# Problem Solving Percentage The phrase of what number represents the whole and is the unknown quantity.

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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.

Identify: 18 is the part and will replace IS in our proportion.

75% means that 75 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.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.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.Substitute: Now we can substitute these values into our proportion.becomes Solve: Cross multiply and we get: 40x = 18(100) or 40x = 1800 Divide both sides by 40 to solve for x and we get: x = 45 Solution: 18 is 40% of 45 Problem 3: What is 20% of 45?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?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.