# Solving Proportion Problems

becomes Solve: Cross multiply and we get: 100p = 52(25) or 100p = 1300 Divide both sides by 100 to solve for p and we get: p = 13 Solution: 13 is 25% of 52 Note that we could restate this problem as, "Find 25% of 52", and get the same answer.However, in the interest of consistency, we will use proportions to solve percent problems throughout this lesson.

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Note that in all three percent statements, the whole always follows the word "of" and the part always precedes the word "is".

This is not surprising since our original statement is, "One number is some percent of another number." Thus, we can revise our proportion as follows: becomes Let's solve some more percent problems using proportions. Identify: 25% means that 25 will replace PERCENT in our proportion.

But what would you do if you given this problem: 8 is what percent of 20?

In this problem, the percent is the unknown quantity! Looking at this problem, it is clear that 8 is the part and 20 is the whole.

Identify: The phrase what is means represents the part and is the unknown quantity.

20% means that 20 will replace percent in our proportion. Substitute: Now we can substitute these values into our proportion.In Problem 1 we were asked 8 is what percent of 20?and we found the solution by substituting into a proportion.PERCENT is the unknown quantity in our proportion, to be represented by n.Substitute: becomes Solve: Cross multiply and we get: 56n = 14(100), or 56n = 1400 Divide both sides by 56 and we get: n = 25 Solution: 25% of 56 is 14 Problem 6: 18 is 75% of what number?In Problems 5 through 7, we will use n to represent the unknown quantity. Identify: 56 is the whole and will replace OF in our proportion.14 is the part and will replace IS in our proportion.Thus, we can rewrite the statement above: The statement: "The part is some percent of the whole.", becomes the equation: the part = some percent x the whole Since a percent is a ratio whose second term is 100, we can use this fact to rewrite the equation above as follows: the part = some percent x the whole becomes: the part = x the whole Dividing both sides by "the whole" we get the following proportion: Since percent statements always involve three numbers, given any two of these numbers, we can find the third using the proportion above. Problem 1: If 8 out of 20 students in a class are boys, what percent of the class is made up of boys?Analysis: In this problem, you are being asked 8 is what percent of 20?But how would we solve this problem: 18 is 40% of what number? Identify: The phrase 18 is means that 18 is the part.and how would we solve this problem: What is 20% of 45? 40% means that 40 will replace percent in our proportion.

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