use a proportion to solve the problem

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How to Solve Proportions

Last Updated: February 24, 2023

This article was reviewed by Grace Imson, MA . Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math instructor at the City College of San Francisco and was previously in the Math Department at Saint Louis University. She has taught math at the elementary, middle, high school, and college levels. She has an MA in Education, specializing in Administration and Supervision from Saint Louis University. This article has been viewed 75,566 times.

What is the "vertical" way to solve a proportion?

How can I solve a proportion with the "horizontal" method?

How do I solve a proportion step by step by cross-multiplying?

How do you find the missing value in a proportion with a table of ratios?

• Each column in this table represents a fraction. All of the fractions in this table are equal to each other.

• The two answers are the same, which means your answer is correct.

How do you solve percent proportions?

How do you solve proportions algebraically?

• You can change the left hand side of the equation, as long as you do the same math to the right hand side.

• To get rid of the fraction on the left, multiply both sides by 27:

How do you solve a proportion with a variable on both sides?

• Warning : This is a difficult example. If you haven't learned about quadratic equations yet, you might want to skip this part.

• You can now solve this as a quadratic equation , using any method that you've learned.

Proportions Calculator, Practice Problems, and Answers

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Community Q&A

• The algebraic method above works with any proportion. But for a specific proportion, there is often a faster way to use algebra to find the answer. As you learn more algebra, this will get easier. Thanks Helpful 0 Not Helpful 0

Video . By using this service, some information may be shared with YouTube.

• ↑ http://www.mathvillage.info/node/72
• ↑ https://www.youtube.com/watch?v=nwsDiID7UtQ
• ↑ https://www.youtube.com/watch?v=Uo8HgcyfRFI
• ↑ https://www.purplemath.com/modules/ratio2.htm

About This Article

To solve proportions, start by taking the numerator, or top number, of the fraction you know and multiplying it with the denominator, or bottom number, of the fraction you don’t know. Next, take that number and divide it by the denominator of the fraction you know. Now you can replace x with this final number. For example, to figure out “x” in the problem 3/4 = x/8, multiply 3 x 8 to get 24, then divide 24 / 4 to get 6, or the value of x. To learn how to use proportions to determine percentages, read on! Did this summary help you? Yes No

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How do you write a problem that can be solved using proportions. Write the proportion and solve the problem?

How Do You Solve a Word Problem Using a Proportion?

This tutorial provides a great real world application of math. You'll see how to use the scale from a blueprint of a house to help find the actual height of the house. This tutorial shows you how to use a proportion to solve!

• word problem
• translate words to algebra
• cross multiplication
• means extremes

Background Tutorials

Ratio definitions.

What's a Ratio?

Ratios are everywhere! The scale on a map or blueprint is a ratio. Ingredients sometimes need to be mixed using ratios such as the ratio of water to cement mix when making cement. Watch this tutorial to learn about ratios. Then think of some ratios you've encountered before!

Determining Proportionality

What's a Proportion?

The idea of proportions is that a ratio can be written in many ways and still be equal to the same value. That's why proportions are actually equations with equal ratios. This is a bit of a tricky definition, so make sure to watch the tutorial!

Evaluating Expressions

What is a Variable?

You can't do algebra without working with variables, but variables can be confusing. If you've ever wondered what variables are, then this tutorial is for you!

Further Exploration

Solving proportions.

How Do You Set Up a Proportion from a Word Problem?

Sometimes the hardest part of a word problem is figuring out how to turn the words into an equation you can solve. This tutorial let's you see the steps to take in order to turn a word problem involving a blueprint into a proportion. Take a look!

How Do You Solve a Proportion Using Cross Products?

Want to solve a percent proportion? Just use the means extremes property of proportions to cross multiply! Solve for the variable, and you have your answer! Learn how with this tutorial.

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Home / Blog / Math / Learn with Math / Real-life Problem-solving Using Proportion

Real-life Problem-solving Using Proportion

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What is Proportion?

Proportion is the relationship between two similar things. It is often used to describe how one thing relates to another in terms of size, quantity, or degree. For example, if you have a pizza that is 10 inches in diameter and you want to cut it into eight pieces, you would use proportion to figure out how big each piece should be. In this case, you would cut the pizza into eight pieces, each 1.25 inches wide. 1

Proportion can also be used to draw comparisons between things that are not similar in size or quantity. For example, if you want to know what percent of your salary you should save each month, you would use proportion to figure out the answer. In this case, you would compare your monthly salary to your savings goal. Proportion is a useful tool for comparing two things and understanding the relationship between them.

Application of Proportion in Solving Real-Life Problems

In many everyday situations, we need to solve proportionality problems. 

For example, imagine you want to bake a cake for 10 people. However, your recipe is for a cake for six people. How much of each ingredient do you need to use? This is a proportionality problem, which can be solved using a ratio. 

First, we convert the serving size into a fraction of the total recipe. 10 people need two-thirds of the recipe, while six people need half of the recipe. This means that the desired amount of each ingredient is two-thirds of the original amount. So, if you need one cup of sugar for the recipe, you would need to use a two-thirds cup of sugar. In this way, proportionality problems can be solved by finding equivalent ratios.

