Problems on pressure and volume?

In math, an inverse proportion is when an increase in one quantity results in a decrease in another quantity. This video will show how to solve an inverse proportion math problem. Example : The pressure in a piston is 2.0 atm at 25°C and the volume is 4.0L. If the pressure is increased to 6.0 atm at the same temperature, what will be the volume?

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.

Worked out Problems on Ratio and Proportion

Worked out problems on ratio and proportion are explained here in detailed description using step-by-step procedure. Solved examples involving different questions related to comparison of ratios in ascending order or descending order, simplification of ratios and also word problems on ratio proportion. Sample questions and answers are given below in the worked out problems on ratio and proportion to get the basic concepts of solving ratio proportion.

1. Arrange the following ratios in descending order. 

        2 : 3, 3 : 4, 5 : 6, 1 : 5  Solution:   Given ratios are 2/3, 3/4, 5/6, 1/5  The L.C.M. of 3, 4, 6, 5 is 2 × 2 × 3 × 5 = 60 

Now, 2/3 = (2 × 20)/(3 × 20) = 40/60           3/4 = (3 × 15)/(4 × 15) = 45/60           5/6 = (5 × 10)/(6 × 10) = 50/60           1/5 = (1 × 12)/(5 × 12) = 12/60  Clearly, 50/60 > 45/60 > 40/60 > 12/60  Therefore, 5/6 > 3/4 > 2/3 > 1/5  So, 5 : 6 > 3 : 4 > 2 : 3 > 1 : 5

2. Two numbers are in the ratio 3 : 4. If the sum of numbers is 63, find the numbers. Solution: Sum of the terms of the ratio = 3 + 4 = 7 Sum of numbers = 63 Therefore, first number = 3/7 × 63 = 27 Second number = 4/7 × 63 = 36 Therefore, the two numbers are 27 and 36.

3. If x : y = 1 : 2, find the value of (2x + 3y) : (x + 4y) Solution: x : y = 1 : 2 means x/y = 1/2 Now, (2x + 3y) : (x + 4y) = (2x + 3y)/(x + 4y) [Divide numerator and denominator by y.] = [(2x + 3y)/y]/[(x + 4y)/2] = [2(x/y) + 3]/[(x/y) + 4], put x/y = 1/2 We get = [2 (1/2) + 3)/(1/2 + 4) = (1 + 3)/[(1 + 8)/2] = 4/(9/2) = 4/1 × 2/9 = 8/9 Therefore the value of (2x + 3y) : (x + 4y) = 8 : 9

More solved problems on ratio and proportion are explained here with full description.

4.  A bag contains $510 in the form of 50 p, 25 p and 20 p coins in the ratio 2 : 3 : 4. Find the number of coins of each type. 

Solution:   Let the number of 50 p, 25 p and 20 p coins be 2x, 3x and 4x.  Then 2x × 50/100 + 3x × 25/100 + 4x × 20/100 = 510 x/1 + 3x/4 + 4x/5 = 510 (20x + 15x + 16x)/20 = 510  ⇒ 51x/20 = 510 x = (510 × 20)/51  x = 200 2x = 2 × 200 = 400  3x = 3 × 200 = 600  4x = 4 × 200 = 800.  Therefore, number of 50 p coins, 25 p coins and 20 p coins are 400, 600, 800 respectively. 

5. If 2A = 3B = 4C, find A : B : C Solution: Let 2A = 3B = 4C = x So, A = x/2 B = x/3 C = x/4 The L.C.M of 2, 3 and 4 is 12 Therefore, A : B : C = x/2 × 12 : x/3 × 12 : x/4 = 12 = 6x : 4x : 3x = 6 : 4 : 3 Therefore, A : B : C = 6 : 4 : 3

6. What must be added to each term of the ratio 2 : 3, so that it may become equal to 4 : 5? Solution: Let the number to be added be x, then (2 + x) : (3 + x) = 4 : 5 ⇒ (2 + x)/(5 + x) = 4/5 5(2 + x) = 4(3 + x) 10 + 5x = 12 + 4x 5x - 4x = 12 - 10 x = 2

7. The length of the ribbon was originally 30 cm. It was reduced in the ratio 5 : 3. What is its length now? Solution: Length of ribbon originally = 30 cm Let the original length be 5x and reduced length be 3x. But 5x = 30 cm x = 30/5 cm = 6 cm Therefore, reduced length = 3 cm = 3 × 6 cm = 18 cm

More worked out problems on ratio and proportion are explained here step-by-step. 8. Mother divided the money among Ron, Sam and Maria in the ratio 2 : 3 : 5. If Maria got $150, find the total amount and the money received by Ron and Sam. Solution: Let the money received by Ron, Sam and Maria be 2x, 3x, 5x respectively. Given that Maria has got $ 150. Therefore, 5x = 150 or, x = 150/5 or, x = 30 So, Ron got = 2x                    = $ 2 × 30 = $60 Sam got = 3x               = 3 × 60 = $90

Therefore, the total amount $(60 + 90 + 150) = $300 

9. Divide $370 into three parts such that second part is 1/4 of the third part and the ratio between the first and the third part is 3 : 5. Find each part. Solution: Let the first and the third parts be 3x and 5x. Second part = 1/4 of third part.                     = (1/4) × 5x                     = 5x/4 Therefore, 3x + (5x/4) + 5x = 370 (12x + 5x + 20x)/4 = 370 37x/4 = 370 x = (370 × 4)/37 x = 10 × 4 x = 40 Therefore, first part = 3x                                 = 3 × 40                                 = $120 Second part = 5x/4                     = 5 × 40/4                     = $50 Third part = 5x                  = 5 × 40                  = $ 200

10. The first, second and third terms of the proportion are 42, 36, 35. Find the fourth term. Solution: Let the fourth term be x. Thus 42, 36, 35, x are in proportion. Product of extreme terms = 42 ×x Product of mean terms = 36 X 35 Since, the numbers make up a proportion Therefore, 42 × x = 36 × 35 or, x = (36 × 35)/42 or, x = 30 Therefore, the fourth term of the proportion is 30.

More worked out problems on ratio and proportion using step-by-step explanation. 11. Set up all possible proportions from the numbers 8, 12, 20, 30. Solution: We note that 8 × 30 = 240 and 12 × 20 = 240 Thus, 8 × 30 = 12 × 20       ………..(I) Hence, 8 : 12 = 20 : 30       ……….. (i) We also note that, 8 × 30 = 20 × 12 Hence, 8 : 20 = 12 : 30       ……….. (ii) (I) can also be written as 12 × 20 = 8 × 30 Hence, 12 : 8 = 30 : 20       ……….. (iii) Last (I) can also be written as 12 : 30 = 8 : 20       ……….. (iv) Thus, the required proportions are 8 : 12 = 20 : 30 8 : 20 = 12 : 30     12 : 8 = 30 : 20     12 : 30 = 8 : 20

12. The ratio of number of boys and girls is 4 : 3. If there are 18 girls in a class, find the number of boys in the class and the total number of students in the class. Solution: Number of girls in the class = 18 Ratio of boys and girls = 4 : 3 According to the question, Boys/Girls = 4/5 Boys/18 = 4/5 Boys = (4 × 18)/3 = 24 Therefore, total number of students = 24 + 18 = 42.

13. Find the third proportional of 16 and 20. Solution: Let the third proportional of 16 and 20 be x. Then 16, 20, x are in proportion. This means 16 : 20 = 20 : x So, 16 × x = 20 × 20 x = (20 × 20)/16 = 25 Therefore, the third proportional of 16 and 20 is 25.

●   Ratio and Proportion

What is Ratio and Proportion?

Practice Test on Ratio and Proportion

●   Ratio and Proportion - Worksheets

Worksheet on Ratio and Proportion

8th Grade Math Practice From Worked out Problems on Ratio and Proportion to HOME PAGE

Didn't find what you were looking for? Or want to know more information about Math Only Math . Use this Google Search to find what you need.

• Preschool Activities
• Kindergarten Math
• 1st Grade Math
• 2nd Grade Math
• 3rd Grade Math
• 4th Grade Math
• 5th Grade Math
• 6th Grade Math
• 7th Grade Math
• 8th Grade Math
• 9th Grade Math
• 10th Grade Math
• 11 & 12 Grade Math
• Concepts of Sets
• Probability
• Boolean Algebra
• Math Coloring Pages
• Multiplication Table
• Cool Maths Games
• Math Flash Cards
• Online Math Quiz
• Math Puzzles
• Binary System
• Math Dictionary
• Conversion Chart
• Homework Sheets
• Math Problem Ans
• Free Math Answers
• Printable Math Sheet
• Funny Math Answers
• Employment Test
• Math Patterns
• Link Partners
• Privacy Policy

Recent Articles

Dividing 3-Digit by 1-Digit Number | Long Division |Worksheet Answer

Apr 24, 24 03:46 PM

Symmetrical Shapes | One, Two, Three, Four & Many-line Symmetry

Apr 24, 24 03:45 PM

Mental Math on Geometrical Shapes | Geometry Worksheets| Answer

Apr 24, 24 03:35 PM

Circle Math | Terms Related to the Circle | Symbol of Circle O | Math

Apr 24, 24 02:57 PM

Fundamental Geometrical Concepts | Point | Line | Properties of Lines

Apr 24, 24 12:38 PM

© and ™ math-only-math.com. All Rights Reserved. 2010 - 2024.

