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Lesson 29 WHAT PERCENT?The Method of ProportionsIn this Lesson we hope to give the student an understanding, rather than just a mechanical method to get an answer. In Lesson 13, we saw how to use a calculator to find what percent one number is of another. And in Lesson 17 we first introduced the method of proportions. In this Lesson we will not only see problems that should not require a calculator, we will see problems that someone who understands percent can do mentally. They are those in which the Base — the number that follows "of" — is either a divisor of 100 (20, 25, 50), a multiple of it (200, 300, 400), or in which we can easily name the ratio of the Amount to the Base: a third, a quarter, three quarters. 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, a percent is a number of hundredths. 5% means 5 hundredths. 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. As a proportion problem, 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. What percent is 24 out of 50? Answer. Again, percent is how many out of 100. But if there are 24 out of, or for each, 50 --
-- then out of 100, there are 2 × 24: 48. 24 is 48% of 50. Proportionally, 24 for each 50 is equal to 48 for each 100. Now without even having to write that, the student should see that to go from 50 to 100, we must multiply by 2. Therefore we must multiply 24 by 2, also. 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. That is a method for someone who does 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 25 students, 11 studied French. What percent studied French? Solution. The student should understand that this means, 11 out of 25 studied French. To make the base 100, we must multiply 25 by 4. Therefore we must also multiply 11 by 4: 44% studied French. 11 is to 25 as 44 is to 100. While it is possible to do this problem by writing the
the student should not have to write that. On seeing, or even hearing, the phrase "11 out of 25," the student should see that we must multiply both 11 and 25 by 4. Seeing that is skill in arithmetic. 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? That is, 9 is what percent of 20? Those questions mean the same. In each one, the base 20 follows "of." 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. 7 out of 12 people voted Yes. Approximately what percent voted Yes? Solution. How many 12's are there in 100? 8 × 12 = 98. And since 8 × 7 = 56, approximately 56% voted Yes. (For the exact calculation, see Section 2, Example 2.) The ratio of the Amount to the Base Example 10. 8 is what percent of 40? Answer. It is not obvious how to make 40 into 100. Therefore we will look directly at the ratio of 8 to 40. 8 is the fifth part of 40. And since 20% is the fifth part of 100%, 8 is 20% of 40. 8 is to 40 as 20 is to 100. To master the subject of percent, the student must master Problem 1 of Lesson 28. Example 11. What percent of 8 is 40? Answer. Here, 8 is the Base -- it follows "of." We might see that better if we express the question in standard form: 40 is what percent of 8? Now, 40 is five times 8. And as a percent, five times is 500%. 40 is 500% of 8. Example 12. In a class of 28 students, 7 got A. What percent of the class got A? Answer. The question is: 7 is what percent of 28? Again, it is not obvious how to make 28 into 100. But 7 is the fourth part, or a quarter, of 28. And as a percent, a quarter of a number is 25%. 7 is 25% of 28. Example 13. 12 is what percent of 18? Answer. 12 has what ratio to 18? They have a common divisor 6. "6 goes into 12 two times and into 18 three times." 12 is to 18 as 2 is to 3 -- 12 is two thirds of 18. Each number says its name.
Example 14. What percent of 35 is 14? Answer. In standard form, 14 is what percent of 35? They have a common divisor 7. Therefore, on dividing each term by 7: 14 is to 35 as 2 is to 5. 2 is two fifths of 5. As a percent, two fifths of 100% is 40%. (One fifth is 20%.) Therefore, 14 is 40% of 35. Example 15. 20 is what percent of 8? Answer. Their common divisor is 4. Therefore they have the same ratio that 5 has to 2, which is two and a half times: 20 is 250% of 8. Example 16. Clearing of decimals. .8 is what percent of 4? Answer. To solve any percent problem, the numbers must be whole numbers .8 is to 4 as 8 is to 40. 8 is the fifth part of 40. That tells us, .8 is 20% of 4. 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 17. What percent of 75 is 21? Solution. In standard form, 21 is what percent of 75? They have a common divisor 3. And on dividing each one by 3: 21 is to 75 as 7 is to 25. We can now change 25 to 100 by multiplying by 4. Therefore, on also multiplying 7 by 4 we find: 21 is 28% of 75. Example 18. 742 people were surveyed, and 213 responded No. What percent responded No? Solution. 213 is what percent of 742? Use your calculator 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 © 2001-2010 Lawrence Spector Questions or comments? E-mail: themathpage@nyc.rr.com |
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100% is the whole thing.
