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Solve Percent Problems Using Proportions

Introduction

The word percent means “out of 100” and a every percent is simply a ratio whose denominator is 100.  Instead of writing the (percent) ratio as a fraction over 100, it is typically written with the symbol %.

 

Consider the problem     “what number is 35% of 80”

 

A previous lesson showed how to solve this problem by translating all the words into math terms.  However, many students find it easier and faster to solve this type of problem using a proportion.  In this lesson, you’ll use proportions to solve percent problems.

 

Solve__problems_w_proportions_vis_1

 

The two solutions above are actually very similar.  The main difference is where to include the number 100 in the calculation.  Although your teacher may favor one method or the other, both are acceptable ways to complete percent problems.

Lesson

When solving percent problems using a proportion, just how does it work?  The answer is simple… it is a matter of determining how the words in the initial equation relate to each other.  The proportion you can use to start the process is:

Solve__problems_w_proportions_vis_2

So every proportion will contain the number 100.  Generally two of the remaining numbers are given and the final one is not given.  Example 1 shows how to set up these type of problems.

 

Example 1:  Set up each problem correctly using a proportion.

Solve__problems_w_proportions_vis_3

Solution: 

Solve__problems_w_proportions_vis_4 

 

Using colors (or parenthesis) to determine where to place the numbers in the proportion is an excellent way to start.  Once the numbers are placed, simply cross multiply and find the missing value.

 

Example 2:  Solve each percent problem using a proportion.

Solve__problems_w_proportions_vis_5

Solution: 

Solve__problems_w_proportions_vis_6

 

Solve__problems_w_proportions_vis_7Example 2 shows that once each proportion is constructed, the cross multiplication and simplification can be done rather quickly.  Note that once you’ve got your answer it is a good habit to take a few seconds and see that your answer makes sense. 

  • In a, the answer will be a fraction of the original 75.  18 seems reasonable.
  • In b, 240 is actually more than 100% of the final answer, so we would expect that the final answer to be slightly smaller than 240.  200 seems reasonable.
  • In c, 12 is a little less than half of 30, so our answer should be a little less than 50%.  Our final answer of 40% is reasonable.

Review

Use a proportion to find the answer to each percent problem.

1)  What percent of 24 is 15?

2)  102 is what percent of 120?

3)  15 is 40% of what number?

4)  What number is 64% of 300?

 

 

Review: Find each the answer to each percent problem by translating.

5)  21 is 70% of what number?

6)  0.5% of 5,000 is equal to what number?

 

 

 

 

Scroll down for answers:

 

 

 

 

 

 

 

 

Answers:

#1-4 done using a proportion.

Solve__problems_w_proportions_vis_8

1)  Solve__problems_w_proportions_vis_9

 

2)  Solve__problems_w_proportions_vis_10 

 

3)  Solve__problems_w_proportions_vis_11 

 

4)  Solve__problems_w_proportions_vis_12

 

#5-6 done by translating the words.  The chart below contains all the potential translations for these problems.

Solve__problems_w_proportions_vis_13

 

5)  Solve__problems_w_proportions_vis_14 

 

6)  Solve__problems_w_proportions_vis_15

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