Home / TEAS Test Review Guide / Solve Problems Involving Proportions: TEAS

Solve Problems Involving Proportions: TEAS

Basic terms and terminology relating to solving problems involving proportions, ratios and their meaning, calculations using ratio and proportion, the conversion of percentages into ratios and converting ratios into percentages, the conversion of fractions into ratios and converting ratios into fractions.

• Ratio: The relationship of two numbers
• Proportion: Two ratios that are equal to each other

Simply stated, a ratio is the relationship of two numbers and proportions are two ratios that are equal to each other.

The picture above is a ratio; this ratio could indicate that there are 4 boys for every 3 girls, that there are 4 pears for every 3 oranges or that there are $ 4 in the piggy bank for every 3 dollars in the drawer. As you can see with these examples, ratios give limited information. For example, a ratio does not tell you how many pears or oranges you actually have; a ratio does not tell you how many boys and how many girls there actually are; and, a ratio does not tell you how much money you have in the piggy bank or in the drawer.

Ratios are read as " 4 is to 3".

In order to determine how many pears or oranges you actually have, how many boys and how many girls there actually are and how much money you have in the piggy bank or in the drawer, you would have to perform ratio and proportion to answer these questions.

The different ways to express ratios are:

When comparing ratios, they should be written as fractions. The fractions must be equal. If they are not equal they are NOT considered a ratio. For example, the ratios 3/8 and 6/16 are equal and equivalent.

Proportions are two ratios that are equal to each other and these ratio and proportion problems are calculated and solved as shown below.

If there is $12 in the drawer and the ratio of money in the drawer compared to money in the piggy bank is 4 :3, how much money is in the piggy bank?

4/3 = 12x OR 12x = 4/3

12×3/4 = 36/4 = $9

Answer: There is $9 in the drawer.

If you have 8 oranges and the ratio of oranges compared to pears is 4 : 3, how many pears do you have?

4:3 = 8:x OR 4/3 = 8x OR 8x = 4/3

8×3/4 = 24/4 = 6

Answer: You have 6 pears.

If there are 24 boys in the group and there are 4 boys to 3 girls in the group, how many girls are in the group?

4:3 = 24:x OR 4/3 = 24x OR 24x = 4/3

24×3/4 = 72/4 = 18

Answer: There are 18 girls in this group.

Here are some ratio and proportion problems that entail different measurement systems and converting between different measurement systems:

Knowing that there are 2.2 pounds in one kilogram, how many kilograms will you weigh when your current weight is 156 pounds?

1.2 pounds : 1 kilogram = 156 pounds : x kilograms

2.2/1 = 156/x

2.2x/2.2 = 156/2.2

x = 156/2.2

x = 70.9 kg

Answer: You weigh 70.9 kilograms when you weigh 156 pounds.

Knowing that there are 2.2 pounds in one kilogram, how many pounds will you weigh when your current weight is 65 kilograms?

2.2 pounds : 1 kilogram = x pounds : 65 kilograms

2.2/1 = x/65

2.2x/2.2 = 65/2.2

x = 65 x 2.2

x = 143 pounds

Answer: You weigh 143 pounds when you weigh 65 kilograms.

Knowing that there are 60 drops in 1 teaspoon, how many teaspoons are in 74 drops?

60 drops : 1 teaspoon = 74 drops : x teaspoons

60/1 = 74/x

60x/60 = 74/60 = x

74/60 = 1.2 teaspoons

Answer: There are 1.2 teaspoons in 74 drops

Proportions are often used in the calculation of dosages and solutions in pharmacology and the preparation of medications by nurses, pharmacists and pharmacy technicians as well as us, in our everyday lives.

These different measurement systems will be fully discussed below in the section entitled "Measurement and Data: M 2; Objective 5 : Converting Within and Between Standard and Metric Systems", however for the moment, we would like you to see some of the most commonly used measurement system conversion factors.

The most frequently used conversions are shown below. It is suggested that you memorize these.

• 1 gr =60 mg
• 1 kg = 2.2 lb.
• 1 mg = 1,000 mcg
• 1 g = 1,000 mg
• 1 kg = 1,000 g
• 1 tbsp. = 3 tsp
• 1 tbsp. = 15 mL
• 1 tsp = 5 mL
• 1 l = 1,000 mL
• 1 oz. = 30 mL
• 1L = 1000 cc

The conversion of percentages into ratios can also be done.

The method for this is to place the percentage number and then : (colon) and then 100. A ratio is read as 12 is to 100 when you see 12 : 100, for example.

Here are some examples:

• 12% = 12 : 100 or 12 is to 100
• 120% = 120 : 100 or 120 is to 100
• 220% = 220 : 100 or 220 is to 100
• 2222% = 2222 : 100 or 2222 is to 100

Converting ratios into percentages is based on, again, the fact that ratios reflect parts of 100.

• 23 : 100 = 23% or 23 is to 100
• 567 : 100 = 567% or 567 is to 100
• 1,222 : 100 = 1,222% or 1,222 is to 100
• 32,678 : 100 = 32,678% or 32,678 is to 100
• 1 : 100 = 1% or 1 is to 100

As stated above, ratios can be expressed in three different ways as follows:

The conversion of fractions into ratios is done in the following manner. The numerator becomes the first number before the colon and the denominator is the number after the colon.

The ratio is 2 : 10 or 2 is to 10

The ratio is 23 : 56 or 23 is to 56

The ratio is 19 : 45 or 19 is to 45

The ratio is 2 : 99 or 2 is to 99

The ratio is 16 : 789 or 16 is to 789

The ratio is 1 : 1 or 1 is to 1

The ratio is 100 : 100 or 100 : 100

Here are some examples of converting percentages into word ratios:

The ratio is 123 : 100

The ratio is 34 : 100

The ratio is 1 : 100

The ratio is 100 : 100

The ratio is 1,222 : 100

RELATED TEAS NUMBERS & ALGEBRA CONTENT :

• Converting Among Non Negative Fractions, Decimals, and Percentages
• Arithmetic Operations with Rational Numbers
• Comparing and Ordering Rational Numbers
• Solve Equations with One Variable
• Solve One or Multi-Step Problems with Rational Numbers
• Solve Problems Involving Percentages
• Applying Estimation Strategies and Rounding Rules for Real-World Problems
• Solve Problems Involving Proportions (Currently here)
• Solve Problems Involving Ratios and Rates of Change
• Translating Phrases and Sentences into Expressions, Equations and Inequalities
• Recent Posts

Ratio and Proportion Word Problems — Examples & Practice - Expii

Ratio and proportion word problems — examples & practice, explanations (4).

Ratio and Proportion Word Problems

Ratio and proportion word problems are really applicable to real life when you need to calculate costs, determine time limits, or even convert between measurement systems ! Here's a breakdown of what a ratio and proportion word problem might look like. We'll also see how to understand what the problem is asking, and how to solve it!

Image by Caroline Kulczycky

It's important to note that in Step Two, when you plug the appropriate values into the proportion, the units in both the numerators must be the same, and the units in both the denominators must be the same. For example, in the graphic above, the units in the numerators are books. The units in the denominators are hours. If the units given to you in the word problem are not consistent, you must somehow make them match. This next problem will show you how. For certain units we will use conversion factors to help make the units the same. We don't always have conversion factors, but it's good to know what ones we do have. Here are some examples:

• Units of length: convert between feet and meters
• Units of mass: convert between grams and kilograms
• Units of time: convert between minutes and days

Once that's done, the numerators will have the same units as each other. Likewise for the denominators. Then we'll be able to solve the proportion just using algebra. The equation will have a variable times a number on one side, and a number on the other side. Thus we can solve the equation in one step using division .

Practice Problem

Trisha can make 5 pairs of earrings in 1 week. Assuming she works at this constant rate, how many complete pairs of earrings can she make in 10 days?

Related Lessons

Math in Music: Rhythm

Now that the basic concepts of ratios and proportions have been covered, we are going to dive a little deeper into working with them through the subject of music.

Ratios are found in music within two main concepts: Rhythm/Polyrhythm and Pitch Intervals . These may seem a little foreign right now, but I promise they won't be too hard to understand. This explanation will focus on Rhythm and Polyrhythm, but the explanation on Pitch Intervals has been linked above.

