direct proportion problem solving

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?

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Direct Proportion

Direct proportion is a mathematical comparison between two numbers where the ratio of the two numbers is equal to a constant value. The proportion definition says that when two ratios are equivalent, they are in proportion. The symbol used to relate the proportions is "∝". Let us learn more about direct proportion in this article.

Direct Proportion Definition

The definition of direct proportion states that "When the relationship between two quantities is such that if we increase one, the other will also increase, and if we decrease one the other quantity will also decrease, then the two quantities are said to be in a direct proportion". For example, if there are two quantities x and y where x = number of candies and y = total money spent. If we buy more candies, we will have to pay more money, and we buy fewer candies then we will be paying less money. So, here we can say that x and y are directly proportional to each other. It is represented as x ∝ y. Direct proportion is also known as direct variation.

Some real-life examples of direct proportionality are given below:

• The number of food items is directly proportional to the total money spent.
• Work done is directly proportional to the number of workers.
• Speed is in direct proportion to the distance w.r.t a fixed time.

Direct Proportion Formula

The direct proportion formula says if the quantity y is in direct proportion to quantity x, then we can say y = kx, for a constant k. y = kx is also the general form of the direct proportion equation.

• k is the constant of proportionality .
• y increases as x increases.
• y decreases as x decreases.

Direct Proportion Graph

The graph of direct proportion is a straight line with an upward slope . Look at the image given below. There are two points marked on the x-axis and two on the y-axis, where (x) 1 < (x) 2 and (y) 2 < (y) 2 . If we increase the value of x from (x) 1 to (x) 2 , we observe that the value of y is also increased from (y) 1 to (y) 2 . Thus, the line y=kx represents direct proportionality graphically.

Direct Proportion Vs Inverse Proportion

There are two types of proportionality that can be established based on the relation between the two given quantities. Those are direct proportion and inverse proportional. Two quantities are directly proportional to each other when an increase or decrease in one leads to an increase or decrease in the other. While on the other hand, two quantities are said to be in inverse proportion if an increase in one quantity leads to a decrease in the other, and vice-versa. The graph of direct proportion is a straight line while the inverse proportion graph is a curve. Look at the image given below to understand the difference between direct proportion and inverse proportion.

Topics Related to Direct Proportion

Check these interesting articles related to the concept of direct proportion.

• Constant of Proportionality
• Inversely Proportional
• Percent Proportion

Direct Proportion Examples

Example 1: Let us assume that y varies directly with x, and y = 36 when x = 6. Using the direct proportion formula, find the value of y when x = 80?

Using the direct proportion formula, y = kx Substitute the given x and y values, and solve for k. 36 = k × 6 k = 36/6 = 6 The direct proportion equation is: y = 6x Now, substitute x = 80 and find y. y = 6 × 80 = 480

Answer: The value of y is 480.

Example 2: If the cost of 8 pounds of apples is $10, what will be the cost of 32 pounds of apples?

It is given that, Weight of apples = 8 lb Cost of 8 lb apples = $10 Let us consider the weight by x parameter and cost by y parameter. To find the cost of 32 lb apples, we will use the direct proportion formula. y=kx 10 = k × 8 (on substituting the values) k = 5/4 Now putting the value of k = 5/4 when x = 32 we have, The cost of 32 lb apples = 5/4 × 32 y =5×8 y = 40

Answer: The cost of 32 lb apples is $40 .

Example 3: Henry gets $300 for 50 hours of work. How many hours has he worked if he got $258?

Solution: Let the amount received by Henry be treated as y and the number of hours he worked as x. Substitute the given x and y values in the direct proportion formula, we get, 300 = k × 50

⇒ k=300/50 k = 6 The equation is: y = 6x. Now, substitute y = 258 and find x. 258 = 6 × x

⇒ x = 258/6 = 43 hours Therefore, if Henry got $258, he worked for 43 hours.

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Direct Proportion Practice Questions

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FAQs on Direct Proportion

What is direct proportion in maths.

Two quantities are said to be in direct proportion if an increase in one also leads to an increase in the other quantity, and vice-versa. For example, if a ∝ b, this implies if 'a' increases, 'b' will also increase, and if 'a' decreases, 'b' will also decrease.

What Is the Symbol ∝ Denotes in Direct Proportion Formula?

In the direct proportion formula, the proportionate symbol ∝ denotes the relationship between two quantities. It is expressed as y ∝ x, and can be written in an equation as y = kx, for a constant k.

What is Direct Proportion and Inverse Proportion?

Direct proportion, as the name suggests, indicates that an increase in one quantity will also increase the value of the other quantity and a decrease in one quantity will also decrease the value of the other quantity. While inverse proportion shows an inverse relationship between the two given quantities. It means an increase in one will decrease the value of the other quantity and vice-versa.

How do you Represent the Direct Proportional Formula?

The direct proportional formula depicts the relationship between two quantities and can be understood by the steps given below:

• Identify the two quantities which vary in the given problem.
• Identify the variation as the direct variation .
• Direct proportion formula: y ∝ kx.

What is a Direct Proportion Equation?

The equation of direct proportionality is y = kx, where x and y are the given quantities and k is any constant value. Some examples of direct proportional equations are y = 3x, m = 10n, 10p = q, etc.

How to Solve Direct Proportion Problems?

To solve direct proportion word problems, follow the steps given below:

• Make sure that the variation is directly proportional.
• Form an equation in terms of y = kx and find the value of k base on the given values of x and y.
• Find the unknown value by putting the values of x and the known variable.

How to Show Relationship Between Two Quantities Using Direct Proportion Formula?

The directly proportional relationship between two quantities can figure out using the following key points.

• Identify the two quantities given in the problem.
• If x/y is constant then the quantities have a directly proportional relationship.

Directly Proportional and Inversely Proportional

Directly proportional: as one amount increases, another amount increases at the same rate.

Example: you are paid $20 an hour

How much you earn is directly proportional to how many hours you work

Work more hours, get more pay; in direct proportion.

This could be written:

Earnings ∝ Hours worked

• If you work 2 hours you get paid $40
• If you work 3 hours you get paid $60

Constant of Proportionality

The "constant of proportionality" is the value that relates the two amounts

Example: you are paid $20 an hour (continued)

The constant of proportionality is 20 because:

Earnings = 20 × Hours worked

This can be written:

Where k is the constant of proportionality

Example: y is directly proportional to x, and when x=3 then y=15. What is the constant of proportionality?

They are directly proportional, so:

Put in what we know (y=15 and x=3):

Solve (by dividing both sides by 3):

The constant of proportionality is 5:

When we know the constant of proportionality we can then answer other questions

Example: (continued)

What is the value of y when x = 9?