How to Use Proportion to Solve a Word Problem?

Proportion is a way of comparing two ratios to see if they are equal. If they are, then the two ratios are proportional. You can use this idea of proportionality to solve many types of word problems. 2 Let’s say you’re trying to find out how long it will take you to mow the lawn if it takes you 30 minutes to mow a quarter of an acre. To solve this problem, you would set up a proportion using the known ratio (30 minutes for one-fourth acre) and the unknown ratio ( x minutes for one acre). This would give you the equation 30/x=1/4. To solve for x , you would multiply both sides by 4, giving you 120=1x . This would mean that it would take you 120 minutes, or 2 hours, to mow 1 acre of land. As you can see, proportion can be a very useful tool for solving word problems. Just be sure to set up your equation carefully and solve it systematically.

How can Proportion be Used in Business?

Businesses use proportion in math all the time. For example, when a company is trying to increase sales, they might use proportions to figure out how many people need to be exposed to their product for them to make more sales. They might also use proportions when they’re trying to figure out how much raw material they need to create a certain number of finished products. In short, proportion is a very important tool that businesses use to make sure that they’re using resources efficiently and effectively.

How can Proportion be Used in the Medical Field? 3

In the medical field, it is often necessary to calculate proportions to ensure that patients receive the correct dosages of medication. For example, when a patient is prescribed a certain amount of medication, it is important to consider their weight to determine the correct dosage. This can be done by using a proportion: first, the weight of the patient is divided by the standard weight (such as 150 lbs.), and then this number is multiplied by the standard dosage. This ensures that patients receive the correct amount of medication, regardless of their weight. In addition, proportions can also be used to calculate how much of a certain medication needs to be mixed with another substance, such as water or saline solution. By using proportions, medical professionals can ensure that patients receive the correct dosages of medication, every time.

What is the Most Difficult Part About Working With Proportions?

Working with proportions can be challenging for several reasons. First, it can be difficult to visualize proportions when they are expressed as numbers. For instance, if you read that a recipe needs half a cup of sugar, you may not be able to figure out how much that is. Second, it can be tricky to adjust proportions when substituting ingredients. Imagine that a recipe requires half a cup of milk, but you only have one-fourth. How many other ingredients do you need to use to make up the difference? 

Finally, even when you are using the same ingredients in the same quantities, it can be difficult to achieve consistent results when working with proportions. This is because even slight variations in ingredient amounts can significantly change the final product. As a result, working with proportions requires careful planning and attention to detail to achieve the desired results.

We hope the blog helps you understand the use of math in your daily life. If you want to learn more about math and the different branches of math, you must consider reading BYJU’s FutureSchool Blog . 

• How To Cut A Pizza Into 6, 8, 10, or 12 Slices | Generation Ready . (2022, August 15). How To Cut A Pizza Into 6, 8, 10, or 12 Slices | Generation Ready. Retrieved November 15, 2022, from https://foodsguy.com/how-to-cut-pizza-slices/
• 7 Ways to Solve Proportions | Generation Ready . (2021, November 9). 7 Ways to Solve Proportions | Generation Ready. Retrieved November 15, 2022, from https://www.wikihow.com/Solve-Proportions
• Dose Calculation Ratio and Proportion Method – StatPearls – NCBI Bookshelf . (2022). Retrieved December 5, 2022, from https://www.ncbi.nlm.nih.gov/books/NBK499884/

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Proportions: Simple Exercises

Ratios Proportions Proportionality Solving Word Problems Similar Figures Sun's Rays / Parts

Solving proportions is simply a matter of stating the ratios as fractions, setting the two fractions equal to each other, cross-multiplying (as demonstrated below):

...and solving the resulting equation. The exercise set will probably start out by asking for the solutions to straightforward simple proportions, but they might use the "odds" notation, something like this:

Content Continues Below

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Solving Proportions

• Find the unknown value in the proportion: 2 :  x = 3 : 9 .

Okay; they've given me to ratios, in "odds" notation, and set them equal. This creates a proportion:

2 : x = 3 : 9

My first step will be to convert the colon-based odds-notation ratios to fractional form, so I get an equation with two fractions:

Then I'll solve the proportion. I could cross-multiply, and then divide back off, but I think I'll use the informal shortcut, like was illustrated earlier:

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Using this method, I always multiply across in the direction that has regular numbers on either end. In this case, I'll multiply the 2 and the 9 (going from the top-left to the bottom-right), and then I'll divide by the 3 (going from the top-right around to the bottom-left, where the variable is):

x = [(2)(9)]/3

x = 18/3 = 6

If I have to "show my work", I'll include my fractional equation with the arrows. My answer is:

• Find the unknown value in the proportion: (2 x  + 1) : 2 = ( x  + 2) : 5

Okay; this proportion has more variables than I've seen previously, and they're in expressions, rather than standing by themselves. So this is gonna be a cross-multiplying solution.