Rhythm/Polyrhythm

All rhythms are referred to by names based off of fractions/ratios. There are whole notes (44 of a measure), half notes (24 of a measure), quarter notes (14 of a measure), eighth notes (18 of a measure), sixteenth notes (116 of a measure), etc. and the name denotes how much of a 4 beat measure the note takes up, as shown below. For our purposes, we are always going to use a 4 beat measure to describe our ratios.

In order, Whole Note, Half Note, Quarter Note, Eighth Note, Sixteenth Note, all taking up the same amount of time

Not all notes and rhythms are based on groups of 4, notes can be based on any division! Triplets can be used to split a four beat measure into groups of 3 or 6 instead of groups of 2 or 4.

Three half note triplets take up a measure, 6 quarter note triplets, 12 eighth note triplets, etc.

The brackets over the notes tell how many notes are played in the span of the bracket. You can see the 3 placed above all of the notes. This is because each group of 3 takes a set amount of time. The half note triplets take up 4 beats (one full measure) with 3 notes, the quarter note triplets take up 2 beats (one half of a measure) with 3 notes, and the eighth note triplets take up 1 beat (one quarter of a measure) with 3 notes. As ratios, these could be written as 3:4, 3:2, and 3:1. If we wanted to write these notes as fractions, they could be called 13, 16, and 112 notes.

When we start combining these rhythms to be played at the same time, we get what are called polyrhythms. These are normally defined by ratio of how many notes are being played in the same amount of time.

The ratios in this case are going to be written in terms of the top line against the bottom line. The first measure is a 4:3 while the second measure is a 3:2. These same polyrhythms can be written without needing a second line to compare by once again using the brackets above the notes.

The groupings on the top line both take up an entire measure, 4 beats. That means that the first measure on the top line shows a 5:4 ratio (aka five-lets), and the second shows a 7:4 (aka seven-lets) ratio. The bottom groupings take up 2 beats , so they show a 5:2 and 7:2 ratio.

We're gonna try a practice question that builds on some of these concepts.

Brandon is playing a trombone solo that has 24 evenly spaced notes. How many evenly spaced notes would Libby need to play in her counter melody to make the entire section a 4:3 polyryhthm?

(Video) Proportion Examples and Word Problems

by Math Meeting

This video by Math Meeting explains proportions and works through a few examples.

A proportion is two ratios that are set equal to one another.

An example of a proportion would be: 1 dollar10 pesos=10 dollars100 pesos As you can see, they are equivalent. The units are the same, and then we notice the fractions are equivalent . The fractions are equivalent because 10100 can be reduced to 110.

If you multiply the numerator and denominator of the first ratio by 10, you get the second ratio. If you have 10 times more dollars, you will have 10 times more pesos.

The standard proportion problem is as follows: 5x=103 You are given two proportions and you want to solve for x.

The best way to solve these types of equations is with cross multiplication .

To cross multiply, you take the product of the numerator of the first ratio and the denominator of the second ratio. Set that equal to the product of the numerator of the second ratio and the denominator of the first ratio.

Let's solve this problem now. 5x=103⇒5⋅3=x⋅1015=10x1510=10x10315210=10x1032=x

Another Example

Here's a word problem using proportions.

The currency in Spain is the euro. If I exchange $250 at the airport, how many euros will I get in return? (1 euro = 1.38 dollars)

The parentheses at the bottom give us our first ratio. 1 euro1.38 dollars

Within the question itself is our second ratio. x euros250 dollars

Set them equal to each other. Remember that the same units have to be in the top/bottom. 11.38=x250

To solve, we cross multiply. 11.38=x250⇒1⋅250=1.38⋅x250=1.38x2501.38=1.38x1.382501.38=1.38x1.38181.16=x

You will get approximately 181.16 euros!

Math in Music: Pitch

Ratios are found in music within two main concepts: Rhythm/Polyrhythm and Pitch Intervals . These may seem a little foreign right now, but I promise they won't be too hard to understand. This explanation is going to cover Pitch and Intervals, and Rhythm/Polyrhythm is linked above.

Pitch and Intervals

When two notes are being played at once, there is an Interval between the two pitches . Every pitch can be written as both a note (think A,B,C, and their placings on a staff) or as a measurement of Hertz (Hz). We are going to use A=440 Hz as a baseline to demonstrate how ratios are used in intervals and tuning in music.

As notes move in octaves, there is always a 2:1 ratio between the notes. Based on A at 440 Hz, the A one octave higher is going to be measured at 880 Hz, and the A one octave lower is going to be measured at 220 Hz.

This pattern continues in both directions (up and down) from A in the second space until eventually the note is outside of the range of human hearing.

The ratios between notes exist for every interval and are the basis for tuning many instruments as well as playing in tune with another musician. For example, were going to use the next two most common intervals, the perfect fifth and the major third. Using A=440 Hz as a basis again, we are going to use the intervals of 3:2 to find the measurement of the perfect fifth (E), and a ratio of 5:4 to find the major third (C# or C Sharp).

In this case we are going to put 440 as a denominator because we want to find the Hz of a note that is higher than our original A.

32=x4402x=1320x=660

Our perfect fifth of E will be at 660 Hz when it is perfectly in tune. Now lets calculate the pitch of the major third, C#. Once again, the 440 is going to be in the denominator as we are finding a note higher than A.

54=x4404x=2220x=550

Our major third of C# is going to be at 550 Hz when perfectly in tune. Combining all of these notes, you get a nice consonant A major chord. Just to make sure our musical brains are keeping up with the math, here is what this looks like on a staff.

Let's try a practice problem to see if everything is sticking.

Brandon and Libby are trying to practice their duet. If Brandon is playing a low A at 220 Hz, and Libby wants to play a perfectly in tune F# that is one octave and a major sixth (5:3) higher, what pitch should she play?

366.67 Hz

733.33 Hz

440 Hz

550 Hz

[FREE] Fun Math Games & Activities Packs

Always on the lookout for fun math games and activities in the classroom? Try our ready-to-go printable packs for students to complete independently or with a partner!

In order to access this I need to be confident with:

Here you will learn about ratios, including how to write a ratio, simplifying ratios, unit rate math and how to solve problems involving ratios and rates.

Students will first learn about ratios as part of ratios and proportions in 6 th grade and 7 th grade.

What is a ratio?

A ratio is a multiplicative relationship between two or more quantities.

Ratios are written in the form a : b, which is read as “a to b”, where a and b are normally integers, fractions, or decimals.

The order of the quantities in the ratio is important.

For example,

If there are 10 boys in a class and 15 girls, the ratio of boys to girls is 10 : 15 which is read as “10 to 15.” This is an example of a part to part ratio. You could also say the ratio of total students to girls is 15 : 25. This is an example of a part to whole ratio.

Step-by-step guide: How to write a ratio

Since a ratio represents a relationship, there is always more than one way to show it.

This includes unit rate math – which creates equivalent ratios where one part of the ratio is 1.

You can use unit rates to compare different quantities.

A grocery store sells a bag of 6 bananas for \$ 2.34 and a bag of 4 bananas for \$ 1.44.

Which bag has the better unit price?

Unit price means the price per 1 unit. In this case, the units are bananas. Divide each ratio to find the price for 1 banana.

The bag of 4 bananas is \$ 0.36 per banana, which is cheaper than the bag of 6 bananas which is \$ 0.39 per banana.

Step-by-step guide: Unit rate math

Unit rates are not the only types of equivalent ratios. When simplifying fractions, use the common factors to divide all the numbers in a ratio until they cannot be divided further to write the ratio in lowest terms.

The ratio of red counters to blue counters is 16 : 12.

You can simplify the ratio to lowest terms by finding the greatest common factor \textbf{(GCF)} of each of the numbers in the ratio.

Factors of 16 \text{:} \, 1, 2, 4, 8, 16

Factor of 12 \text{:} \, 1, 2, 3, 4, 6, 12

The greatest common factor is 4. To simplify the ratio, you divide both sides by 4.

Step-by-step guide: Simplifying ratios

Another way to write ratios is by using fraction notation.

Fraction notation can be used to show a part to whole ratio relationships.

The bar model below shows the ratio of blue : red as 3 : 2 (3 to 2). There are 3 blue blocks, 2 red blocks and 5 blocks in total.

This part to whole relationship allows us to make statements like…

• \cfrac{3}{5} of the blocks are blue
• \cfrac{2}{5} of the blocks are red
• \cfrac{5}{5} of the blocks are blue or red

The ratio of blue : red as 3 : 2 can also be shown as a part to part fraction…

The fractions show the ratio relationship BETWEEN the blue and red blocks. This allows us to make statements like…

• The number of blue blocks is \cfrac{3}{2} larger than red
• The number of red blocks is \cfrac{2}{3} the amount of blue

Step-by-step guide: Ratio to fraction

Ratios can also be written with percents.

The ratio of pencils to crayons is 4 : 6.