What is the value of x when y = 2?

Inversely Proportional

Example: speed and travel time.

Speed and travel time are Inversely Proportional because the faster we go the shorter the time.

• As speed goes up, travel time goes down
• And as speed goes down, travel time goes up

Example: 4 people can paint a fence in 3 hours. How long will it take 6 people to paint it? (Assume everyone works at the same rate)

It is an Inverse Proportion:

• As the number of people goes up, the painting time goes down.
• As the number of people goes down, the painting time goes up.

We can use:

• t = number of hours
• k = constant of proportionality
• n = number of people

"4 people can paint a fence in 3 hours" means that t = 3 when n = 4

So now we know:

And when n = 6:

So 6 people will take 2 hours to paint the fence.

How many people are needed to complete the job in half an hour?

So it needs 24 people to complete the job in half an hour. (Assuming they don't all get in each other's way!)

Proportional to ...

It is also possible to be proportional to a square, a cube, an exponential, or other function!

Example: Proportional to x 2

A stone is dropped from the top of a high tower.

The distance it falls is proportional to the square of the time of fall.

The stone falls 19.6 m after 2 seconds, how far does it fall after 3 seconds?

• d is the distance fallen and
• t is the time of fall

When d = 19.6 then t = 2

And when t = 3:

So it has fallen 44.1 m after 3 seconds.

Inverse Square

Inverse Square : when one value decreases as the square of the other value.

Example: light and distance

The further away we are from a light, the less bright it is.

In fact the brightness decreases as the square of the distance. Because the light is spreading out in all directions.

So a brightness of "1" at 1 meter is only "0.25" at 2 meters (double the distance leads to a quarter of the brightness), and so on.

Direct Proportion: Definition, Formula, Symbol, Examples, FAQs

What is direct proportion in math, difference between direct proportion and inverse proportion, how to use direct proportion to solve problems, solved examples of direct proportion, practice problems on direct proportion, frequently asked questions about direct proportion.

Direct proportion or direct variation is a type of proportion in which the ratio of two quantities stays constant. 

Suppose a variable y varies directly with x, then it means that if x increases, y increases by the same factor. If x decreases, then there is a proportionate decrease in y also.

Direct Proportion Example: 

The cost of the candy increases as the number of the same increases.

Direct Proportion Symbol

The “directly proportional symbol” or “direct proportional symbol” is $\propto$. 

We read x ∝ y as “x is directly proportional to y.” 

It means that x is dependent on y.

We read y ∝ x as “y is directly proportional to x.” 

It means that y is dependent on x.

Direct Proportion Formula

If y is directly proportional to x, then the direct proportion formula or the direct proportion equation is given by y=kx, where k is a constant of proportionality.

Constant of Proportionality

From the direct proportion formula, we have y = kx.

The fixed value $k = \frac{y}{x}$ is called the constant of proportionality and it represents the constant ratio between the two quantities that are in direct proportion. k can be any non-zero real number.

Direct Proportion Graph

If you construct a graph of a direct proportion, it always comes out to be a straight line passing through the origin (0, 0).

The slope of this line is k.

• If k is negative, the line goes down from left to right.
• If k is positive, the line rises from left to right.

Direct proportion and inverse proportion are two kinds of proportional relationships in math describing the relation between two variables. Here are their key differences in the table below:

Let’s understand this with the help of an example.

Example: If 20 pens cost $25, what would be the cost of 100 pens?

Here, the cost of pens is directly proportional to the number of pens.

Note down the given values of x and y. 

Since the quantities are in direct proportion, their ratio is constant.

$\frac{20}{25} = \frac{100}{?}$

By cross multiplying, we get

$20 \times ?= 100 \times 25$

? $= \$125$

100 pens will cost $\$125$.

Facts about Direct Proportion

• Prior to $\propto$ (symbol), a double colon (::) was used.
• The proportionality symbol $(\propto)$ was used by William Emerson (London, 1768) for the first time.
• The link between the two variables is no longer a direct proportion if the proportionality ratio changes.

In this article, we learned about direct proportion, its graph, formula, equation, and examples. Let’s solve a few examples and practice problems to understand the concept better.

1. If 8 rooms are required for 24 guests, how many rooms would be required to accommodate 12 guests?

Here, $\frac{24}{8} = \frac{w}{12}$

Cross-multiply:

$w \times 8 = 12 \times 24$

Therefore, for 12 guests, a total of 36 rooms will be required. 

2. If 4 tasks take 8 hours for completion, how many tasks can be completed in 20 hours?

Let’s write the constant ratios.

$\frac{4}{8} = \frac{n}{20}$

In 20 hours, 10 tasks can be completed.

3. What will 10 movie tickets cost if the price of 6 tickets is $\$36$ ?

6 tickets cost $\$36$.

Let 10 movie tickets cost $y.

$\frac{6}{36} = \frac{10}{6}$

$y \times 6 = 36 \times 10$

4. If a task is completed in 20 days by 5 workers, how long will it take 10 workers to perform the same task?

Solution:  

This problem can again be solved using the direct proportion formula.

Let the number of days for 10 workers to complete the task be x.

$\frac{5}{10} = \frac{20}{x}$

Cross-multiply

$5x = 10 \times 20$

$x = \frac{(10 \times 20)}{5}$

Therefore, 10 workers will require 40 days to complete the same task.

5. How far can John run in 60 minutes if he can cover 9 miles in 90?

Solution: 

Use the direct proportion formula for this problem.

Let the distance John covers in 60 minutes be x.

$\frac{9}{90} = \frac{x}{60}$

Cross-multiply.

$\frac{9 \times 60} = 90x$

$x = \frac{(9 \times 60)}{90}$

Therefore, John can run 6 miles in an hour.

Attend this quiz & Test your knowledge.

If x is directly proportional to y, we write it as

If a car travels 75 miles on 3 gallons of gas, how many miles can it travel on 5 gas gallons, the graph for direct proportionality is, on the graph of a direct proportion, the constant of proportionality represents the.

Is y being inversely proportional to x the same thing as y being directly proportional to $\frac{1}{x}$ ?

Yes, y being inversely proportional to x the same thing as y being directly proportional to the reciprocal of x, which is $\frac{1}{x}$.

How do we know whether any two variables are directly proportional or not?

We can confirm the direct proportionality of any two variables if their ratio remains constant. This is expressed mathematically as $\frac{y}{x} = k$. Here, k is the constant of proportionality.

What are the uses of direct proportion?

