# Solving Proportion Problems

The phrase what percent tells us that percent is the unknown quantity.This unknown quantity will be represented by x in our proportion. Substitute: Now we can substitute these values into 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?

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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.

In Problem 1 we let x represent the unknown quantity "what percent"; in Problem 2 we let x represent the unknown quantity "of what number"; and in Problem 3 we let x represent the unknown quantity "What is." Thus, we solved three different percent problems, where in each problem, two numbers were given and we were asked to find the third.

We did this by letting a variable represent the unknown quantity and then substituting the given values into a proportion to solve for the unknown quantity.

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.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 previous lessons, you were shown how to convert a decimal to a percent and a percent to a decimal.Thus, if you were asked to Find 15% of 120, you would multiply .15 by 120, to get an answer of 18.In Problem 1 we were asked 8 is what percent of 20?and we found the solution by substituting into a proportion.becomes Solve: Cross multiply and we get: 100x = 45(20) or 100x = 900 Divide both sides by 100 to solve for x and we get: x = 9 Solution: 9 is 20% of 45 In Problems 1, 2 and 3 we are given two numbers and asked to find the third by using a proportion.However, the unknown quantity was different for each problem. Red is used for the unknown quantity in each problem.The phrase of what number represents the whole and is the unknown quantity.We will let variable x represent this unknown quantity in our proportion.

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