problem solving in one word

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Synonyms for Problem Solving

Meaning of Problem Solving

The term “problem solving” refers to the process of identifying, analyzing, and resolving issues or challenges. This process often involves critical thinking, creativity, and resourcefulness. In this article, we will explore general synonyms for “problem solving,” as well as those specifically used in academic writing.

General Synonyms for Problem Solving

• Troubleshooting
• Conflict resolution
• Decision making
• Critical thinking
• Resourcefulness
• Adaptability

Synonyms for Problem Solving in Academic Writing

In academic writing, it is essential to use precise and appropriate language. When discussing the problem-solving process or skills, consider the following synonyms, which are better suited for academic contexts:

Synonyms, Definitions, and Examples

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problem-solving

Definition of problem-solving

Examples of problem-solving in a sentence.

These examples are programmatically compiled from various online sources to illustrate current usage of the word 'problem-solving.' Any opinions expressed in the examples do not represent those of Merriam-Webster or its editors. Send us feedback about these examples.

Dictionary Entries Near problem-solving

Cite this entry.

“Problem-solving.” Merriam-Webster.com Dictionary , Merriam-Webster, https://www.merriam-webster.com/dictionary/problem-solving. Accessed 7 Sep. 2024.

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Problem Solving or Problem-Solving?

The term problem-solving with a hyphen is an adjective that describes nouns in a sentence. E.g., “The problem-solving abilities of this program are excellent.” Furthermore, the term problem solving without a hyphen is a noun. E.g., “He is excellent at problem solving.”

The term problem solving can appear as two different word types, a noun or a compound adjective.

When problem-solving has a hyphen, it is an adjective describing a noun .

For example, in this sentence, problem-solving describes capabilities , which is a noun. Therefore, you need to include a hyphen.

• Her problem-solving capabilities are outstanding.

However, when you use problem solving as a noun , the term is usually not hyphenated because it doesn’t modify anything.

• She is an expert at all types of problem solving .

Furthermore, the rule of using a hyphen for the adjective but not for the noun is correct grammar according to AP Style and the Chicago Manual of Style .

Sometimes in English, the rules concerning punctuation, such as hyphens, are vague. For example, in the Cambridge and Oxford dictionaries, the noun problem-solving appears with a hyphen.

Essentially, you should always use a hyphen for the term problem-solving as an adjective. However, for problem solving as a noun, unless you follow a specific style guide like AP that does not use a hyphen, you can choose whether to hyphenate the noun. Just make sure that you are consistent with your choice.

In addition, Google Ngram shows that problem solving without a hyphen is slightly more popular in the US than with a hyphen. However, in the UK , the hyphenated version is slightly more frequent.

Now that you have learned the basics concerning the term problem solving, please keep reading the rest of the article to learn more about using the two variations of problem solving .

Problem Solving

The term problem solving as two words without a hyphen is the noun form.

In terms of use, it relates to resolving problems, which can relate to problems in a specific discipline or more general problems.

In the following examples, the term problem solving refers to the act of solving problems. Therefore, we do not need a hyphen.

• Her inability to perform problem solving on any task means that she loses her temper quickly.
• You should include problem solving as a skill on your resume.
• In team projects, effective problem solving can lead to innovative solutions and successful outcomes.
• Problem solving is not just about finding immediate answers but understanding the root cause of the issue.
• Her approach to problem solving often involves breaking down complex tasks into manageable steps.

Furthermore, you should use the above rule when following both AP Style or the Chicago Manual of Style.

Problem-Solving

The term problem-solving as one word with a hyphen is a compound word that appears before a noun. Hence, it modifies the noun .

For example, in the following sentence, the word problem-solving modifies the noun skills.

• His problem-solving skills are the best in the company.
• Her problem-solving approach is both methodical and creative, making her a valuable asset to the team.
• The workshop focuses on problem-solving techniques to address everyday challenges.
• Many employers prioritize candidates with strong problem-solving abilities in dynamic work environments.
• The game challenges children to use their problem-solving instincts to navigate various puzzles.

Furthermore, you will come across some sentences in which the noun problem solving has a hyphen.

As shown in these examples:

• You need to work on your problem-solving if you want to become an engineer.
• Improving your problem-solving is essential for success in mathematics.
• Mastering problem-solving will greatly benefit you in software development.