Direct proportion is a universal concept profoundly used in different fields to explain and predict relationships between variables. The direct proportion has multiple practical uses in various fields, including mathematics, science, engineering, finance, art, population rates, and many others.

What is an independent variable and dependent variable in direct proportion?

For given two quantities, the cause and effect relationship decides the independent variable and the dependent variable. The independent variable is the cause, and the dependent variable is the variable whose value depends on the independent variable. We can define them based on the context.

is $x \propto y$ same as $y \propto x$ ?

No, these two statements are not the same. 

$x \propto y$ means that x is directly proportional to y.

$y \propto x$ means that y is directly proportional to x.

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5.2: Applications of Proportionality

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In the previous section, we studied proportions, and used them to solve problems involving ratios. In this section, we continue our study of proportions, and investigate two different types of proportionality.

In this section, you will learn to:

Recognize direct proportionality relationships, and use them to answer questions involving direct proportionality

• Compute and interpret the constant of proportionality in context
• Recognize inverse proportionality relationships, and use them to answer questions involving inverse proportionality

There are two main types of proportionality. We will learn about the more common one first.

Direct Proportionality

Definition: directly proportional.

Two quantities are directly proportional if, as one quantity increases, the other quantity also increases at the same rate.

Direct proportionality describes all of the proportion problems we've seen before. Here is another example that shows how direct proportionality works, and introduces the next important notion.

Example \(\PageIndex{1}\)

At an hourly wage job, you work for \(5\) hours, and get paid \(\$83.75\). How much money will you earn if you work \(7\) hours? (Assume that you are not making overtime pay, or any other sort of special pay rate.)

This is a situation described by direct proportionality. Since this is an hourly wage job, and we're told that there is no overtime pay or other special pay rate, we can safely assume that we make the same amount per hour in this job. In other words, the rate of pay will be the same no matter how many hours we work. That means that if the number of hours worked increases, the amount you're paid increases at the same rate, no matter the number of hours worked.

We can solve this using a proportion, using the same techniques as in the last section. We will set up the following proportion equation: \[\frac{\$83.75}{5 \text{ hours}} = \frac{\$x}{7 \text{ hours}}\]

Notice that we have picked a variable, \(x\), to denote the answer we are trying to find -- the number of dollars earned for working \(7\) hours. Also notice how we've labeled our units, and have made sure that the corresponding quantities are together. If you do this every time -- label your units, and make sure corresponding quantities stay together -- you can solve any direct proportion problem.

Now for the process that will actually help us solve for \(x\). First, rewrite the equation without labels: \[\frac{83.75}{5} = \frac{x}{7}\]

Next, apply Cross Multiplication \[5x = 83.75 \times 7\]

Next, simplify the right side (using a calculator): \[5x = 586.25\]

Finally, apply Division undoes Multiplication to find \(x\): \[x = \frac{586.25}{5} = 117.25\]

That means that if you work \(7\) hours, you will make \(\$117.25\). Think for a moment to see if that's reasonable: it's more, but not too much more, than you made working for \(5\) hours. So, it seems like a sensible answer.

The approach above works just fine to find the desired answer. However, what if you wanted to know how much you'd make working for \(3\) hours? Or \(4\) hours? Or \(10\) hours? You could just reproduce the work above each time, you wanted. But you may also find the following approach quicker:

Definition: Constant of Proportionality

In a situation involving directly proportional quantities, the constant of proportionality is the common ratio that describes the comparison of any two corresponding quantities. In other words, it is the constant rate of change between the two quantities.

Let's see how to find a constant of proportionality, and how to interpret it.

You're in the same situation as the previous example: you work for \(5\) hours, and earn \($83.75\). What is the constant of proportionality in this example, and what does it mean in context?

The constant of proportionality is the common ratio that describes the comparison of any two corresponding quantities. In this situation, we actually have two sets of corresponding quantities, one of which we found in the previous example. You know that you'll make \($83.75\) working for \(5\) hours, and we calculated above that you'll make \(\$117.25\) if you work for \(7\) hours. Let's look at these two ratios:

\[\frac{\$83.75}{5 \text{ hours}} \quad \text{ and } \quad \frac{\$117.25}{7 \text{ hours}}\]

Both of these ratios are fractions, and we can simply divide the top by the bottom to reduce them to a single number. Using a calculator, we can see that

\[\frac{\$83.75}{5 \text{ hours}} = \$83.75 \div 5 \text{ hours} = \$16.75 \text{ per hour}\]

\[\frac{\$117.25}{7 \text{ hours}} = \$117.25 \div 7 \text{ hours} = \$16.75 \text{ per hour}\]

These are the same answer! Do you see why? We originally found our value for \(x\) in the previous example by setting the two ratios equal. So, they must give the same answer when divided.

This shared rate -- \(\$16.75\) per hour -- is the constant of proportionality in this situation. It is the shared value of all ratios described by this problem, where the top of the ratio is money earned, and the bottom is hours worked.

What does this constant of proportionality mean in this situation? In this case, the constant of proportionality is your hourly pay rate. In other words, it's how much you make per hour.

A few more comments on the example above: now that you know this rate, it's quite simple to find how much money you'll make if you work \(3, 4,\) or \(10\) hours. You just multiply your hourly rate by the number of hours worked. For example, if you worked \(4\) hours, you could calculate:

\[\underset{\text{hours}}{4} \times \$16.75 \text{ per hour} = \$67.00\]

This means you would make \(\$67.00\) working for \(4\) hours. Notice that if we reverse the process -- in other words, if we try to extract the constant of proportionality knowing that we make \(\$67.00\) in \(4\) hours, we get:

\[\frac{\$67.00}{4 \text{ hours}} = \$67.00 \div 4 \text{ hours} = \$16.75 \text{ per hour}\]

It's the same rate we found before. This is why it's called a constant of proportionality -- it stays the same, even as the corresponding quantities change.

Constants of proportionality will change in meaning depending on the context of the problem. For example, you might ask: If \(5\) people eat a total of \(10\) slices of pizza, how many slices would \(7\) people eat? In this case, the constant of proportionality could be found by: \[\frac{10 \text{ slices}}{5 \text{ people}} = 2 \text{ slices per person}\]

In this case, a correct interpretation would be: "The constant of proportionality is \(2\) slices per person, which means that each person eats \(2\) slices of pizza." When asked for an interpretation, you should write a sentence similar to the previous -- your goal is the explain the meaning of the constant of proportionality in context of the situation. You will need to read carefully and use critical thinking to deduce a meaningful interpretation. As with many questions in this class, there are multiple good answers to these types of questions!

Problems that involve rates, ratios, scale models, etc. can be solved with proportions. When solving a real-world problem using a proportion, be consistent with the units.