The ratio has 10 parts, so the fractions are

\cfrac{4}{10} : \cfrac{6}{10}.

The numerator represents the numbers of the ratio, which show how many pencils or crayons there are. The denominator represents the total number of pencils and crayons.

You may be able to recognize what the fractions are as percents or you may need to use long division to help convert your fractions.

\cfrac{4}{10}=40 \%, so 40 \% are pencils.

\cfrac{6}{10}=60 \%, so 60 \% are crayons.

Step-by-step guide: Ratio to percent

Solving problems with ratios is common in the real world. One place that this shows up is in calculating exchange rates. An exchange rate is the rate at which the money of one country can be exchanged for the money of another country.

Using a currency’s exchange rate you can convert between US dollars and foreign currencies.

To convert from US dollars (USD) to Japanese yen (JPY), you must multiply by the exchange rate.

So \$ 15 \; USD would be ¥2,134.35 \; JPY because,

\$ 15 \; USD \times 142.29=¥ 2,134 .35 \; JPY.

Step-by-step guide: How to calculate exchange rates

All the skills above are examples of ratio problem solving. When solving problems with ratios, it is important to ask:

• What is the ratio involved?
• What order are the quantities in the ratio?
• What is the total amount / what is the part of the total amount known?
• What are you trying to calculate ?

In the classroom, ratio problem solving often comes in the form of real world scenarios or word problems.

\cfrac{8}{10} students are right handed. What is the ratio of left handed students to right handed students? (2 : 8)

Step-by-step guide: Ratio problem solving

[FREE] Ratio Worksheet (Grade 6 and 7)

Use this quiz to assess your 6th and 7th grade students’ understanding of ratios. Covers 10+ questions with answers on ratio topics to identify areas of strength and support!

Common Core State Standards

How does this relate to 6 th grade math and 7 th grade math?

• Grade 6 – Ratios and Proportions (6.RP.A.1) Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2 : 1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”
• Grade 6 – Ratios and Proportions (6.RP.A.2) Understand the concept of a unit rate \cfrac{a}{b} associated with a ratio a : b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is \cfrac{3}{4} cup of flour for each cup of sugar.” “We paid \$ 75 for 15 hamburgers, which is a rate of \$ 5 per hamburger.”
• Grade 6 – Ratios and Proportions (6.RP.A.3) Use ratio and rate reasoning to solve real-world and mathematical problems, for example, by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
• Grade 6 – Ratios and Proportions (6.RP.A.3b) Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?
• Grade 7 – Ratios and Proportions (7.RP.A.1) Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks \cfrac{1}{2} mile in each \cfrac{1}{4} hour, compute the unit rate as the complex fraction \cfrac{\cfrac{1}{2}}{\cfrac{1}{4}} miles per hour, equivalently 2 miles per hour.

How to work with a ratio

There are a lot of ways to work with a ratio. For more specific step-by-step guides, check out the ratio pages linked in the “What are ratios?” section above or read through the examples below.

Ratio examples

Example 1: how to write a ratio.

Write the ratio of apples to pears.

• Identify the different quantities being compared and their order.

There are 5 pears and 2 apples.

The order of the ratio is apples to pears.

2 Write the ratio using a colon.

Apples : Pears

\hspace{0.7cm} 5 : 2

3 Check if the ratio can be simplified.

5 and 2 only have a common factor of 1, so this ratio is already in its lowest terms (simplest form).

Example 2: unit rate calculation – decimal

A car travels 303 miles in 6 hours. If the car travels the same number of miles each hour, what is the miles per hour rate?

Write the original rate.

303 miles in 6 hours → 303 : 6.

Use multiplication or division to create a unit rate.

The miles ‘per hour’ refers to 1 hour. Divide each side of the rate by 6, to create a rate for 1 hour.

Use the unit rate to answer the question.

The car travels 50.5 miles each hour.

Example 3: simplifying ratios

Write the ratio 48 : 156 in lowest terms.

Calculate the greatest common factor of the parts of the ratio.

Factors of 48=1, 2, 3, 4, 6, 8, 12, 16, 24, 48

Factors of 156=1, 2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156

GCF(48,156)=12

Divide each part of the ratio by the greatest common factor.

4 : 13 is in lowest terms.

Example 4: solving a problem involving ratio to percents

The ratio of adults to children in a park is 11 : 14.

One-fourth of the adults are women. What percent of the people in the park are men?

Add the parts of the ratio for the denominator of the fractions.

11+14=25. There are 25 parts in total. The denominator is 25.

Convert each part of the ratio to a fraction.

11 : 14 becomes \cfrac{11}{25} : \cfrac{14}{25}.

Convert the fractions to percents.

\begin{aligned}& \cfrac{11}{25}=\cfrac{44}{100}=44 \% \\\\ & \cfrac{14}{25}=\cfrac{56}{100}=56 \%\end{aligned}

You now know that 44 \% of the people are adults.

One-fourth of the adults are women.

\cfrac{1}{4} of 44 \%=11 \%.

11 \% of the people in the park are women and therefore 44-11=33 \% of the people are men.

Example 5: converting from KRW / USD

₩5,000 \; KRW is equal to \$ 3.85 \; USD. What is the exchange rate from \$ \; (USD) to ₩ \; (KRW)?

Use the information given to set up a rate.

When calculating the currency exchange rate from ₩ \; (KRW), you want to know how many ₩ \; (KRW) are equal to \$ 1 \; (USD). This is the ratio of ₩ to \$, so set up the rate as \cfrac{₩ 5,000}{\$ 3.85}.

Divide both parts by the base currency.

In this case, the base currency is \$ \; (USD), so divide both parts by 3.85, rounding the ₩ \; KRW is to the nearest whole:

\cfrac{₩ 5,000 \div 3.85}{\$ 3.85 \div 3.85}=\cfrac{₩ 1,299}{\$ 1}.

State the final exchange rate with the correct currency symbols.

The exchange rate from \$ \; (USD) to ₩ \; (KRW) is 1,299.

Example 6: ratio problem solving – mixed numbers

Fruit Salad Recipe:

• 2 \cfrac{1}{2} cups of blueberries
• 2 \cfrac{1}{5} cups of orange slices
• 1 \cfrac{1}{4} cups of strawberries
• 2 cups of apple slices

Write the ratio of the total cups of berries for every 1 cup of strawberries in the salad.

Identify key information within the question.

There are 1 \cfrac{1}{4} cups of strawberries and 2 \cfrac{1}{2} cups of blueberries.

Know what you are trying to calculate.

You need to create the ratio of the total cups of berries (strawberries and blueberries) for every 1 cup of strawberries.

Use prior knowledge to structure a solution.

First add 1 \cfrac{1}{4}+2 \cfrac{1}{2} to find the total cups of berries.

\begin{aligned}& 1 \cfrac{1}{4}+2 \cfrac{1}{2} \\\\ & =\cfrac{5}{4}+\cfrac{5}{2} \\\\ & =\cfrac{5}{4}+\cfrac{10}{4} \\\\ & =\cfrac{15}{4}\end{aligned}

Then write the ratio of total cups of berries to cups of strawberries.

\cfrac{15}{4} : \cfrac{5}{4}

Now multiply both sides of the ratio by \cfrac{4}{5}, to calculate the ratio of 1 cup of strawberries.

\cfrac{15}{4} \times \cfrac{4}{5} : \cfrac{5}{4} \times \cfrac{4}{5}

There are 3 total cups of berries for every 1 cup of strawberries.

*Note: To solve, you can also write the ratio \cfrac{15}{4} : \cfrac{5}{4} as the complex fraction \cfrac{\cfrac{15}{4}}{\cfrac{5}{4}} and find the quotient of the numerator and denominator.

Teaching tips for ratio

• There are many ways to engage students in ratios. One way is to introduce the golden ratio (based on Fibonacci’s sequence) and challenge students to look for it in the real world. Keep a chart or wall in the classroom for students to add any examples of this ratio that they find.
• Incorporate as many examples of ratios in the classroom as you can – even across subjects. For example, have students write ratios about the “Word of the day” – from an English or Science class. Such as “Write the ratio of nouns to adjectives” or “Write the ratio of words with the letter ‘e’ to total words.”
• As students work with ratios in different ways, keep track of successful solving strategies on a bulletin board or on chart paper. This allows students to see and utilize another students’ strategy, make connections between strategies and feel ownership in any ideas they help create.
• Be mindful of how to progress with ratio topics. Typically whole number ratios are introduced first, then ratios with rational numbers. Ratios that involve compare only fractional (or decimal values), such as \cfrac{2}{3} : \cfrac{5}{6} or 0.45 : 0.34 are the most difficult for students. As always, be mindful of your state’s curriculum when making decisions on when to introduce certain ratio topics.