First, I convert the colon-based odds-notation ratios to fractional form:

Then I solve the proportion, starting by cross-multiplying:

5(2 x + 1) = 2( x + 2)

10 x + 5 = 2 x + 4

8 x = −1

x = −1 / 8

I wasn't expecting a fraction, but it's a perfectly valid answer (which I can check, if I want, by plugging it back into the original equation).

Once you've solved a few proportions, you'll likely then move into word problems where you'll first have to invent the proportion, extracting it from the word problem, before solving it.

• If twelve inches correspond to 30.48 centimeters, how many centimeters are there in thirty inches?

I will set up my ratios with "inches" on top (just because; there's no logic or particular reason for it), and will use the letter " c " to stand for the number of centimeters for which they've asked me. So here's my set-up:

Once I have my proportion, I can solve. In this case, I can use the shortcut method:

c = [(30)(30.48)]/12

Looking back at how I defined the variable, I see that c stands for "the number of centimeters". The question asked for "how many centimeters?", not "what value?", so I need the unit "centimeters" on my answer.

I could have used any letter I liked for my variable. I chose to use " c " because this helped me to remember what the variable was representing; namely, "centimeters". An x would only tell me that I'm looking for "some unknown value"; a c can remind me that I'm looking for " c entimeters".

Don't fall into the trap of thinking that you have to use x for everything. You can use whatever variable you find most helpful.

• A metal bar ten feet long weighs 128 pounds. What is the weight of a similar bar that is two feet four inches long?

First, I'll need to convert the "two feet four inches" into a feet-only measurement. Since one foot contains twelve inches, then four inches is four-twelfths, or one-third, of a foot. So the length, converted to feet only, is:

2 feet + 4 inches

I will set up my ratios with the length values on top (because I happened to pick that ordering, probably because the length info came before the weight info in the exercise). I'll use this set-up to make sure that I write out my proportion correctly, and then I'll solve for the required weight value. By the way, since I'm looking for a weight, I'm going to use w as my variable.

I can use the shortcut method for solving this proportion, so I'll multiply crossways in the one direction, and then hook back and divide going in the other direction:

w = [(7/3)(128)]/10

w = (7/3)(128/1)(1/10) = 448/15

Since this is a "real world" word problem, I should probably round or decimalize my exact fractional solution to get a practical "real world" sort of number. To be on the safe side, though, I'll give both the "exact" (fractional) form and also the rounded (more real-world) form:

• The tax on a property with an assessed value of $70,000 is $1,100 . What is the assessed value of a property in the same district, if the tax on that property is $1,400 ?

The logical basis for answering this question is the unstated assumption that all properties in the same district are assessed at the same percentage rate.

If this question were being asked in the homework for the section on " percent of " word problems, then I would have the tax rate as a percentage from the info they gave me for the first property; and then I would have back-solved, using the rate I'd just found, for the value of the second property. However, since this question is being asked in the section on proportions, I'll solve using a proportion.

They've given me two categories of things: assessed values of properties, and the amounts of taxes paid. My ratios will then use these two categories. I will set up my ratios with the assessed valuation on top (because that's what I read first in the exercise), and I will use " v " to stand for the value that I need to find.

I'll use the shortcut method for solving, multiplying the 70,000 and the 1,400 in one direction, and then dividing by the 1,100 going in the other direction:

v = [(70,000)(1,400)]/1,100

v = 98,000,000/1,100

v = 89,090.9090909...

Since the solution is a dollars-and-cents value, I must round the final answer to two decimal places; the "exact" form (whether repeating decimal or fraction) wouldn't make sense in this context. So my answer is:

The assessed value is $89,090.91 .

• One piece of pipe 21 meters long is to be cut into two pieces, with the lengths of the pieces being in a 2 : 5 ratio. What are the lengths of the pieces?

I'll label the length of the short piece as " s ". Then the long piece, being what was left of the original piece after I cut off s meters, must have a length of 21 −  s . Then my ratio, in fractional (rather than in odds) format, is:

short/long: 2/5 = s /(21 − s )

Because there are two parts of this proportion that contain variables, I can't use the shortcut to solve. Instead, I'll cross-multiply, and proceed from there.

2(21 − s ) = 5 s

42 − 2 s = 5 s

Referring back to my set-up for my equation, I see that I defined " s " to stand for the length of the s horter piece, with the unit of length being meters. Then the length of the longer piece is given by:

21 − s = 21 − 6 = 15

Now that I've found both required values, I can give my answer, complete with the correct units:

6 meters and 15 meters

In the last exercise above, if I had not defined what I was using " s " to stand for, I could easily have overlooked the fact that " s  = 6 " was not the answer the exercise was wanting. Try always to clearly define and label your variables. Also, be sure to go back and re-check the word problem for what it actually wants. This exercise did not ask me to find "the value of a variable" or "the length of the shorter piece". By re-checking the original exercise, I was able to provide an appropriate response, being the lengths of each of the two pieces, including the correct units of meters.

URL: https://www.purplemath.com/modules/ratio4.htm

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