Inverse Proportionality

Let's start this section with a question to illustrate the main concept. Before reading ahead, try to answer this question on your own:

Example \(\PageIndex{2}\)

Suppose it takes \(6\) sanitation workers, all working simultaneously, \(4\) hours to pick up the trash and recycling in a given neighborhood. How many sanitation workers would it take to pick up the trash and recycling in the same neighborhood in \(3\) hours? (Assume that all sanitation workers work at the same rate, and can work independently.)

Did you try to answer the question yourself? What did you come up with? If you're like many students who carefully read the previous sections, you might have written this down:

\[\frac{6 \text{ sanitation workers}}{4 \text{ hours}} = \frac{x \text{ sanitation workers}}{3 \text{ hours}}\]

Then you'd apply Cross Multiplication to get:

\[4x = 18\]

and then you'd use Division which undoes Multiplication to get

\[x = \frac{18}{4} = 4.5\]

Now, the numerical answer \(4.5\) is a bit nonsensical, because it's talking about a number of people. So you could round up to \(5\), and say "It would take 5 sanitation workers to pick up the trash and recycling in \(3\) hours."

But wait a second: this does not make sense! Think about it: if it takes \(6\) workers \(4\) hours to accomplish this task, shouldn't it take \(5\) workers more time than \(4\) hours? After all, there is the same amount of work to be done, but fewer people to do it! So the answer "5 workers" cannot possibly be correct. We expect a number of workers that is larger than 6 to get the task done in a shorter amount of time.

We can learn two things from the previous discussion:

• It is important to evaluate whether or not an answer to a question makes sense in context by asking: What sort of answer would I expect to get? Does my answer seem reasonable?
• Not all problems can be solved using direct proportionality!

The good news is that this type of problem can be solved in a relatively simple way. We define the main concept in this section to see how these problems work.

Definition: Inversely Proportional

Two quantities are inversely proportional if, as one quantity increases, the other quantity decreases at the same rate.

Note how similar this definition is to the previous definition of direct proportionality. The only difference here is that one quantity increases while the other decreases:

In order to determine what type of problem you're working on, you'll need to think critically about the quantities involved, and use clues from your experience and the context of the problem to determine how the quantities are related. Things like the previous problem -- when a group of people are working together to accomplish a specific task -- are one of the primary examples of inverse proportionality. Let's see the same example again, and this time find the correct answer.

The way to approach this is to find the number of worker hours needed to accomplish the task of picking up the trash and recycling in this neighborhood. A worker hour is defined to be an hour of work done by a worker, and that number will remain constant no matter the number of workers used.

To find the number of worker hours needed for this particular neighborhood, we simply multiply the known number of workers by the known number of hours: \[\underset{\text{workers}}{6} \times \underset{\text{hours}}{4} = \underset{\text{worker hours}}{24}\]

That means that it will require \(24\) worker hours to pick up the trash and recycling in this neighborhood.

To find the number of workers needed to pick up the trash and recycling in \(3\) hours, we divide the number of worker hours by the number of hours to find the number of workers:

\[\frac{24 \text{ worker hours}}{3 \text{ hours}} = \underset{\text{worker hours}}{24} \div \underset{\text{hours}}{3} = \underset{\text{workers}}{8}\]

This means it will take \(8\) workers \(3\) hours to pick up the trash and recycling. This makes sense -- it's larger than \(6\), which was the number of workers needed to accomplish the task in \(4\) hours.

All inverse proportionality problems work this way -- multiply the two known corresponding quantities, and then divide to find the answer. As always, label your units, and check to see if your answers make sense!

Solve Similar Figure Applications

When you shrink or enlarge a photo on a phone or tablet, figure out a distance on a map, or use a pattern to build a bookcase or sew a dress, you are working with similar figures . If two figures have exactly the same shape, but different sizes, they are said to be similar. One is a scale model of the other. All their corresponding angles have the same measures and their corresponding sides are in the same ratio.

Definition: SIMILAR FIGURES

Two figures are similar if the measures of their corresponding angles are equal and their corresponding sides are in the same ratio.

For example, the two triangles in Figure are similar. Each side of ΔABC is 4 times the length of the corresponding side of ΔXYZ.

This is summed up in the Property of Similar Triangles.

Definition: PROPERTY OF SIMILAR TRIANGLES

If ΔABC is similar to ΔXYZ

To solve applications with similar figures we will follow the Problem-Solving Strategy for Geometry Applications we used earlier.

Definition: SOLVE GEOMETRY APPLICATIONS.

• Read the problem and make all the words and ideas are understood. Draw the figure and label it with the given information.
• Identify what we are looking for.
• Name what we are looking for by choosing a variable to represent it.
• Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
• Solve the equation using good algebra techniques.
• Check the answer in the problem and make sure it makes sense.
• Answer the question with a complete sentence.

Example \(\PageIndex{8}\)

ΔABC is similar to ΔXYZ

TRy it \(\PageIndex{15}\)

ΔABC is similar to ΔXYZ. The lengths of two sides of each triangle are given in the figure.

Find the length of side a

TRy it \(\PageIndex{16}\)

The next example shows how similar triangles are used with maps.

Example \(\PageIndex{9}\)

On a map, San Francisco, Las Vegas, and Los Angeles form a triangle whose sides are shown in the figure below. If the actual distance from Los Angeles to Las Vegas is 270 miles find the distance from Los Angeles to San Francisco.

TRY IT \(\PageIndex{17}\)

On the map, Seattle, Portland, and Boise form a triangle whose sides are shown in the figure below. If the actual distance from Seattle to Boise is 400 miles, find the distance from Seattle to Portland.

Try it \(\PageIndex{18}\)

Using the map above, find the distance from Portland to Boise.

We can use similar figures to find heights that we cannot directly measure.

Example \(\PageIndex{10}\)

Tyler is 6 feet tall. Late one afternoon, his shadow was 8 feet long. At the same time, the shadow of a tree was 24 feet long. Find the height of the tree.

Try it \(\PageIndex{19}\)

A telephone pole casts a shadow that is 50 feet long. Nearby, an 8 foot tall traffic sign casts a shadow that is 10 feet long. How tall is the telephone pole?

Try it \(\PageIndex{20}\)

A pine tree casts a shadow of 80 feet next to a 30-foot tall building which casts a 40 feet shadow. How tall is the pine tree?

Make sure that when you are asked to interpret something, you write a complete sentence describing the meaning of your numerical answer in the context of the problem.