Easy mistakes to make

• Writing the ratio in the wrong order A common error is to write the parts of the ratio in the wrong order. For example, The number of dogs to cats is given as the ratio 12 : 13 but the solution is incorrectly written as 13 : 12.
• Confusing ratios and fractions You can write a ratio with fraction notation. A part to whole fraction will have the same fractional language as a fraction. However, a part to part fraction will not. For example, The ratio of boys to girls is 2 : 3. Two ways to express this ratio are \cfrac{2}{3} or \cfrac{2}{5}. However, you must be careful how you read these fractions. You can say “ \cfrac{2}{5} of the kids are boys” but you cannot say “ \cfrac{2}{3} of the kids are boys.” Instead, say “The number of boys is \cfrac{2}{3} the number of girls.”
• Not fully simplified A common error is to not find the greatest common factor when simplifying a ratio. For example, Simplify the ratio 12 : 18. Dividing both numbers by only 2 leaves a ratio of 6 : 9, which is not fully simplified. This can be simplified further by dividing by 3 to get the ratio 2 : 3, which is the correct answer. By dividing both numbers by the greatest common factor, 6, would get the ratio 2 : 3 in one step.

Practice ratio questions

1. 500 people attended a concert. There were 240 boys. What is the ratio of boys to girls who went to the concert?

There are 500 people and 240 boys.

500-240=260. There are 260 girls.

The order of the ratio is boys to girls.

Boys : Girls

2. A musical requires 200 costumes. 140 costumes are for the background dancers. The rest are for the lead roles. Write the ratio of the costumes for lead roles to background dancers in the simplest form.

There are 200 costumes. 140 costumes are for background dances.

200-140=60 lead role costumes

The ratio order of the ratio is lead roles to background dancers

Lead roles : Background dancers

\hspace{1cm} 60 : 140

The greatest common factor of 60 and 140 is 20\text{:}

3. A shop is selling the same pencils in two different packs.

Which statement correctly compares the packs?

Pack \textbf{A}\text{:} \; 5 pens cost \$ 6.20

Pack \textbf{B}\text{:} \; 4 pens cost \$ 4.88

Pack A is \$ 1.32 cheaper per pencil than Pack B.

Pack B is \$ 1.32 cheaper per pencil than Pack A.

Pack A is \$ 0.02 cheaper per pencil than Pack B.

Pack B is \$ 0.02 cheaper per pencil than Pack A.

Offer A\text{:} \; 5 pencils cost \$ 6.20 → 5 : \$ 6.20

Each pencil in Pack A costs \$ 1.24.

Offer B\text{:} \; 4 pencils for \$ 4.88 → 4 : \$4.88

Each pencil in Pack B costs \$ 1.22.

\$ 1.24-\$ 1.22=\$ 0.02.

Offer B costs \$ 0.02 cheaper than Offer A.

4. The fraction of bananas in a bowl is \cfrac{13}{20}. Calculate the ratio of bananas to other pieces of fruit in the bowl.

The total number of pieces of fruit is 20. The number of bananas is 13.

As a bar model, this looks like

The number of other pieces of fruit is therefore 7 (this is calculated by 20-13=7 or counting the number of red bars above).

The ratio of bananas to other pieces of fruit is therefore 13 : 7.

5. Given the exchange rate between US dollars (USD) and New Zealand dollars (NZD) is \$ 1 \; USD=\$ 1.63 \; NZD, convert \$ 50 \; USD to New Zealand dollars (NZD). Round to the nearest cent.

\$ 50 \, USD=\$ \rule{0.5cm}{0.15mm} \, NZD

Since each US dollar is equal to \$ 1.63 \, NZD, multiply the USD by 1.63 to find the number of \$ \, (NZD).

\$ 50 \, USD=\$ 81.50 \, NZD

6. A soap is made by combining lavender soap with lemon soap. Each bar of soap weighs 330 \, g. If the ratio of lavender to lemon is 4 : 7,   how many grams of lemon soap are in each bar?

As there are 7+4=11   shares within the ratio

330 \div 11=30 \, g   per share

The amount of Lemon in the soap is equal to 7 \times 30=210 \, g

While the term ratio is used in a variety of ways in the real world, the definition of ratio in math is the comparison of two or more values that have a constant relationship. Some examples of ratios are “ 2 dogs to 5 cats” or “ 24 miles per hour.”

A rate is a special type of ratio that compares different units. They are not synonyms, since not all ratios are rates. However, all rates are ratios, so they can be called by either name.

Ratio understanding is expanded to include more complex comparisons that involve exponents, variables and/or polynomials. This extends to include ratio relationships in proportions and linear equations. As students progress in their learning, they will become comfortable graphing, creating tables and equations that represent ratio relationships.

The next lessons are

• Converting fractions, decimals and percentages

Still stuck?

At Third Space Learning, we specialize in helping teachers and school leaders to provide personalized math support for more of their students through high-quality, online one-on-one math tutoring delivered by subject experts.

Each week, our tutors support thousands of students who are at risk of not meeting their grade-level expectations, and help accelerate their progress and boost their confidence.

Find out how we can help your students achieve success with our math tutoring programs .

[FREE] Common Core Practice Tests (Grades 3 to 6)

Prepare for math tests in your state with these Grade 3 to Grade 6 practice assessments for Common Core and state equivalents.

40 multiple choice questions and detailed answers to support test prep, created by US math experts covering a range of topics!

Privacy Overview

Ratio and Proportion Examples With Answers

We may have mastered comparing objects descriptively, but how do we mathematically compare objects?

The answer is through ratio.

The ratio is the mathematical tool we use to compare quantities using division. The concept of ratios leads to the concept of proportions, which has a lot of applications in our daily lives. For example, when we convert currencies, estimate the volume of gasoline required for a car to cover a certain distance, calculate the cost of items bought, and so on.

This article features ratio and proportion examples with answers to help you effectively learn these concepts and how to apply them in your daily life.

Click below to go to the main reviewers:

Ultimate UPCAT Reviewer

Ultimate NMAT Reviewer

Ultimate Civil Service Exam Reviewer

Ultimate PMA Entrance Exam Reviewer

Ultimate PNP Entrance Exam Reviewer

Ultimate LET Reviewer

Table of Contents

A ratio shows how an object’s quantity is related to another object’s quantity . For example, if there are 15 male and 23 female students in a classroom, we can compare these quantities using a ratio, particularly 15 : 23.

We usually write it in the format below to express two quantities being compared as a ratio. Note that we use a colon (:) to express a ratio.

<first number> : <second number>

Example : Aling Bela has 4 chickens and 8 pigs on her small farm. What is the ratio of her chickens to her pigs?

Solution : We can express the ratio of chickens to pigs that Aling Bela owns as 4 : 8

Ratio as a Fraction

We can write a ratio into its equivalent fractional form . We write the first number as the numerator, then the second as the denominator.

For instance, using our example above about Aling Nena’s chickens and pigs, we can express the ratio of her chickens to her pigs (i.e., 4 : 8 ) as 4⁄8.

Example 1 : For every 4 burgers you buy, you must pay ₱128 . What is the ratio of the number of burgers bought to the price you must pay? Express the ratio in fractional form.

Solution : We can express the ratio of the number of burgers bought to the price you must pay as 4 : 128. In fractional form, we can write this as 4⁄128.

Example 2: There are 15 science teachers in a public high school. In that same high school, there were 10 math teachers. What is the ratio of science teachers to math teachers in that public high school? Express the ratio in fractional form .

Solution : We can express the ratio of science teachers to math teachers in that public high school as 15 : 10. In fractional form, we write it as 15⁄10.

Using Ratio to Compare a Part to a Whole

We have already defined what ratios are. However, the ratios that we tackled in our previous sections pertain to comparing an object’s quantity to a different object’s. 

This time, let us use the ratio to compare a part of a whole to the whole itself.

Suppose you and your friends bought a pizza and sliced it into 8 equal parts. Suppose that you’re able to take 2 slices from it. What is the ratio of the slices of pizza you have (a portion of the whole pizza) to the total number of slices (the whole pizza)?

The given situation above might ring a bell to you. Yes, we can use fractions to show that comparison. In particular, fraction 2⁄8 can be expressed as 2 : 8.

This means that to use the ratio to compare a part of a whole to the whole itself, we can use this format:

<portion of the whole> : <total number portions of the whole>

Example : In a classroom, 15 students are male while 20 are female. What is the ratio of female students to the total number of students in the classroom?

Solution : There are 20 female students in the classroom. Meanwhile, the total number of students in the classroom is the sum of the number of male students and the number of female students. There are 15 + 20 = 35 students in that classroom.

Therefore, the ratio of female students to the total number of students in that classroom can be expressed as 20 : 35

A proportion indicates that the two ratios are equal. In other words, proportions are equivalent ratios . Hence, if we say that ratios are proportional, we mean that those ratios are equal in value.

Let’s say we have two ratios, 1 : 2 and 2 : 4. We illustrate these ratios as shown below:

It’s seen that the ratios represent the same parts. It implies that these ratios are equivalent.