This is often a stylistic choice . However, in writing that follows AP Style , you should not use a hyphen with the noun form of problem solving.

Problemsolving

The word problemsolving as a single word with no hyphen or space is incorrect , and you shouldn’t use it in this format.

There are two ways you can use the term problem solving .

The first is as a noun, in which case there is no hyphen, but there is a space.

• Correct: He is excellent at problem solving because he never gives up. (noun)
• Incorrect: He is excellent at problemsolving because he never gives up. (noun)
• Correct : She has a knack for problem solving , especially in high-pressure situations. (noun)
• Incorrect : She has a knack for problemsolving , especially in high-pressure situations. (noun)

Also, you can write problem-solving as an adjective, in which case you need a hyphen.

• Correct: He is excellent at problem solving because he never gives up. (adjective)
• Incorrect: He is excellent at problemsolving because he never gives up. (adjective)
• Correct : They often turn to her for advice on problem solving in challenging situations. (adjective)
• Incorrect : They often turn to her for advice on problemsolving in challenging situations. (adjective)

That’s all you need to know about the grammar rules concerning the words problem solving , problem-solving , and problemsolving . Rest assured that your sentences will be correct if you follow these!

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Problem solving.

• noun the thought processes involved in solving a problem see more see less types: show 14 types... hide 14 types... enquiry , inquiry , research a search for knowledge calculation , computation , figuring , reckoning problem solving that involves numbers or quantities nature study the study of animals and plants in the natural world (usually at an elementary level) experiment , experimentation the testing of an idea empirical research an empirical search for knowledge investigation , probe an inquiry into unfamiliar or questionable activities canvass , opinion poll , poll , public opinion poll , survey an inquiry into public opinion conducted by interviewing a random sample of people heraldry the study and classification of armorial bearings and the tracing of genealogies extrapolation (mathematics) calculation of the value of a function outside the range of known values interpolation (mathematics) calculation of the value of a function between the values already known conversion a change in the units or form of an expression: "conversion from Fahrenheit to Centigrade" approximation , estimate , estimation , idea an approximate calculation of quantity or degree or worth derivative , derived function , differential , differential coefficient , first derivative the result of mathematical differentiation; the instantaneous change of one quantity relative to another; df(x)/dx integral the result of a mathematical integration; F(x) is the integral of f(x) if dF/dx = f(x) type of: cerebration , intellection , mentation , thinking , thought , thought process the process of using your mind to consider something carefully
• noun the area of cognitive psychology that studies the processes involved in solving problems see more see less type of: cognitive psychology an approach to psychology that emphasizes internal mental processes

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Problem-Solving Strategies and Obstacles

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• Application
• Improvement

From deciding what to eat for dinner to considering whether it's the right time to buy a house, problem-solving is a large part of our daily lives. Learn some of the problem-solving strategies that exist and how to use them in real life, along with ways to overcome obstacles that are making it harder to resolve the issues you face.

What Is Problem-Solving?

In cognitive psychology , the term 'problem-solving' refers to the mental process that people go through to discover, analyze, and solve problems.

A problem exists when there is a goal that we want to achieve but the process by which we will achieve it is not obvious to us. Put another way, there is something that we want to occur in our life, yet we are not immediately certain how to make it happen.

Maybe you want a better relationship with your spouse or another family member but you're not sure how to improve it. Or you want to start a business but are unsure what steps to take. Problem-solving helps you figure out how to achieve these desires.

The problem-solving process involves:

• Discovery of the problem
• Deciding to tackle the issue
• Seeking to understand the problem more fully
• Researching available options or solutions
• Taking action to resolve the issue

Before problem-solving can occur, it is important to first understand the exact nature of the problem itself. If your understanding of the issue is faulty, your attempts to resolve it will also be incorrect or flawed.

Problem-Solving Mental Processes

Several mental processes are at work during problem-solving. Among them are:

• Perceptually recognizing the problem
• Representing the problem in memory
• Considering relevant information that applies to the problem
• Identifying different aspects of the problem
• Labeling and describing the problem

Problem-Solving Strategies

There are many ways to go about solving a problem. Some of these strategies might be used on their own, or you may decide to employ multiple approaches when working to figure out and fix a problem.