• How many gallons of gas will you need to go 400 miles?
• Is this situation described by direct or inverse proportionality, and why? Give a one-sentence answer.
• What is the constant of proportionality in this situation, and how would you interpret it?
• How many professors would it take to grade the same exams in 4 hours?
• A family drinks 2 gallons of milk every 9 days. How many gallons of milk will they use in 2 weeks? Be careful with units here! (Round to one decimal place.)
• At a rate of 30 miles per hour, a certain trip takes 2 hours. How long would the same trip take at 40 miles per hour? (Round to one decimal place or give a fractional answer.)
• Think of a real-world example of direct proportionality that is different than ones we've covered in this section. Give a 2-3 sentence description of the quantities involved, and why you think they are directly proportional. If you use a source, please cite it by providing a URL.
• Think of a real-world example of inverse proportionality that is different than ones we've covered in this section. Give a 2-3 sentence description of the quantities involved, and why you think they are inversely proportional. If you use a source, please cite it by providing a URL.

Exercises \(\PageIndex{1}\)

Tonisha drove her car \(320\) miles and used \(12.5\) gallons of gas. At this rate, how far could she drive using \(10\) gallons of gas?

• Marcus worked \(14\) hours and earned $ \(210\). At the same rate of pay, how long would he have to work to earn $ \(300\)?

• At a fast-food restaurant, a 22-ounce chocolate shake has 850 calories. How many calories are in their 12-ounce chocolate shake? Round your answer to nearest whole number.

7. \(256\) miles

8. \(20\) hours

9. \(190\) pixels wide

10. 464 calories

11. 590.4 Euros

Direct Proportion problems

Understanding direct proportion.

• Direct proportion is a relationship between two variables where if one variable is multiplied by a factor, the other variable is multiplied by the same factor.
• This is often noted as ‘y is directly proportional to x’, which can be written as y ∝ x .
• If y is directly proportional to x, you can write this as y = kx , where k is called the constant of proportionality.
• You can find this constant k by rearranging the formula into k = y/x .

Solving Direct Proportion Problems

To solve a direct proportion problem, start by identifying which quantities are in direct proportion.

Find the scale factor or constant of proportionality, k .

Use this scale factor to find the unknown quantity by using the formula y = kx.

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Direct and Inverse Proportion: Worksheets with Answers

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• Directly Proportional – Explanation & Examples

Directly Proportional – Explanation & Examples

What does directly proportional mean.

Direct proportion is the relationship between two variables whose ratio is equal to a constant value. In other words, direct proportion is a situation where an increase in one quantity causes a corresponding increase in the other quantity, or a decrease in one quantity results in a decrease in the other quantity. 

Sometimes, the word proportional is used without the word direct, just know that they have a similar meaning.

Directly Proportional Formula

Direct proportion is denoted by the proportional symbol (∝). For example, if two variables x and y are directly proportional to each other, then this statement can be represented as x ∝ y.

When we replace the proportionality sign (∝) with an equal sign (=), the equation changes to:

x = k * y or x/y = k, where k is called non-zero constant of proportionality.

In our day-to-day life, we often encounter situations where a variation in one quantity results in a variation in another quantity. Let’s take a look at some of the real-life examples of directly proportional concept.

• The cost of the food items is directly proportional to the weight.
• Work done is directly proportional to the number of workers. This means that, more workers, more work and les workers, less work accomplished.

The fuel consumption of a car is 15 liters of diesel per 100 km. What distance can the car cover with 5 liters of diesel?

• Fuel consumed for every 100 km covered = 15 liters
• Therefore, the car will cover (100/15) km using 1 liter of the fuel

If 1 liter => (100/15) km

• What about 5 liters of diesel

= {(100/15) × 5} km

= 33.3 Therefore, the car can cover 33.3 km using 5 liters of the fuel.

The cost of 9 kg of beans is $ 166.50. How many kgs of beans can be bought for $ 259?

• $ 166.50 = > 9 kg of beans
• What about $ 1 => 9/166.50 kg Therefore the amount of beans purchased for $259 = {(9/166.50) × 259} kg
• =14 kg Hence, 14 kg of beans can be bought for $259

The total wages for 15 men working for 6 days are $ 9450. What is the total wages for 19 men working for 5 days?

Wages of 15 men in 6 days => $ 9450 The wage in 6 days for 1 worker = >$ (9450/15) The wage in 1 day for 1 worker => $ (9450/15 × 1/6) Wages of 19 men in a day => $ (9450 × 1/6 × 19)

The total wages of 19 men in 5 days = $ (9450 × 1/6 × 19 × 5) = $ 9975 Therefore, 19 men earn a total of $ 9975 in 5 days.

Practice Questions

Previous lesson  |  main page | next lesson.

Direct Proportion Questions

Students can quickly understand the concept of “Direct Proportion” by using the direct proportion questions and answers. The questions provided here can help students grasp the concept more quickly. To help them comprehend more clearly, we have also included some practice questions. Additionally, you may double-check your answers with comprehensive explanations provided on our page for each question. To learn more about direct proportion, click here .

Direct Proportion Questions with Solutions

1. Determine whether the following quantities vary directly or inversely with each other?

(i) Number of books purchased (a) and their price (b).

(ii) Area of land (x) and the cost of the land (y).

(i) The number of books purchased (a) and their price (b)

As we know, if the number of books purchased increases, their price will also increase. Similarly, if we purchase fewer books, their price will decrease. Hence, the number of books purchased and their price are directly proportional.

Thus, a ∝ b.

(ii) Area of land (x) and the cost of the land (y) .

The cost of the land will be more if its area increases. Similarly, the cost of land will be lesser if its area decreases. Hence, the area of land and the cost of the land is directly proportional.

Therefore, x ∝ y.

2. Given that both x and y vary directly from each other. If x = 10 and y = 15, which of the following pairs is not possible with respect to the value of x and y?

• x = 2 and y = 3
• x = 8 and y = 12
• x = 15 and y = 20
• x = 25 and y = 37.5

Given that x and y are directly proportional.

Hence, x/y = k(constant)

So, x/y = 10/15 = 2/3 …(1)

Now, check with the options provided here.

(a) x = 2 and y = 3

x/y = 2/3 …(2)

Hence, (1) = (2)

(b) x = 8 and y = 12

x/y = 8/12 = 2/3 …(3)

Hence, (1) = (3)

(c) x = 15 and y = 20

x/y = 15/20 = 3/4 …(4)

Hence, (1) ≠ (4)

(d) x = 25 and y = 37.5

x/y = 25/37.5 = 2/3 …(5)

Hence, (1) = (5)

Therefore, option (c) x = 15 and y = 20 should not be a possible pair with respect to the values x and y.