Hence, 1 : 2 = 2 : 4 is a proportion.

You might have realized that proportions are similar to the concept of equivalent fractions . Indeed, proportions indicate equivalent fractions since we can write ratios in fractional form.

How to Know if Two Ratios Are Proportional

Two ratios are proportional if they are equal . One way to determine if two ratios are equal is by converting them into fractional form and using the cross-multiplication method we discussed in the Fractions and Decimals review .

For instance, let us use the cross-multiplication method to determine if 1 : 2 = 2 : 4 is proportional.

Since the products are equal (both equal to 4), the ratios are equal. Hence, the ratios are proportional.

Moreover, you might also notice that if we multiply the numbers in a ratio, we can obtain another ratio that is proportional to that ratio. 

For example, if we multiply each number in 5 : 2 by the same number, let’s say 2, we have 10 : 4. Using the cross-multiplication method, you can verify that 5 : 2 = 10 : 4

Example: Give a ratio that is equivalent or proportional to 2 : 9

Solution : We can determine a ratio equivalent to or proportional to 2 : 9 by multiplying each number in 2 : 9 by the same number.

Let us try to multiply the numbers in 2 : 9 by 5.

(2 x 5) : (9 x 5) = 10 : 45

Hence, 2 : 9 = 10 : 45. 

Note : The number you can use to find a proportional ratio to 2 : 9 is arbitrary.  If we multiply the numbers in 2 : 9 by the same number, we will come up with a ratio that is proportional to 2 : 9. In this example, I just arbitrarily used 5. You may use any number and multiply it by the numbers in 2 : 9, and you will come up with a ratio that is proportional to it.  For example, I can multiply the numbers of 2 : 9 by 7 and obtain 14 : 63. 14 : 63 is also proportional to 2 : 9

Parts of a Proportion: Extremes and Means

Suppose a proportion a : b = c : d where a, b, c, and d represent real numbers .

The proportion’s first and last terms (i.e., a and d) are called the extremes. Meanwhile, the second and third terms (i.e., b and c ) are called the means .

Example : Determine the extremes and the means of the proportion 5 : 10 = 20 : 40

Solution : The extremes are the first and last terms of the proportion, which are 5 and 40, respectively. Meanwhile, the means are the second and third terms of the proportion, which are 10 and 20, respectively.

Properties of Proportion

Using the fact that proportions are equivalent ratios, we can mathematically derive their properties. These properties are beneficial when solving problems involving ratios and proportions.

Here are the properties of proportion:

Property #1: The Product of the Means Is Equal to the Product of the Extremes

For every proportion a : b = c : d, then a x d = b x c

This property tells us that we will obtain the same number if we multiply the means and the extremes of a proportion.

For example, suppose the proportion 4 : 3 = 12 : 9. 

If we multiply the means: 3 x 12 = 36

If we multiply the extremes: 4 x 9 = 36

Note that the products of the means and the extremes equal 36.

Example 1: What must be N so that N : 8 = 2 : 16 is a proportion?

Solution : Let us use the fact that the product of the means of a proportion equals the product of the extremes.

Multiplying the means, we have: 8 x 2 = 16

Multiplying the extremes, we have 16 x N 

Now, by the first property, 16 x N = 16. What must be multiplied by 16 so that it will be 16? That number should be 1.

Hence, N = 1.

Therefore, the proportion should be 1 : 8 = 2 : 16.

Example 2: Four kilos of chicken cost ₱640. How many kilos of chicken can you buy with ₱ 3 200?

Solution : The ratio of the kilos of chicken that can be bought to the cost is 4 : 640. Now, let’s use N to represent the number of kilos of chicken that can be bought with ₱3200. Thus, we have the ratio N : 3200.

4 : 640 = N : 3200

Let us apply that the product of the means is equal to the product of extremes so we can determine N.

Multiplying the means of the ratio: 640 x N 

Multiplying the extremes of the ratio: 4 x 3200 = 12800

Since the product of the means is equal to the product of the extremes:

640 x N = 12800

What must be multiplied by 640 to obtain 12800? We determine that number by dividing 12800 by 640.

N = 12800 ÷ 640 = 20

Therefore, you can buy 20 kilos of chicken for ₱3200.

Property #2: The Reciprocals of the Ratios in a Proportion Are Equal

Recall that the reciprocal of a fraction is its multiplicative inverse or simply the same fraction but with the positions of the numerator and the denominator reversed .

For example, the reciprocal of ⅖ is 5⁄2.

Given a proportion, say a : b = c : d , we can express it in fractional form as a⁄b = c⁄d

If we get the reciprocal of both fractions in a⁄b = c⁄d, we have:

We can express  b⁄a = d⁄c in ratio as b : a = d : c

This property states that if we take the reciprocal of each ratio in proportion, the ratios are still proportional. In symbols:

a : b = c : d → b : a = d : c

Example: If 5 : 4 = 35 : 28, what should be N so that 4 : 5 = 28 : N

Solution : Since the ratios in the proportion are reciprocated, we can use the second property of proportions. Using the second property, N = 35.

Property #3: Switching the Means or the Extremes in a Proportion Will Result in a Proportion

Suppose the proportion 1 : 7 = 3 : 21. If we try to switch the positions of the means of this proportion, we have 1 : 3 = 7 : 21 . You can verify using cross-multiplication that 1 : 3 = 7 : 21 is true (that is, 1 : 3 and 7 : 21 are equivalent ratios or proportional ).

Now, let us try switching the extremes of 1 : 7 = 3 : 21. We obtain 21 : 7 = 3 : 1 . Again, you can verify using cross-multiplication that 21 : 7 = 3 : 1 is true.

Hence, for every proportion a : b = c : d , switching the means or the extremes will still result in a proportion.

a : b = c : d → a : c = b : d and d : b = c : a

Example: When A is divided by 5, the result will equal the result when you divide B by 2. What is the result if you divide A by B?

Solution : The problem sounds tricky since we have no idea what the values of A and B are. However, we can determine the result by dividing A by B using the third property of proportion.

A divided by five can be written as A⁄5, which can then be expressed into a ratio of A : 5 .

Meanwhile, B divided by 2 can be written as B⁄2, which can then be expressed into a ratio of B : 2.

Since the problem states that if A is divided by 5, the result will be equal to the result if B is divided by 2, then 

A : 5 = B : 2

We want to know what will be the result when we divide A by B or A⁄B or, as a ratio, A : B

So, from A : 5 = B : 2, how can we obtain A : B?

We can apply the property that if we switch the means of a proportion, the result is still a proportion.

Let us now switch the means of A : 5 = B : 2

We obtain A : B = 5 : 2. Expressing into a fractional form:

Therefore, if A is divided by B, the result is 5⁄2 or 2.5.

How To Solve Problems Involving Ratio and Proportion

In this section, let us try to solve some real-life word problems that can be solved using the concepts of ratio and proportion.

Problem 1: Suppose that on a specific date, 1 US dollar equals ₱50. How many US dollars is equivalent to ₱650?

Solution: The ratio of US dollars to the Philippine Peso can be expressed as 1 : 50. Let N be the number of dollars we can obtain from ₱650.

Hence, we have this proportion: 1 : 50 = N : 650

The product of the means is equal to the product of extremes. Thus:

What must be multiplied by 50 to obtain 650? 

N = 650 ÷ 50 = 13

Therefore, ₱650 is equal to 13 US dollars.

Problem 2 : Leonor loves animals. He has a lot of dogs and cats in his house. The ratio of dogs to his cats is 1 : 3. The total number of dogs and cats is 8. How many cats does Leonor own?

Solution: The ratio of dogs to cats is 1 : 3. This doesn’t mean Leonor has 1 dog and 3 cats. 1 : 3 is just a ratio used to compare the number of dogs to cats. To find Leonor’s actual number of dogs and cats, we need to find two numbers with a sum of 8 that, when expressed as a ratio, will be proportional to 1 : 3.

Recall that we can obtain a ratio proportional to 1 : 3 if we multiply 1 and 3 by the same number.

Let us multiply the parts of the ratio with a number a.

(1 x a) : (3 x a) 

This means that we have two numbers, 1 x a and 3 x a . 1x a represents the total number of dogs Leonor has, while 3 x a represents the total number of cats Leonor has.

Since the total number of dogs and cats that Leonor has is 8:

(1 x a) + (3 x a) = 8

We can simplify the expression above as:

(4 x a) = 8

What must be multiplied by 4 to obtain 8? Simple, that number must be 2.

Hence, a = 2.

Recall that 1 x a represents the number of dogs Leonor has; since we have computed that a = 2, Leonor has 1 x (2) = 2 dogs.

We are doing the same thing to find Leonor’s number of cats: 3 x (2) = 6.

Therefore, Leonor has six cats.