An algorithm is a step-by-step procedure that, by following certain "rules" produces a solution. Algorithms are commonly used in mathematics to solve division or multiplication problems. But they can be used in other fields as well.

In psychology, algorithms can be used to help identify individuals with a greater risk of mental health issues. For instance, research suggests that certain algorithms might help us recognize children with an elevated risk of suicide or self-harm.

One benefit of algorithms is that they guarantee an accurate answer. However, they aren't always the best approach to problem-solving, in part because detecting patterns can be incredibly time-consuming.

There are also concerns when machine learning is involved—also known as artificial intelligence (AI)—such as whether they can accurately predict human behaviors.

Heuristics are shortcut strategies that people can use to solve a problem at hand. These "rule of thumb" approaches allow you to simplify complex problems, reducing the total number of possible solutions to a more manageable set.

If you find yourself sitting in a traffic jam, for example, you may quickly consider other routes, taking one to get moving once again. When shopping for a new car, you might think back to a prior experience when negotiating got you a lower price, then employ the same tactics.

While heuristics may be helpful when facing smaller issues, major decisions shouldn't necessarily be made using a shortcut approach. Heuristics also don't guarantee an effective solution, such as when trying to drive around a traffic jam only to find yourself on an equally crowded route.

Trial and Error

A trial-and-error approach to problem-solving involves trying a number of potential solutions to a particular issue, then ruling out those that do not work. If you're not sure whether to buy a shirt in blue or green, for instance, you may try on each before deciding which one to purchase.

This can be a good strategy to use if you have a limited number of solutions available. But if there are many different choices available, narrowing down the possible options using another problem-solving technique can be helpful before attempting trial and error.

In some cases, the solution to a problem can appear as a sudden insight. You are facing an issue in a relationship or your career when, out of nowhere, the solution appears in your mind and you know exactly what to do.

Insight can occur when the problem in front of you is similar to an issue that you've dealt with in the past. Although, you may not recognize what is occurring since the underlying mental processes that lead to insight often happen outside of conscious awareness .

Research indicates that insight is most likely to occur during times when you are alone—such as when going on a walk by yourself, when you're in the shower, or when lying in bed after waking up.

How to Apply Problem-Solving Strategies in Real Life

If you're facing a problem, you can implement one or more of these strategies to find a potential solution. Here's how to use them in real life:

• Create a flow chart . If you have time, you can take advantage of the algorithm approach to problem-solving by sitting down and making a flow chart of each potential solution, its consequences, and what happens next.
• Recall your past experiences . When a problem needs to be solved fairly quickly, heuristics may be a better approach. Think back to when you faced a similar issue, then use your knowledge and experience to choose the best option possible.
• Start trying potential solutions . If your options are limited, start trying them one by one to see which solution is best for achieving your desired goal. If a particular solution doesn't work, move on to the next.
• Take some time alone . Since insight is often achieved when you're alone, carve out time to be by yourself for a while. The answer to your problem may come to you, seemingly out of the blue, if you spend some time away from others.

Obstacles to Problem-Solving

Problem-solving is not a flawless process as there are a number of obstacles that can interfere with our ability to solve a problem quickly and efficiently. These obstacles include:

• Assumptions: When dealing with a problem, people can make assumptions about the constraints and obstacles that prevent certain solutions. Thus, they may not even try some potential options.
• Functional fixedness : This term refers to the tendency to view problems only in their customary manner. Functional fixedness prevents people from fully seeing all of the different options that might be available to find a solution.
• Irrelevant or misleading information: When trying to solve a problem, it's important to distinguish between information that is relevant to the issue and irrelevant data that can lead to faulty solutions. The more complex the problem, the easier it is to focus on misleading or irrelevant information.
• Mental set: A mental set is a tendency to only use solutions that have worked in the past rather than looking for alternative ideas. A mental set can work as a heuristic, making it a useful problem-solving tool. However, mental sets can also lead to inflexibility, making it more difficult to find effective solutions.