3. Check whether the values “x” and “y” given the table are directly proportional.

In the given table, the value of y is three times the value of x.

Hence, in all columns, we can observe that y = 3x, which is equal to y/x = 3

y/x = 21/7 = 3

y/x = 27/9 = 3

x/y = 39/13 = 3

x/y = 63/21 = 3

x/y = 75/25 = 3

Hence, the values “x” and “y” presented in the table are directly proportional.

4. If “a” varies directly as “b”, then find the value of “k” in the following table.

Given that “a” and “b” are directly proportional.

Hence, a/b = k (constant)

From the given table,

a/b = 12/48 = 1/4.

Similarly in second column,

a/b = 6/k = 1/4.

Hence, the value of k is 24.

5. Fill in the missing values in the table, such that “a” is directly proportional to “b”.

Given that, “a” and “b” are directly proportional.

Let the unknown values be “x” and “y”

From the given table, we can write

a/b = 3/x = 5/20 = 7/28 = 9/y

Now, compare 3/x = 5/20

Cross multiplying the above equation, we get

3(20) = 5(x)

x = 60/5 = 12.

Similarly, compare 7/28 = 9/y.

So, we get 7y = 28(9)

Hence, y = 252/7

Hence, the unknown values are x = 12 and y = 36.

Also, read: Direct and Inverse Proportion .

6. Ramya purchased 97 meters of cloth that cost Rs. 242.50. What will the length of the cloth be if she purchased it for Rs. 302.50.

As we know, the length of the cloth and its costs are directly proportional. Because if we purchase more, the cost will be higher. Similarly, if we purchase less, the cost will decrease.

Hence, we get

Now, we have to find the value of “x”.

Since the length and cost of cloth are directly proportional, we can write

97/242.50 = x/302.50

Now, cross multiply the above equation, we get

242.50x = 97(302.50)

242.50x = 29342.5

Hence, x = 29342.5 / 242.50 = 121.

Hence, the length of the cloth is 121 meters, if she purchased it for Rs. 302.50.

7. The area occupied by 10 postal stamps is 50 square centimeters. Hence, find the total area occupied by 100 such postal stamps.

Given that,

The area occupied by 10 postal stamps = 50 cm 2

Hence, the area occupied by 1 postal stamp = 50/10 = 5 cm 2 .

Therefore, the area occupied by 100 postal stamps = 100 × 5 = 500 cm 2 .

8. State whether the given statement is true or false:

‘If “a” and “b” are in direct proportion, then (a – 1) and (b – 1) are also in direct proportion”.

The given statement is “ False ”

Justification:

We know that, if “a” and “b” are in direct proportion, we can write

Let us assume that a = 2 and b = 4

Hence, a/b = 2/4 = 1/2 …(1)

So, (a – 1) / (b – 1) = (2 – 1) / (4 – 1) = 1/3 …(2)

Thus, (1) ≠ (2)

Hence, the given statement “If ‘a’ and ‘b’ are in direct proportion, then (a – 1) and (b – 1) are also in direct proportion” is false.

9. A housemaid is paid Rs. 800 for 8 days. If she works for 25 days, how much will she get?

According to the given question, we get the following:

Here, we have taken the unknown income to be “x”.

If the housemaid works for many days, her income will be more. Similarly, if she works for fewer days, she will get less income.

Hence, the number of days worked and income is directly proportional.

Therefore, we can write

8/800 = 25/x

1/100 = 25/x

Hence, x = 25(100)

Therefore, if she works for 25 days, she will get Rs. 2500.

10. If a woman earns Rs. 805 per week, how much will she earn in 16 days?

Solution: Rs. 1840

As we know, 1 week = 7 days.

Thus, the income of a woman in 7 days = 805.

Hence, the income of a woman in 1 day = 805/7 = 115.

So, the income of a woman in 16 days = 16 × 115 = 1840.

Therefore, a woman will earn Rs. 1840 in 16 days.

Explore More Articles

• Divisibility Rules Questions
• Direct and Inverse Proportion Questions
• Class 11 Maths Questions
• Volume Questions
• Area Questions
• Perimeter Questions

Practice Questions

Solve the following direct proportion questions:

1. If the cost of 18 dolls is Rs. 630, how many dolls can be purchased for Rs. 455?

2. Find the missing values in the below-given table, if p is directly proportional to q.

3. Fill in the blanks in each of the given statements, such that the statement becomes true.

• If x = 3y, then x and y vary ___ with each other.
• If “a” and “b” are said to vary directly with each other, then ____ = k, where “k” is a positive number.

4. Rahul purchased 12 books for Rs. 156. Then find the cost of 7 such books.

5. If 2 kg sugar contains 7 × 10 6 crystals, then find how many sugar crystals are present in 4 kg of sugar?

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Direct And Indirect Proportion

Here we will learn about direct and indirect proportion, including what direct proportion is and what indirect proportion is. We will look at solving some real life word problems involving these different proportional relationships. We will also look at some GCSE maths revision and exam style questions (which are also in the IGCSE).

There are also direct and indirect proportion worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What are direct and indirect proportion?

Direct and indirect proportion are two different proportional relationships. They are two ways in which quantities are related to each other.

• Direct proportion is a relationship between two quantities where as one quantity increases, so does the other quantity. 

For example,

The cost of a banana is 70p. As the number of bananas increases, so does the cost; 3 bananas would cost 3 times the cost of one banana (£2.10).

If y is directly proportional to x \ (y\propto{x}), then y=kx where k is the constant of proportionality.

Step-by-step guide: Direct proportion

• Indirect proportion (inverse proportion) is a relationship between two quantities where as one quantity increases, the other quantity decreases and vice-versa.

For example, it takes 1 worker 9 hours to dig a hole. As the number of workers increases, the number of hours it takes to dig the same hole decreases. 3 workers would take a third of the time ( 3 hours).

To calculate indirect proportion problems we need to appreciate that multiplication and division are inverse operations of each other.

Indirect proportion is sometimes known as inverse variation.

If y is indirectly proportional to x \ (y\propto\frac{1}{x}), then y=\frac{k}{x} where k is the constant of proportionality.

Step-by-step guide: Inverse proportion

How to use direct and indirect proportion

In order to answer word problems involving direct and inverse proportion:

Determine the type of proportionality relationship between the two quantities.

Calculate the constant of proportionality, k.

Calculate the unknown value.

Write the solution.

Direct and inverse proportion worksheet

Get your free direct and inverse proportion worksheet of 20+ questions and answers. Includes reasoning and applied questions.

Direct and indirect proportion examples

Example 1: direct proportion.