Next topic:  Algebraic Expressions

Previous topic: Percentage

Return to the main article:  The Ultimate Basic Math Reviewer

Download Printable Summary/Review Notes

Download printable flashcards, test yourself, 1.  practice questions [free pdf download], 2.  answer key [free pdf download], 3.  math mock exam + answer key.

Written by Jewel Kyle Fabula

in Civil Service Exam , College Entrance Exam , LET , NAPOLCOM Exam , NMAT , PMA Entrance Exam , Reviewers , UPCAT

Last Updated May 3, 2023 06:49 PM

Jewel Kyle Fabula

Jewel Kyle Fabula is a Bachelor of Science in Economics student at the University of the Philippines Diliman. His passion for learning mathematics developed as he competed in some mathematics competitions during his Junior High School years. He loves cats, playing video games, and listening to music.

Browse all articles written by Jewel Kyle Fabula

Copyright Notice

All materials contained on this site are protected by the Republic of the Philippines copyright law and may not be reproduced, distributed, transmitted, displayed, published, or broadcast without the prior written permission of filipiknow.net or in the case of third party materials, the owner of that content. You may not alter or remove any trademark, copyright, or other notice from copies of the content. Be warned that we have already reported and helped terminate several websites and YouTube channels for blatantly stealing our content. If you wish to use filipiknow.net content for commercial purposes, such as for content syndication, etc., please contact us at legal(at)filipiknow(dot)net

Home » Math » Middle School Grades 6th-8th » How to Solve Problems Involving Ratios and Proportions

How to Solve Problems Involving Ratios and Proportions

Ratios and proportions are essential concepts in mathematics. They are used to compare quantities and find relationships between them. Ratios compare two quantities by dividing one by the other, while proportions compare two ratios. Understanding ratios and proportions is crucial in various fields, including finance, science, and engineering. In this article, we will explore how to solve problems involving ratios and proportions, specifically how to solve for unknown values.

Table of Contents

Solving for unknown values made easy

Solving for unknown values in ratios and proportions may seem daunting, but it is relatively simple. The key is to remember that ratios and proportions are fractions, and fractions represent part of a whole. Therefore, when we solve for an unknown value, we are finding the missing part of the whole.

The process of solving for an unknown value involves cross-multiplying the ratios or proportions. This means multiplying the numerator of one ratio by the denominator of the other ratio. Then, we equate the products and solve for the unknown value. The following section will illustrate this process using an example.

Example problem: 2/5 = x/10

Let's consider the following problem: Determine the value of x in the proportion 2/5 = x/10. We are given a proportion with two ratios, and we need to find the value of x. To solve this problem, we will use the cross-multiplication method.

Step-by-step solution process

• Cross-multiply the ratios: 2 x 10 = 5 x x
• Simplify: 20 = 5x
• Solve for x: x = 4

Therefore, the value of x in the proportion 2/5 = x/10 is 4.

Tips and tricks for ratio and proportion problems

Here are some tips and tricks to help you solve ratio and proportion problems:

• Always make sure that the units of measurement are the same for both ratios before solving for unknown values.
• When dealing with fractions, it is essential to simplify them before cross-multiplying.
• If the proportion has more than two ratios, cross-multiply each pair of ratios until you have a single ratio with the unknown value.

Now you're a ratio and proportion pro!

In conclusion, solving problems involving ratios and proportions is a fundamental skill in mathematics. By following the cross-multiplication method and using the tips and tricks outlined in this article, you can easily solve for unknown values in proportions. Remember, ratios and proportions are fractions, and fractions represent part of a whole. So, when you solve for an unknown value, you are finding the missing part of the whole. With practice, you will become a ratio and proportion pro in no time!

The Summer Math Bridge Grades 6-7

The Summer Math Bridge: A workbook for Grades 6 to 7: Fractions, Exponents, Roots, Equations, Expressions, Ratio, Proportion, Area, Perimeter, Volume, and more with Step by Step guide and Answers Key.

• Chapter. 01: Fractions
• Chapter. 02: Pre-Algebra
• Chapter. 03: Ratio and Proportion
• Chapter. 04: Geometry
• Chapter. 05: Metric Weights and Unit Conversion
• Chapter. 06: Statistics

Summer Math Workbook 6-7

Summer Math Success: Summer Math Workbook 6-7: 180 Worksheets of Percent, Beginning Statistics, Geometry, Pre Algebra and More

• Multiplication with Whole Numbers
• Division with Whole Numbers
• Mixed Fractions – Multiplication
• Mixed Fractions- Division
• Fractions: Multiple Operations
• Simplifying Fractions
• Percent and Decimals
• Percent – Advanced
• Ratio Conversions
• Cartesian Coordinates
• Cartesian Coordinates With Four Quadrants
• Scientific Notation
• Expressions – Single Step
• Number Problems
• Pre-Algebra Equations (One Step) Addition and Subtraction
• Pre-Algebra Equations (One Step) Multiplication and Division
• Pre-Algebra Equations (Two Sides)
• Simplifying Expressions
• Inequalities – Addition and Subtraction
• Find the Area and Perimeter
• Find the Volume and Surface Area
• Calculate the area of each circle
• Calculate the circumference of each circle
• Measure of Center – Mean
• Measure of Center – Median
• Measure of Center – Mode
• Measure of Variability – Range

Pre Algebra Workbook Grade 6-7

Summer Math Success: Pre Algebra Workbook Grade 6-7: 6th and 7th Grade Pre Algebra Workbook: Solving Single and Double Step Equations and More

• Pre-Algebra: Order of Operations
• Find Numbers Using Algebraic Thinking
• Evaluating Expressions
• Solving Equations
• Equations (One Variable)
• Solving Equations (Two Steps)

Math Word Problems Workbook Grades 6-8

Summer Math Success: Math Word Problems Workbook Grades 6-8: Fractions, Numbers, Multi Step, and Percent Word Problems with Answers

• Fractions Addition Word Problems
• Fractions Subtraction Word Problems
• Fractions Multiplication Word Problems
• Fractions Division Word Problems
• Numbers Word Problems
• Mixed Word Problems
• Percent Word Problems

Leave a Comment Cancel reply

Ratio and Proportion Questions & Word Problems | GMAT GRE Maths

Whether you are using units from the Metric system (as we do in this post) or US measurement system (the GMAT being an American test), the concepts don’t change.  

Introduction to Ratios

Ratio is the quantitative relation between two amounts showing the number of times one value contains or is contained within the other.

( Reference : Oxford dictionary )  

Notation : Ratio of two values a and b is written as a:b or a/b or a to b.

For instance, the ratio of number of boys in a class to the number of girls is 2:3. Here, 2 and 3 are not taken as the exact count of the students but a multiple of them, which means the number of boys can be 2 or 4 or 6…etc and the number of girls is 3 or 6 or 9… etc. It also means that in every five students, there are two boys and three girls.

  Question : In a certain room, there are 28 women and 21 men. What is the ratio of men to women? What is the ratio of women to the total number of people?

Men : women = 21 : 28 = 3:4

Women : total number of people = 28 : 49 = 4 : 7

Question : In a group, the ratio of doctors to lawyers is 5:4. If the total number of people in the group is 72, what is the number of lawyers in the group?

Let the number of doctors be 5x and the number of lawyers be 4x.

Then 5x+4x = 72 → x=8.

So the number of lawyers in the group is 4*8 = 32.

Question : In a bag, there are a certain number of toy-blocks with alphabets A, B, C and D written on them. The ratio of blocks A:B:C:D is in the ratio 4:7:3:1. If the number of ‘A’ blocks is 50 more than the number of ‘C’ blocks, what is the number of ‘B’ blocks?

Let the number of the blocks A,B,C,D be 4x, 7x, 3x and 1x respectively

4x = 3x + 50 → x = 50.

So the number of ‘B’ blocks is 7*50 = 350.

If the ratio of chocolates to ice-cream cones in a box is 5:8 and the number of chocolates is 30, find the number of ice-cream cones.

Let the number of chocolates be 5x and the number of ice-cream cones be 8x.

5x = 30 → x = 6.

Therefore, number of ice-cream cones in the box = 8*6 = 48.

Introduction to Proportion

A lot of questions on ratio are solved by using proportion.  

Definition & Notation

A proportion is a comparison of two ratios. If a : b = c : d, then a, b, c, d are said to be in proportion and written as a:b :: c:d or a/b = c/d.

a, d are called the extremes and b, c are called the means.

For a proportion a:b = c:d, product of means = product of extremes → b*c = a*d.

Let us take a look at some examples:

In a mixture of 45 litres, the ratio of sugar solution to salt solution is 1:2. What is the amount of sugar solution to be added if the ratio has to be 2:1?

Number of litres of sugar solution in the mixture = (1/(1+2)) *45 = 15 litres.

So, 45-15 = 30 litres of salt solution is present in it.