How to Improve Your Problem-Solving Skills

In the end, if your goal is to become a better problem-solver, it's helpful to remember that this is a process. Thus, if you want to improve your problem-solving skills, following these steps can help lead you to your solution:

• Recognize that a problem exists . If you are facing a problem, there are generally signs. For instance, if you have a mental illness , you may experience excessive fear or sadness, mood changes, and changes in sleeping or eating habits. Recognizing these signs can help you realize that an issue exists.
• Decide to solve the problem . Make a conscious decision to solve the issue at hand. Commit to yourself that you will go through the steps necessary to find a solution.
• Seek to fully understand the issue . Analyze the problem you face, looking at it from all sides. If your problem is relationship-related, for instance, ask yourself how the other person may be interpreting the issue. You might also consider how your actions might be contributing to the situation.
• Research potential options . Using the problem-solving strategies mentioned, research potential solutions. Make a list of options, then consider each one individually. What are some pros and cons of taking the available routes? What would you need to do to make them happen?
• Take action . Select the best solution possible and take action. Action is one of the steps required for change . So, go through the motions needed to resolve the issue.
• Try another option, if needed . If the solution you chose didn't work, don't give up. Either go through the problem-solving process again or simply try another option.

You can find a way to solve your problems as long as you keep working toward this goal—even if the best solution is simply to let go because no other good solution exists.

Sarathy V. Real world problem-solving .  Front Hum Neurosci . 2018;12:261. doi:10.3389/fnhum.2018.00261

Dunbar K. Problem solving . A Companion to Cognitive Science . 2017. doi:10.1002/9781405164535.ch20

Stewart SL, Celebre A, Hirdes JP, Poss JW. Risk of suicide and self-harm in kids: The development of an algorithm to identify high-risk individuals within the children's mental health system . Child Psychiat Human Develop . 2020;51:913-924. doi:10.1007/s10578-020-00968-9

Rosenbusch H, Soldner F, Evans AM, Zeelenberg M. Supervised machine learning methods in psychology: A practical introduction with annotated R code . Soc Personal Psychol Compass . 2021;15(2):e12579. doi:10.1111/spc3.12579

Mishra S. Decision-making under risk: Integrating perspectives from biology, economics, and psychology . Personal Soc Psychol Rev . 2014;18(3):280-307. doi:10.1177/1088868314530517

Csikszentmihalyi M, Sawyer K. Creative insight: The social dimension of a solitary moment . In: The Systems Model of Creativity . 2015:73-98. doi:10.1007/978-94-017-9085-7_7

Chrysikou EG, Motyka K, Nigro C, Yang SI, Thompson-Schill SL. Functional fixedness in creative thinking tasks depends on stimulus modality .  Psychol Aesthet Creat Arts . 2016;10(4):425‐435. doi:10.1037/aca0000050

Huang F, Tang S, Hu Z. Unconditional perseveration of the short-term mental set in chunk decomposition .  Front Psychol . 2018;9:2568. doi:10.3389/fpsyg.2018.02568

National Alliance on Mental Illness. Warning signs and symptoms .

Mayer RE. Thinking, problem solving, cognition, 2nd ed .

Schooler JW, Ohlsson S, Brooks K. Thoughts beyond words: When language overshadows insight. J Experiment Psychol: General . 1993;122:166-183. doi:10.1037/0096-3445.2.166

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

“Problem-Solving” Or “Problem Solving”? Learn If It Is Hyphenated

Is it problem-solving or problem solving? Hyphenation rules seem to be a little confusing when you’re first picking up a language. Don’t worry, though. They’re not nearly as complicated as the language may have led you to believe!

Problem-Solving Or Problem Solving – Hyphenated Or Not?

When we discuss the problem-solving hyphen rule, we learn that problem-solving is hyphenated when used to modify a noun or object in a sentence. We keep the two words separated when using them as their own noun and not modifying anything else in the sentence.

Examples Of When To Use “Problem-Solving”

Now that we’re into the whole debate of problem-solving vs problem solving, let’s look through some examples of how we can use “problem-solving” with a hyphen. As stated above, we use “problem-solving” when modifying a noun or object in a sentence. It’s the most common way to write “problem-solving.” Even the spelling without a hyphen is slowly being pushed out of common language use!

• This is a problem-solving class.
• I hold a problem-solving position at my workplace.
• My manager put me in charge of the problem-solving accounts.
• They say I have a problem-solving mind.
• We’re known as problem-solving children.