A t-shirt costs £4. How much do 5 t-shirts cost?

As the number of t-shirts increases, so does the cost. This is a direct proportion problem.

2 Calculate the constant of proportionality, k.

For direct proportion, the constant of proportionality k is the cost of one t-shirt. As this is already given (a t-shirt costs £4 ) we can say k=4 and so y=4x where y would be the total cost of x number of t-shirts.

3 Calculate the unknown value.

Substituting x=5 into y=4x, we have

4 Write the solution.

The cost of 5 t-shirts is £20.

Example 2: direct proportion

7 bags of sweets weigh 350 grams. How much do 10 bags of sweets weigh?

As the number of bags of sweets increases, so does the weight. This is a direct proportion problem.

For direct proportion, the constant of proportionality k is the weight of one bag of sweets.

Using k=\frac{y}{x} where y is the weight of a bag of sweets and x is the number of bags of sweets, we can calculate the value of k.

k=\frac{350}{7}=50

k=50 and so a bag of sweets weighs 50g and we can say y=50x.

Substituting x=10 into y=50x, we have

y=50\times{10}=500.

The weight of 10 bags of sweets is 500g.

Example 3: direct proportion

8 laps of a race track has a total of 12 \ km. What would the distance be for 20 laps of the race track?

As the number of laps of the track increases, so does the total distance. This is a direct proportion problem.

For direct proportion, the constant of proportionality k is the distance of one lap of the track.

Using k=\frac{y}{x} where y is the distance travelled and x is the number of laps, we can calculate the value of k.

k=\frac{12}{8}=1.5

k=1.5 and so one lap of the track is 1.5 \ km and we can say y=1.5x.

Substituting x=20 into y=1.5x, we have

y=1.5\times{20}=30.

The distance covered in 20 laps is 30 \ km.

Example 4: indirect proportion

A worker takes 10 days to fit a bathroom. How long would it take 2 workers to fit a bathroom?

As the number of workers increases, the time taken to fit a bathroom decreases. This is an indirect proportion problem.

For indirect proportion, the constant of proportionality k is the time it takes one person to fit a bathroom.

As one worker takes 10 days to fit a bathroom, we can say that k=10 and so we have the equation y=\frac{10}{x} where y is the time taken for x number of workers to complete a bathroom.

Substituting x=2 into y=\frac{10}{x}, we have

y=10\div{2}=5.

It takes 2 workers 5 days to complete a bathroom.

Example 5: indirect proportion

An oil tank takes 25 hours to be filled by 3 hose pipes. How long does it take 5 hose pipes to fill the same oil tank?

As the number of hose pipes increases, the time taken to fill the oil tank decreases. This is an indirect proportion problem.

For indirect proportion, the constant of proportionality k is the time it takes one hose to fill the oil tank.

Using k=xy where x is the number of hoses and y is the time taken to fill the oil tank, we can calculate the value of k.

k=3\times{25}=75

k=75 and so one hose would take 75 hours to fill the oil tank and we can say y=\frac{75}{x}.

Substituting x=5 into y=\frac{75}{x}, we have

y=75\div{5}=15.

It takes 5 hoses 15 hours to fill an oil tank.

Example 6: indirect proportion

10 computers can do a task in 15 minutes. How long does it take 3 computers to do the same task?

As the number of computers increases, the time taken to do a task decreases. This is an indirect proportion problem.

For indirect proportion, the constant of proportionality k is the time it takes one computer to complete a task.

Using k=xy where x is the number of computers and y is the time taken to complete the task, we can calculate the value of k.

k=10\times{15}=150

k=150 and so one computer would take 150 hours to complete a task and we can say y=\frac{150}{x}.

Substituting x=3 into y=\frac{150}{x}, we have

y=150\div{3}=50.

It takes 3 computers 50 hours to complete a task.

Common misconceptions

• Modelling assumption

Whenever you solve word problems for proportion you assume everything has the same value. If the question involves the costs of pencils, we assume each pencil costs the same. If the question involves the number of people working, we assume all the workers work at the same rate.

• Indirect proportion has a negative rate of change

Direct proportion is referred to as “as one value increases, so does the other”. Indirect proportion is therefore considered to be the opposite where “as one value decreases, so does the other”. This is not true. An indirectly proportional relationship shows that when one value increases, the other decreases. As a graph, this would look like a reciprocal graph.

• Indirect proportion is treated as direct proportion

For example, if 3 people take 12 hours to build a wall, 6 people take 24 hours to build the same size wall. This is not true as we assume everyone works at the same rate and so the wall should be built in less time if more people are building it. As the number of people increases, the time taken to build the wall decreases and so if we have 6 builders (double the original amount), the time it takes to build the wall should be 6 hours (half of the original amount). The type of proportion must be determined for every proportionality question.

• There may be several ways to solve problems involving proportion

There may be several ways to get the correct answer for proportion questions. Some ways are more efficient than others depending on the numbers involved.

• Take care with writing money

Money is used in some proportional word problems. If an answer is 4.1 you may be tempted to write it as £4.1, but the correct way of writing it would be £4.10.

• Take care with writing time

Time is used in some proportional word problems. If an answer is 7.25 you may be tempted to write it as 7 hours 25 minutes, but it would be 7 hours 15 minutes. (Remember there are 60 minutes in an hour).

Related lessons on direct and indirect proportion

Direct and indirect proportion is part of our series of lessons to support revision on proportion . You may find it helpful to start with the main proportion lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

• Directly proportional graph / inversely proportional graph
• Direct proportion
• Direct proportion formula
• Inverse proportion
• Inverse proportion formula

Practice direct and indirect proportion questions

1. One tennis ball weighs 57 grams. Find the weight of 4 tennis balls.

14.25 grams

k=57 and y=kx where y is the weight of x number of tennis balls. This means that y=57x. When x=4 ,

2. One worker takes 30 hours to build a wall. Find the time it would take 5 workers to build a similar wall.

k=30 and y=\frac{k}{x} where y is the time taken to build a wall with x number of people. This means that y=\frac{30}{x}. When x=5,

y=30\div{5}=6 hours.

3. 4 computer games cost £18. Find the cost of 5 computer games.

If 4 computer games cost £18 , 1 computer game will cost £4.50 .

k=4.5 and y=kx where y is the cost of x number of computer games. This means that y=4.5x. When x=5 ,

4. 7 workers take 20 weeks to build a house. How long would it take 10 workers to build the same house?

k=xy. When x=7, \ y=20 and so k=7 \times 20=140. This means that it would take 1 person 140 weeks to build the house and so y=\frac{140}{x}.

y=140 \div 10=14 weeks.

