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# Proportions

Proportions are comparisons of two equal ratios. Ratios are comparisons of two numbers and can be written as a fraction (ex. $\frac{5}{7}$ ) or as a colon (ex. $5:7$ ). In order to solve a proportion, ratios generally take the form of fractions. Here is an example of a proportion:

$\frac{a}{b}=\frac{x}{y}$

The proportion above reads a is to b as x is to y. It's good to note that, in this equation, b and x are called the means, and a and y are called the extremes.

## How to solve proportions

When solving a proportion, you'll take the cross product (also known as using cross multiplication or the means-extremes property) and solve the resulting equation. As an example, let's solve the following proportion for x:

$\frac{x}{4}=\frac{6}{8}$

Take the cross product then divide both sides by 8:

$8x=6\cdot 4$

$8x=24$

Answer: $x=3$

## Solving proportions as word problems

Now, suppose that you've been given a word problem to solve. You might receive a question like the following:

If you have 45 T-shirts in 5 suitcases, how many suitcases will you need for 90 T-shirts?

First, you will write out a proportion. In this case, the numbers of t-shirts are numerators and the numbers of suitcases are the denominators:

$\frac{45}{5}=\frac{90}{x}$

Now, take the cross product and divide both sides by 45:

$45x=90\cdot 5$

$45x=450$

Answer: $x=10$

## Practice questions on solving proportions

a. Solve the following proportion: $\frac{y}{12}=\frac{3}{4}$

$4y=12\cdot 3$

$4y=36$

$y=9$

b. Solve the following proportion: $\frac{25}{x}=\frac{5}{2}$

$5x=25\cdot 2$

$5x=50$

$x=10$

c. Solve the following proportion: $\frac{360}{40}=\frac{a}{9}$

$40a=360\cdot 9$

$40a=3240$

$a=81$

d. Solve the following proportion: $\frac{4}{11}=\frac{8}{b}$

$4b=11\cdot 8$

$4b=88$

$b=22$

e. Solve the following word problem: If you can fit 300 pieces of candy in 15 containers, how many containers will you need for 960 pieces of candy?

$\frac{300}{15}=\frac{960}{x}$

$300x=960\cdot 15$

$300x=14,400$

$x=48$

f. Solve the following word problem: If a family taking a road trip needs 8 vehicles for 32 people, how many vehicles will be needed for 60 people?

$\frac{32}{8}=\frac{60}{x}$

$32x=60\cdot 8$

$32x=480$

$x=15$

## Get assistance learning how to solve proportions

Understanding proportions can sometimes be challenging, particularly if your student is being asked to solve word problems. Fortunately, there are tutors ready to answer any questions your student might have. Find out how working with a tutor can make a major difference in your student's math journey. Contact our Educational Directors at Varsity Tutors to learn more today.

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