Examples Of When To Use “Problem Solving”

Though much less common to be seen written as a phrase noun, it is still worth mentioning. It’s grammatically correct to use “problem solving” at the end of a sentence or clause without a hyphen. However, as we stated above, many people are beginning to prefer the ease of sticking to the hyphenated spelling, meaning that it’s slowly phasing out of existence even in this form.

• I’m good at problem solving.
• This requires a lot of problem solving.
• We are all trained in problem solving.
• My job asks for problem solving.
• Did you say you were good at problem solving?

Is Problem-Solving Hyphenated AP Style?

Have you had a look through the rules in the AP stylebook before? Even if you haven’t, there’s a good explanation for hyphens there. As we stated above, we use hyphens when linking close words that modify a noun or object in a sentence. They’re used to help a reader better understand what is going on through the modification of the clause.

Should I Capitalize “Solving” In The Word “Problem-Solving”?

The question of “is problem-solving hyphenated” was answered, but now we’ve got a new question. What happens to capitalization rules when we add a hyphen to a title. It depends on your own title choices, so let’s look a little further into the three potential options. The first option capitalizes only the first word and any proper nouns in a title. In this case, neither word in “problem-solving” is capitalized.

The second option capitalizes all words except for short conjunctions, short prepositions, and articles. In this case, you will always capitalize “problem” but always leave “solving” uncapitalized. The final option capitalizes every single word in a title. No matter what, you’ll capitalize both words in “problem-solving” when using this style to write your titles.

Does The Rule Also Apply To “Problem Solver” Vs “Problem-Solver”?

The same rule does apply when we use “problem solver” instead of “problem solving.” However, it’s not often that we’ll see a “problem-solver” modifying a noun or object (unless it’s a problem-solver robot or something). So, it’s most likely you’ll write “problem solver.”

Alternatives To “Problem-Solving”

If you’re still struggling with the hyphen rule of whether it’s problem solving or problem-solving, there’s one last thing we can help you with. We can give you some alternatives that have the same meanings but don’t require a hyphen. This way, you can be safe in your own knowledge without having to worry about getting the rules wrong.

• interpretive

Quiz – Problem-Solving Or Problem Solving?

We’ll finish with a quiz to see how much you’ve learned from this article. The answers are all multiple choice, so you should have a blast with them! We’ll include the answers at the end to reference as well.

• I’ve been told that I’m good at (A. problem-solving / B. problem solving).
• I hold my (A. problem-solving / B. problem solving) skills close to my heart.
• We aren’t great at (A. problem-solving / B. problem solving).
• These are all the best (A. problem-solving / B. problem solving) subjects.
• Can we have a go at a (A. problem-solving / B. problem solving) puzzle?

Quiz Answers

Martin holds a Master’s degree in Finance and International Business. He has six years of experience in professional communication with clients, executives, and colleagues. Furthermore, he has teaching experience from Aarhus University. Martin has been featured as an expert in communication and teaching on Forbes and Shopify. Read more about Martin here .

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Meaning of problem-solving in English

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• problem-solver

Examples of problem-solving

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kick something into the long grass

to delay dealing with something, especially because you want people to forget about it

Like a bull in a china shop: talking about people who are clumsy

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Thesaurus for Problem-solving

Related terms for problem-solving - synonyms, antonyms and sentences with problem-solving, similar meaning.

• troubleshooting
• problem solving
• investigative
• brainstorming
• pinpointing
• critical thinking
• well grounded
• discovering
• figuring something out
• finding a solution to
• finding an answer to
• racking brains
• deliberating

Opposite meaning

• complicating
• disregarding
• putting off
• aggravation
• shortcoming
• learning parrot-fashion
• learning off by heart
• nonrational
• nonsensical
• unmethodical
• unreasonable
• unsystematic
• complexification
• convolution
• elaboration

Common usage

• science teaching
• proper solution
• creative thinking

Sentence Examples

Proper usage in context.

