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Lesson 29 WHAT PERCENT?The Method of ProportionsIn Lesson 13, we saw how to use a calculator to find what percent one number is of another. In this Lesson, we will see which problems should not require a calculator. They are those problems in which the Base -- the number the follows "of" -- is either a divisor of 100 (20, 25, 50), a multiple of it (200, 300, 400), or in which the Amount is a simple part of the Base (half, a third, a quarter). With a little practice, the student should be able to do such problems easily and mentally. In this Lesson, we will answer the following:
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5 is 5% of 100. 12 is 12% of 100. 250 is 250% of 100. For, percent is how many for each 100, which is to say, percents are a number of hundredths. 5 is 5 hundredths of 100. That is the ratio of 5 to 100. When the percent is less than or equal to 100%, then we can say "out of" 100. 5% is 5 out of 100. 12% is 12 out of 100. But 250% cannot mean 250 out of 100 -- that makes no sense. It means 250 for each 100, which is two and a half times (Lesson 15). Example 1. $42.10 is what percent of $42.10? Answer. 100% The method of proportions Example 2. 24 out of 100 is 24%. But what percent is 24 out of 200? Answer. Percent is how many out of 100. But if there are 24 out of 200, then out of each 100 there are half as many: 12. 24 is 12% of 200. If we consider this as a proportion problem, then we must find the missing term: 24 out of 200 is equal to ____ out of 100? When we say "24 out of 200," the Base -- the number that follows "of" -- is 200. But with percent, the Base must be 100. What must we do then to 200 to make it 100? We must take half of it, or equivalently, divide it by 2. Therefore we must also divide 24 by 2. 24 out of 200 is equal to 12 out of 100. But 12 out of 100 is 12%. Therefore, 24 out of 200 is also 12%.
Example 3. 8 out of 25 is what percent? That is, 8 is what percent Those questions mean the same. In each one, the base Solution. In this problem, the Base is 25. But to find the percent, we must make the Base 100. Therefore must multiply both terms by 4. 8 out of 25 is equal to 32 out of 100. That means: 8 is 32% of 25. The student should be able to solve this problem simply by looking at "8 out of 25," and realizing that 25 must be changed to 100. An all too common method these days is to make this an algebra problem.
The student is taught to cross-multiply and solve for x. This is a method for someone who do not understand percent. Whoever does understand percent will either do such a problem mentally, or, if that is too difficult, with a calculator: Percent = Amount ÷ Base. |
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The Base will always follow "of." Example 4. In a class of 50 students, 11 studied French. What percent studied French? Solution. The student should realize that this means, 11 out of 50 studied French. To make the base 100, we must multiply 50 by 2. Therefore we must also multiply 11 by 2: 11 out of 50 is equal to 22 out of 100. 22% studied French. Example 5. In a class of 200 students, 11 studied French. What percent studied French? Solution. In this case, to make 200 into 100, we must divide by 2, or take half. Therefore we must also take half of 11, which is 5½. 5½% studied French. We see that to solve a proportion: We must divide both terms by the same number, Example 6. 9 out of 20 students were able to stop smoking. What percent were able to stop smoking? Answer. 45%. For, to make 20 into 100, we must multiply by 5. Therefore we must multiply 9 by 5, also: 9 out of 20 is equal to 45 out of 100. Example 7. What percent of 400 is 33? Solution. 400 is the Base; it follows "of." To make it 100, we must divide by 4. Therefore we must divide 33 by 4, also.
"4 goes into 33 eight (8) times (32) with 1 left over." 33 is 8¼% of 400. Example 8. 1000 people voted in the recent election, and 763 voted for Jones. What percent voted for Jones? Solution. 763 out of 1000 voted for Jones. To make the Base 100, we must divide by 10. To divide 763 by 10, simply separate one decimal digit (Lesson 3, Question 4): 763 ÷ 10 = 76.3 76.3% voted for Jones. Example 9. Percents are ratios. In a class of 45 students, there were 9 A's. What percent got A? Answer. 9 out of 45 got A. It is not obvious, however, how to make 45 into 100. Therefore we will look directly at the ratio of 9 to 45. 9 is the fifth part of 45. And since 20% is the fifth part of 100%, 20% of the students got A. To master the subject of percent, the student must master Problem 1 of Lesson 28. Example 10. 7 is what percent of 28? Solution. Again, it is not obvious how to make 28 into 100. But 7 is the fourth part of 28. And the fourth part is 25%. 7 is 25% of 28. In summary, then, to find what percent one number is of another: Multiply or divide both terms so that the Base becomes 100. If that is not possible: Look directly at the ratio of the Amount to the Base. Example 11. 21 is what percent of 75? Solution. What must we do to 75 to make it 100? We have to add a third more, which is 25. (75 + 25 = 100.) Therefore we have to add a third more to 21 -- we have to add 7. 21 + 7 = 28. Therefore, 21 is 28% of 75. At this point, please "turn" the page and do some Problems. or Continue on to the next Section. Introduction | Home | Table of Contents Please make a donation to keep TheMathPage online. Copyright © 2001-2009 Lawrence Spector Questions or comments? E-mail: themathpage@nyc.rr.com |
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100% is the whole thing.
