6.2. PERCENT FORMULA MODULE 6. PERCENT

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1 6.2 Percent Formula In this lesson, we will learn 3 types of percent problems, and use 3 methods to solve each type of problems. You can choose your favorite method. It would be great if you can use all methods, so you can have a better understanding of percent problems. We will handle the following 3 types of percent problems: Type I: What is 40% of? Type II: is what percent of? Type III: is 40% of what? The Percent Formula is one popular method to solve the above problems Percent Formula Many textbooks give the Percent Formula as: Part = Percent Whole or something similar. In this lesson, instead of memorizing a formula, I expect you to write out the following example on scratch paper. Recall that the English word "of" can be translated into the multiplication symbol, in situations like "3 is 1 2 of 6." You can verify this by doing: = = 6 2 = 3 It s common sense that 1 2 = 50%, so we have "3 is 50% of 6." Here is the percent formula: Once we change 50% to 0.5, we can verify = 3 with calculator. When you need to use the Percent Formula, instead of writing down a formula, write down the example above. In later lessons, when I refer to the "Percent Formula", I mean: There are three types of percent problems Type I Percent Problem Example What is 40% of? Solution Method 1: Use Percent Formula Assume x is 40% of. We will write down the "Percent Formula" and the problem right next to each other: x = 40% Next, we can solve for x in the equation. In this type of problems, x happens to be alone on one side of the equal sign, so all we need to do is to do the calculation on the other side of the equal sign. Remember that 40% = 0.4. We have: x = 40% x = 0.4 x = 102

2 Method 2: Use Proportion Assume x is 40% of. We can rephrase this sentence as: x out of is like 40 out of. Here is the key: The number following the word "of" corresponds to. Now we can set up and solve a proportion: x = 40 x = 40 x = 00 x = 00 x = To find "50% of 6", we do: 50% 6 = = 3 Similarly, to find "40% of ", we do: 40% = 0.4 = Type II Percent Problem Example is what percent of? Solution Method 1: Use Percent Formula Assume is x (as a percent) of. We will write down the "Percent Formula" and the problem right next to each other: = x (as a percent) = x = x = x 0.4 = x 40% = x Method 2: Use Proportion Assume is x% of. We can rephrase this sentence as: out of is like x out of. Here is the key: The number following the word "of" 103

3 corresponds to. Now we can set up and solve a proportion: = x x = x = 00 x = 00 x = 40 To find "3 is what percent of 6", we do: 3 6 = 0.5 = 50% Similarly, to find " is what percent of ", we do: = 0.4 = 40% Type III Percent Problem Example is 40% of what? Solution Method 1: Use Percent Formula Assume is 40% of x. We will write down the "Percent Formula" and the problem right next to each other: = 40% x = 40% x = 0.4x 0.4 = 0.4x 0.4 = x Method 2: Use Proportion Assume is 40% of x. We can rephrase this sentence as: out of x is like 40 out of. Here is the key: The number following the word "of" corresponds to. Now we can set up and solve a proportion: x = 40 40x = 40x = 00 40x 40 = x = 104

4 To find "3 is 50% of what", we do: = 6 Similarly, to find " is 40% of what", we do: 0.4 = Rounding Sometimes we need to round numbers, like in the next example. Example is what percent of 91? Round your answer to two decimal places. Solution This is a Type II percent problem. We will use the Percent Formula to solve this problem. Assume 45 is x (as a percent) of 91. We will write down the "Percent Formula" and the problem right next to each other: 45 = x (as a percent) = x = 91x = 91x x 4.59% x Conclusion: 45 is approximately 4.59% of More than % The rules are the same when we deal with percents bigger than %. Let s look at a few examples. To save space, we will solve these problems only with the Percent Formula. Example What is 140% of? Solution We know the solution must be bigger than, because the percent is bigger than %. Assume x is 140% of. We will write down the "Percent Formula" and the problem right next to each other: x = 140% Next, we can solve for x in the equation. In this type of problems, x happens to be alone on one side of the equal sign, so all we need to do is to do the calculation on the other side of the equal sign. We have: x = 140% x = 1.4 x = 2 105

5 Conclusion: 2 is 140% of. Example is what percent of? Solution We know the solution must be bigger than %, because 2 is bigger than. Assume 2 is x (as a percent) of. We will write down the "Percent Formula" and the problem right next to each other: Conclusion: 2 is 140% of. Example is 140% of what? 2 = x (as a percent) 2 = x 2 = x 2 = x 1.4 = x 140% = x Solution We know the solution must be smaller than 2, because the percent is bigger than %. Assume 2 is 140% of x. We will write down the "Percent Formula" and the problem right next to each other: 2 = 140% x Conclusion: 2 is 140% of. 2 = 140% x 2 = 1.4x = 1.4x 1.4 = x 106

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