• And we have a problem-solving technique that works
• It says I lack problem-solving skills
• Lectures, problem-solving and so on
• Lectures, problem-solving and so on. It's an extraordinary place
• Practical problem-solving builds a culture of cooperation

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Definition of problem-solving noun from the Oxford Advanced Learner's Dictionary

problem-solving

• to develop problem-solving skills and strategies

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Solve Unitary Method Questions and Word Problems

• Updated on  
• Sep 4, 2024

The Unitary Method is a fundamental mathematical technique used to solve problems by finding the value of a single unit and then using it to determine the value of multiple units. It is widely applicable in solving real-life problems, particularly in areas like ratio and proportion, percentage calculations, and unit conversions. The Unitary method is essential for understanding how to break down complex problems into simpler, more manageable parts. This approach is often used in various types of competitive exams, including those for banking , government jobs , and entrance tests , where questions related to the Unitary Method appear frequently. Key concepts include understanding ratios, proportions, and basic arithmetic operations. This guide covers the definition, types, properties, formulas, and word problems associated with the Unitary Method, providing a comprehensive overview to enhance your problem-solving skills.

Table of Contents

• 1 What is the Unitary Method?
• 2.1 1. Direct Variation (Direct Proportion)
• 2.2 2. Inverse Variation (Inverse Proportion)
• 3 Properties of Unitary Method
• 4 Formulas of Unitary Method
• 5 Types of Unitary Method Word Problems
• 6 Unitary Method Questions: Step-by-Step Solution

What is the Unitary Method?

The Unitary Method is a fundamental mathematical approach used to solve problems by first determining the value of a single unit and then using that value to find the value of multiple units. It operates on the principle that if the value of one unit is known, the value of any number of units can be calculated through multiplication or division.

Important Terms:

• Unit : A single quantity or measurement, such as one item, one kilogram, or one liter.
• Ratio : A relationship between two quantities, showing how many times one value is contained within another.
• Proportion : An equation that states that two ratios are equal.
• Multiplier : A factor by which a unit is multiplied to find the value of multiple units.
• Divisor : A number by which a total is divided to find the value of a single unit.
• Cost per Unit : The price of one unit of an item or service.
• Rate : A measure of a quantity relative to another quantity, such as speed (distance per unit of time) or density (mass per unit of volume).
• Percentage : A ratio or fraction out of 100, used to express proportions.

How It Works:

• If the value of multiple units is known, divide the total value by the number of units to find the value of one unit.
• Example: If 5 apples cost $10, the cost of one apple is $10 ÷ 5 = $2.
• If the value of one unit is known, multiply it by the number of units to find the total value.
• Example: If one apple costs $2, the cost of 8 apples is $2 × 8 = $16.

Types of Unitary Method

The Unitary Method can be categorized into two main types, based on the relationship between the quantities involved:

1. Direct Variation (Direct Proportion)

In direct variation, as one quantity increases, the other quantity also increases in the same proportion, and vice versa. The Unitary Method is applied here to find the value of one quantity when the other is known.

Example : If 5 kilograms of rice cost $20, then the cost of 1 kilogram (the unit) is $20 ÷ 5 = $4. If you want to find the cost of 8 kilograms of rice, you multiply the unit cost by 8: $4 × 8 = $32.

Application : This type of unitary method is used in problems involving prices, wages, and other quantities that increase or decrease in direct proportion to each other.

2. Inverse Variation (Inverse Proportion)

In inverse variation, as one quantity increases, the other quantity decreases proportionally, and vice versa. Here, the Unitary Method is used to find one quantity when the product of two quantities is constant.

Example : If 6 workers can complete a task in 12 days, then the work done by 1 worker (the unit) in 12 days is equivalent to the work done by 6 workers. To find out how long it will take for 3 workers to complete the task, divide the total work (6 × 12 = 72 worker-days) by the number of workers (3), resulting in 72 ÷ 3 = 24 days.

Application : This type is used in problems involving speed and time, work and workers, and other scenarios where an increase in one quantity leads to a proportional decrease in another.

Also Read: Reciprocal: Definition, Meaning and Solved Examples

Properties of Unitary Method

The Unitary Method has several key properties that make it an effective and versatile tool for solving a wide range of mathematical problems. Here are the main properties of the Unitary Method:

1. Proportionality

• Property : If one quantity changes, the other changes in a consistent ratio.
• Example : If 2 pens cost $4, then 4 pens will cost $8 (double the quantity, double the cost).

2. Linearity

• Property : Changes between quantities are linear, meaning direct or inverse proportionality.
• Example : If a car travels 60 miles in 1 hour, it will travel 120 miles in 2 hours (distance is directly proportional to time).