Let the quantity of sugar solution to be added be x litres.

Setting up the proportion,

sugar solution / salt solution = (15+x)/30 = 2/1 → x = 45.

Therefore, 45 litres of sugar solution has to be added to bring it to the ratio 2:1.

A certain recipe calls for 3kgs of sugar for every 6 kgs of flour. If 60kgs of this sweet has to be prepared, how much sugar is required?

Let the quantity of sugar required be x kgs.

3 kgs of sugar added to 6 kgs of flour constitutes a total of 9 kgs of sweet.

3 kgs of sugar is present in 9 kgs of sweet. We need to find the quantity of sugar required for 60 kgs of sweet. So the proportion looks like this.

3/9 = x/60 → x=20.

Therefore, 20 kgs of sugar is required for 60 kgs of sweet.

Problems on Mixtures / Blends

If a 60 ml of water contains 12% of chlorine, how much water must be added in order to create a 8% chlorine solution?

Let x ml of chlorine be present in water.

Then, 12/100 = x/60 → x = 7.2 ml

Therefore, 7.2 ml is present in 60 ml of water.

In order for this 7.2 ml to constitute 8% of the solution, we need to add extra water. Let this be y ml.

Then, 8/100 = 7.2/y → y = 90 ml.

So in order to get a 8% chlorine solution, we need to add 90-60 = 30 ml of water.

There is a 20 litres of a solution which has 20% of bleach. Extra bleach is added to it to make it to 50% bleach solution. How much water has to be added further to bring it back to 20% bleach solution?

This question has 3 parts.

In the first part, there is 20% of bleach in 20 L of solution → 4 L of bleach in 16 L of water = 20 L of solution. Let’s note the details in a table for better clarity and understanding.

In the second part, Extra bleach is added to bring it to 50% of total solution. Let the amount of bleach added be x litres.

Then, (4+x)/(20+x) = 50/100 → x = 12 L of bleach is added.

Now, there is 4+12 = 16 L of bleach in 16 L of water in a total of 32 L of solution.

Now, to bring the bleach percentage back to 20%, extra water is added and the amount of bleach remains the same. Let this extra amount of water be y litres.

16 L of bleach constitutes 20% of the solution →

16/(32+y) = 20/100 → y = 48.

Therefore, 48 litres of water has to be added to the solution if bleach has to be 20% of the whole solution.

1 kg of cashews costs Rs. 100 and 1 kg of walnuts costs Rs. 120. If a mixture of cashews and walnuts is sold at Rs. 105 per kg,then what fraction of the total mixture are walnuts?

For this type of problems, first step is to determine how much each of the items is above or below the target.

Our target price is Rs. 105. Cashews price is Rs. 5 below the target price and walnuts price is Rs. 15 above the target price.

So, for each kg of cashews added, let’s consider it as ‘-5’ and for each kg of walnuts added, let’s consider it as ‘+15’. These two have to be added in such a way that they cancel out each other. Adding ‘-5’ thrice gives a ‘15’ and adding ‘+15’ once results in cancellation of the terms.

This means that adding 3 kgs of cashews and 1 kg of walnuts gives a mixture that can be sold at Rs. 105 per kg.

So, 3 kgs of cashews present for every 1 kg of walnuts. The ratio of cashews to walnuts is 3:1. Fraction of walnuts in the mixture = 1/(3+1) = 1/4 of the total mixture  

Practice Questions in Ratio and Proportion

Problem 1: Click here

On a certain map, 1 cm = 12 km actual distance. If two places are 96 km apart, what is their distance on map?

A. 10 cm B. 12 cm C. 96 cm D. 8 cm

Answer 1: Click here

Explanation :

1cm/12 km = x cm/100 km → x = 8 cm

Problem 2: Click here

A person types 360 words in 4 minutes. How much time does he take to type 900 words? A. 15 B. 90 C. 10 D. 9

Answer 2: Click here

Explanation

4/360 = x/900 → x=10

Mini-MBA | Start here | Success stories | Reality check | Knowledgebase | Scholarships | Services Serious about higher ed? Follow us:                

12 thoughts on “Ratio and Proportion Questions & Word Problems | GMAT GRE Maths”

Hey can you please emphasize on the working of the cashews and walnuts question?

I’m stuck on how we’re assuming this: So, for each kg of cashews added, let’s consider it as ‘-5’ and for each kg of walnuts added, let’s consider it as ‘+15’. These two have to be added in such a way that they cancel out each other. Adding ‘-5’ thrice gives a ‘15’ and adding ‘+15’ once results in cancellation of the terms.

Why do they have to be added to cancel out eaach other?

Would appreciate the help.

I think if they equal zero, then is the only time a ratio can be found. If it isn’t equal to zero you won’t get a ratio.

assume the following Lets the total Qty of the Mixture is 10 KG Lets assume the Qty of Cashew is X Lets assume the Qty of Walnut is 10-X As we know that the purchase qty and sold qty can be equal for instance 10 KG now put this into equation 100(X) + 120(10-X) = 10*105 ” Total purchase = total Sales” X = 7.5 , hence walnut = 2.5 now 2.5/10=25%

given 1 KG of cashews cost 100 and walnuts was 120 given their mixture costs 105 rupees this 105 rupees of mixture cost is for just 1 kg includes X grams of cashews and Y grams of walnuts so, X+Y=1 KG or X+Y=1000 grams——1 1 KG of cashews cost is 100 so ,1000 grams=100 rupees. Hence 1 gram=1/10 rupees same way 1 gram of walnuts costs 12/100 rupees so, X/10 +Y*(12/100)=105——-2 solve equation 1 and 2 . Quantity of X and Y will be 750 grams and 250 grams so their ratio is 3:1 walnuts to the total ratio will be 1/4

Hello I have this peoblem The ratio of the area of the dining room to the family room is 2 to 3. After remodeling the family room is now 1/2 as large as it used to be and has 60 Sq less than the dining room. How many Sq feet is the isning room?

Hi Let the area of the dining room be d and the family room be f. d/2 = b f/3 = k d= 2k f=3k Total area = 5k 5k=3k +60(1/2*3k +1/2*3k + 60) 2k= 60 k = 30 Area of family = 3*30 /2 = 45 Area of dining = 105 45+105 = 150 and total area remains the same

Hi Caterina,

I had a look at Erica’s answer, and unfortunately it doesn’t add up. However, I will stick to the same format she used:

dining room = d family room = f

d : f = 2 : 3

Lets make that ratio easier to handle: d : f = 4 : 6

Now what does 4 : 6 mean? It means that the dining room represents 4 parts whilst the family room represents 6 parts. A part can be any size in a ratio, what matters is the proportion that the ratio describes.

Lets say 1 part = k, therefore d : f = 4k : 6k

Now if the family room halves in size, the ratio becomes 4k : 3k because the family room used to be 6 parts but is now 3 parts.

It is now also 60sq ft less than the dining room, so ‘size of f’ – ‘size of d’ = 60 sq ft therefore 4k – 3k = 60 sq ft i.e 1k = 60sq ft

Remember that the dining room is 4k so its size is:

4k = 4 x 60 sq ft = 240 sq ft

For every 2 boy students there are 3 girl students and for every one teacher there are 10 students whereas for every 4 male teachers there are 5 female teachers. Which of the following is the ratio of number of boy students to the number of male teachers?

For 1 teacher, there are 10 students which contains 4 boys, and 6 girls(because ratio of boys to girls is 2:3)

Therefore ratio of boys to teacher is 4:1

Now for 9 (4 male+5 female) teachers, there are total 90 students which has 36(90*0.4) boys

This means for a group of 36 boys, there are 4 male teachers.

Hence the ratio of boys to male teacher is 36:4 or 9: 1

Three similar lamps use 4 liters of oil in 80 hours. How much oil will 6 lamps of the same kind use in 40 hours?

Worker*Rate*Time=Object

Worker(lamp)=3 Time=80 hours Object = 4 liters

3R80=4. R=4/240 or R=1/60

Worker(lamp)=6 Time=40 hours Object= x

6R40=X (Rate is 1/60 from previous). 6(1/60)40=x (240/60)=x and x=4

Object or Liters is 4 liters.

a farmer has a total yield of 42,000 bu of corn from a 350-acre farmer what total yield should he expect from a similar 560-acre farm?

Leave a Comment Cancel reply

Direct and Inverse Proportion Practice Questions

Click here for questions, click here for answers .

variation, proportionality

GCSE Revision Cards

5-a-day Workbooks

Primary Study Cards

Privacy Policy

Terms and Conditions

Corbettmaths © 2012 – 2024

• Anatomy & Physiology
• Astrophysics
• Earth Science
• Environmental Science
• Organic Chemistry
• Precalculus
• Trigonometry
• English Grammar
• U.S. History
• World History

... and beyond

• Socratic Meta
• Featured Answers

How do you solve ratio and proportion problems?