5. 5 pens cost 65p. Find the cost of 8 pens.

k=\frac{y}{x}. When x=5, \ y=65 and so k=65 \div 5=13. This means that 1 pen costs £0.13 and so y=0.13x.

6. 4 machines take 15 hours to complete a job. Find how long it would take 3 machines to complete the same job.

k=xy. When x=4, \ y=15 and so k=4 \times 15=60. This means that it would take 1 machine 60 hours to complete the job and so y=\frac{60}{x}.

y=60\div{3}=20 hours.

Direct and indirect proportion GCSE questions

1. 5 sacks of potatoes cost £40.

Find the cost of 7 sacks of potatoes.

2. A small town has four rubbish trucks to collect its rubbish.

It takes four trucks 18 hours to collect the rubbish.

One of the trucks breaks down.

Find how long it would take 3 trucks to collect the rubbish in the town.

3. A recipe for lemon cheesecake needs 250 grams of soft cheese.

The lemon cheesecake will have 6 proportions.

Soft cheese is sold in 300 gram packets.

The packets cost £1.25 each.

Samir wants to make enough lemon cheesecake for 15 portions.

Calculate the cost of soft cheese for Samir to make 15 portions of cheese cake.

625\div{300}=2.08\dot{3} and 3 packets.

Learning checklist

You have now learned how to:

• Solve problems involving direct and inverse proportion, including algebraic representations

The next lessons are

• Compound measures
• Best buy maths

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Direct Variation

Direct Variation or Direct Proportion:

Examples on direct variation or direct proportion:.

(i) The cost of articles varies directly as the number of articles. (More articles, more cost) (ii) The distance covered by a moving object varies directly as its speed. (More speed, more distance covered in the same time) (iii) The work done varies directly as the number of men at work. (More men at work, more is the work done in the same time) (iv) The work done varies directly as the working time. (More is the working time, more is the work done)

Solved worked-out problems on Direct Variation:

1. If $ 166.50 is the cost of 9 kg of sugar, how much sugar can be purchased for $ 259? Solution: For $ 166.50, sugar purchased = 9 kg For $ 1, sugar purchased = 9/166.50 kg   [less money, less sugar] For $ 259, sugar purchased = {(9/166.50) × 259} kg                                                                  [More money, more sugar]                                              = 14 kg. Hence, 14 kg of sugar can be purchased for $ 259.

2. If one score oranges cost $ 45, how many oranges can be bought for $ 72? Solution: For $ 45, number of oranges bought = 20 For $ 1, number of oranges bought = 20/45                                                                                                   [less money, less oranges] For $ 72, number of oranges bought = {(20/45) × 72}                                                            [More money, more oranges]                                                            = 32. Hence, the number of oranges bought for $ 72 is 32.

3. If a car covers 82.5 km in 5.5 litres of petrol, how much distance will it cover in 13.2 litres of petrol? Solution: In 5.5 litres of petrol, distance covered = 82.5 km In 1 litre of petrol, distance covered = 82.5/5.5 km                                                                 [less petrol, less distance] In 13.2 litres of petrol, distance covered = {(82.5/5.5) × 13.2} km                                                                 [More petrol, more distance]                                                                  = 198 km. Hence, the car covers 198 km in 13.2 litres of petrol.

More examples on Direct Variation word problems:

4. If 5 men or 7 women can earn $ 875 per day, how much would 10 men and 5 women earn per day? Solution: 5 men = 7 women ⇒ 1 man = 7/5 women ⇒ 10 men = (7/5 × 10) women = 14 women ⇒ (10 men + 5 women) ≡ (14 women + 5 women) = 19 women. Daily earning of 7 women = $ 875 Daily earning of 1 woman = $ (875/7)   [less women, less earning] Daily earning of 19 women = $ (875/7 × 19)                                                               [More women, more earning]                                            = $ 2375 Hence, 10 men and 5 women earn $ 2375 per day.

5. If 3 men or 4 women earn $ 480 in a day, find how much will 7 men and 11 women earn in a day? Solution: One day earning of 3 men = $ 480 One day earning of 1 man = $ (480/3)         [less men, less earning] One day earning of 7 men = $ (480/3 × 7)   [more men, more earning]                                           = $ 1120 One day earning of 4 women = $ 480 One day earning of 1 woman = $ (480/4)   [less women, less earning] One day earning of 11 women = $ (480/4 x 11)                                                                [More women, more earning]                                               = $ 1320 One day earning of 7 men and 11 women = $ (1120 + 1320) = $ 2440.

6. The cost of 16 packets of salt, each weighing 900 grams is $ 84. What will be the cost of 27 packets of salt, each weighing 1 kg? Solution: Cost of 16 packets, each weighing 9/10 kg = $ 84 Cost of 16 packets, each weighing 1 kg = $ (84 × 10/9)                                                    [more weight per packet, more cost] Cost of 1 packet, each weighing 1 kg = $ (84 × 10/9 × 1/16)                                                    [less packets, less cost] Cost of 27 packets, each weighing 1 kg = $ (84 × 10/9 × 1/16 × 27)                                                                = $ (315/2)                                                                = $ 157.50                                                    [more packets, more cost] Hence, the cost of 27 packets, each weighing 1 kg is $ 157.50.

7. If the wages of 15 workers for 6 days are $ 9450, find the wages of 19 workers for 5 days. Solution: Wages of 15 workers for 6 days = $ 9450 Wages of 1 worker for 6 days = $ (9450/15)                                                               [less workers, less wages] Wages of 1 worker for 1 day = $ (9450/15 × 1/6)                                                               [less days, less wages] Wages of 19 workers for 1 day = $ (9450 × 1/6 × 19)                                                               [more workers, more wages]

Wages of 19 workers for 5 day = $ (9450 × 1/6 × 19 × 5)                                                               [more workers, more wages]                                                  = $ 9975 Hence, the wages of 19 workers for 5 days = $ 9975. Direct Variation

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R10a – Solving problems involving direct proportion

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• A knowledge of the four operations from lessons N2a , N2b , N2c , N2d , N2e and N2f is assumed.
• A17a – Solving simple linear equations in one unknown algebraically
• A17c – Finding solutions to linear equations using graphs
• A11a – Identifying roots, intercepts and turning points of quadratic functions graphically

An illustration of quantities in direct proportion

Adjust the dark grey slider to select the cost of 1 metre of rope. Now in this example, the cost of a piece of rope is directly proportional to its length . The longer the rope, the more it will cost. What happens to the cost if you double the length of rope? What if you triple the length? Investigate by adjusting the blue and green sliders.

Double number line activity

Solving problems involving direct proportion.