3. Scalability

• Property : The method can be applied to any number of units.
• Example : If 1 apple costs $2, then 50 apples will cost $2 × 50 = $100.

4. Versatility

• Property : Can be used for various types of problems (e.g., ratios, speed, cost).
• Example : If a worker can complete a task in 5 days, 2 workers can complete it in 2.5 days (work is inversely proportional to the number of workers).

5. Simplicity

• Property : Breaks down complex problems into simpler steps.
• Example : To find the cost of 7 chocolates when 1 chocolate costs $3, simply multiply: $3 × 7 = $21.

6. Consistency

• Property : Provides uniform and reliable results.
• Example : If 10 books cost $50, then 1 book costs $50 ÷ 10 = $5. For 20 books, it’s $5 × 20 = $100.

7. Applicability in Both Direct and Inverse Proportions

• Property : Works for both direct and inverse relationships.
• Example : Direct: 3 meters of cloth costs $30, so 6 meters cost $60. Inverse: 4 workers take 8 hours to complete a task; 8 workers will take 4 hours.

Formulas of Unitary Method

The Unitary Method primarily revolves around finding the value of one unit and then using that value to determine the value of multiple units. The formulas associated with the Unitary Method are straightforward and can be categorized based on direct and inverse proportion problems.

1. Direct Proportion Formula

In direct proportion, as one quantity increases, the other increases proportionally. The key formulas are:

• Finding the value of one unit:

Example : If 5 apples cost $10, then the cost of one apple is: 10/5 = 2 dollars

• Finding the value of multiple units:

Example : If one apple costs $2, then the cost of 8 apples is: 2×8 = 16 dollars

2. Inverse Proportion Formula

In inverse proportion, as one quantity increases, the other decreases proportionally. The key formulas are:

• Finding the value of one unit (work or time):

Example : If 6 workers take 12 days to complete a task, the total worker-days is: 6×12 = 72 worker-days

Example : If 3 workers are to complete the same task, the time taken would be: 72/3 = 24 days

Also Read: All Perfect Cube Numbers

Types of Unitary Method Word Problems

Here are five-word problems that can be solved using the Unitary Method:

1. Cost Calculation

Problem : If 7 notebooks cost Rs 35, what is the cost of 5 notebooks? 

• Cost of 1 notebook = Rs35 ÷ 7 = Rs5
• Cost of 5 notebooks = Rs 5 × 5 = $25

2. Time and Work

Problem : If 8 workers can complete a task in 12 days, how many days will it take for 6 workers to complete the same task? 

• Total work = 8 workers × 12 days = 96 worker-days
• Time taken by 6 workers = 96 worker-days ÷ 6 workers = 16 days

3. Distance and Speed

Problem : A car travels 240 kilometers in 4 hours. How far will it travel in 7 hours at the same speed? 

• Distance covered in 1 hour = 240 km ÷ 4 hours = 60 km/hour
• Distance in 7 hours = 60 km/hour × 7 hours = 420 kilometers

4. Wage Calculation

Problem : If 5 workers earn Rs 400 in a day, how much will 8 workers earn in a day? 

• Earnings of 1 worker = Rs 400 ÷ 5 = $80
• Earnings of 8 workers = Rs 80 × 8 = $640

5. Quantity and Cost

Problem : If 10 kilograms of sugar cost $50, how much will 15 kilograms of sugar cost? 

• Cost of 1 kilogram of sugar = Rs 50 ÷ 10 kg = $5/kg
• Cost of 15 kilograms = Rs 5/kg × 15 kg = $75

Unitary Method Questions: Step-by-Step Solution

Question 1 If 15 workers can complete a construction project in 20 days, how many days will it take for 25 workers to complete the same project, assuming the work rate is consistent?

Solution: Total work = Number of workers × Number of days

Total work = 15 × 20

Total work = 300 worker-days

Days for 25 workers = Total work / Number of workers

Days for 25 workers = 300 / 25

Days for 25 workers = 12 days

Question 2:A machine can produce 1200 units in 8 hours. If the production rate is consistent, how many units can it produce in 15 hours?

Solution: Units per hour = Total units / Number of hours

Units per hour = 1200 / 8

Units per hour = 150

Units in 15 hours = Units per hour × Number of hours

Units in 15 hours = 150 × 15

Units in 15 hours = 2250

Question 3:If 30 students can complete a group project in 10 days, how many students are required to complete the same project in 6 days?

Solution: Total work = Number of students × Number of days

Total work = 30 × 10

Total work = 300 student-days

Number of students for 6 days = Total work / Number of days

Number of students for 6 days = 300 / 6

Number of students for 6 days = 50 students

Question 4:A factory produces 5000 units of a product in 12 hours. How many units will it produce in 20 hours?

Units per hour = 5000 / 12

Units per hour ≈ 416.67

Units in 20 hours = Units per hour × Number of hours

Units in 20 hours = 416.67 × 20

Units in 20 hours ≈ 8333.33

Question 5:If 9 liters of paint cover 72 square meters, how many liters are required to cover 150 square meters?

Solution: Coverage per liter = Total coverage / Number of liters

Coverage per liter = 72 / 9

Coverage per liter = 8 square meters

Liters required = Total area / Coverage per liter

Liters required = 150 / 8

Liters required = 18.75 liters

Question 6:If a car travels 360 kilometers on 40 liters of fuel, how many liters will it need to travel 540 kilometers?

Solution: Distance per liter = Total distance / Number of liters

Distance per liter = 360 / 40

Distance per liter = 9 kilometers

Liters required = Total distance / Distance per liter

Liters required = 540 / 9

Liters required = 60 liters

Question 7:If 5 workers can finish a task in 16 days, how many days will it take for 8 workers to complete the same task?

Total work = 5 × 16

Total work = 80 worker-days

Days for 8 workers = Total work / Number of workers

Days for 8 workers = 80 / 8

Days for 8 workers = 10 days

Question 8:A factory produces 2400 units of an item in 6 hours. How many hours will it take to produce 6000 units?

Solution: Production rate = Total units / Number of hours

Production rate = 2400 / 6

Production rate = 400 units per hour

Hours required = Total units / Production rate

Hours required = 6000 / 400

Hours required = 15 hours

Question 9:If 10 kg of a substance is used to prepare 25 liters of solution, how many kilograms are needed to prepare 75 liters of the same solution?

Solution: Kilograms per liter = Total kilograms / Number of liters

Kilograms per liter = 10 / 25

Kilograms per liter = 0.4 kg per liter

Kilograms required = Kilograms per liter × Number of liters

Kilograms required = 0.4 × 75

Kilograms required = 30 kg

Question 10:A contractor can paint 1500 square meters in 10 hours. If another contractor can paint at twice the speed, how many hours will it take for this second contractor to paint 1500 square meters?

Solution: First contractor’s rate = Total area / Number of hours

First contractor’s rate = 1500 / 10

First contractor’s rate = 150 square meters per hour

Second contractor’s rate = 2 × First contractor’s rate

Second contractor’s rate = 2 × 150

Second contractor’s rate = 300 square meters per hour

Hours required = Total area / Second contractor’s rate

Hours required = 1500 / 300

Hours required = 5 hours

Unitary method is a technique used to solve problems by first finding the value of a single unit, then multiplying it to find the value of the required quantity. For example, if 5 apples cost ₹20, to find the cost of 3 apples, we first find the cost of 1 apple (₹20 ÷ 5 = ₹4) and then multiply by 3 (₹4 × 3 = ₹12).

Unitary method is a technique to solve problems by first finding the value of one unit and then multiplying it by the required number of units. It’s a common method used in mathematics for calculations involving ratios, proportions, and rates.

Unitary method formula is a technique to find the value of a single unit from the value of multiple units, and then use that value to find the value of a different number of units. — Value of a single unit = Total value / Total number of units — Value of required number of units = Value of a single unit x Required number of units.

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Mohit Rajak

Mohit Rajak, a soul entwined with the rhythm of words, finds solace in crafting verses that dance between the lines of poetry. With a pen as his wand, he weaves intricate tales and musings, breathing life into the blank canvas of pages. Through the art of blogging, Mohit embraces the world, sharing his thoughts, emotions, and unique perspective with those who venture into the realm of his written expressions. Each word, a brushstroke painting the canvas of his literary journey.

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