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Lesson 29 PERCENT OF A NUMBER
Every statement of percent involves three numbers. For example, 8 is 50% of 16. 8 is called the Amount. 50% is the Percent. 16 is called the Base. The Base always follows "of." What you see above is the standard form of any statement of percent. The Amount is some Percent of the Base. In a percent problem, we are given two of those numbers and we are asked to find the third. We have already seen how to solve any percent problem with a calculator. The same procedures apply in a written calculation, where we would typically change the percent to a decimal. In Lesson 4 we saw how to take 1% and 10% of a number simply by placing the decimal point. Those should be basic skills. What is more, from 1% we can calculate 2%, 3%, and so on. While from 10% we can easily calculate 20%, 30% and any multiple of 10%. In Lesson 28 we saw how to solve percent problems by understanding that a percent is a ratio. Here, we will see how to find the Amount with a minimum amount of writing. And in Section 3 we will see how to find the Base. In this Lesson, we will answer the following: We begin with the elementary question: |
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32% of 100 is 32. 87.9% of 100 is 87.9. 416% of 100 is 416. For, as we saw in Lesson 4, percent is an abbreviation for the Latin per centum, which means for each 100. (Per means for each.) A percent is a number of hundredths. Example 1. A store paid $100 for a jacket. It then raised the selling price by 28%. But a week later it reduced that price by 10%. What was the final selling price? Solution. 28% of $100 is $28. So the selling price became $128. 10% of that is $12.80. (Lesson 4.) To subtract $12.80 from $128, round it off to $13: $128 − $13 = $115, plus $.20 is $13.20. That was the final selling price. Example 2. How much is 11% of $420? Solution 1. As a written method, 11% is 11 hundredths, which we can represent as the decimal .11. Therefore, 11% of $420 = .11 × 420. (Lesson 27.) Now, 11 × 420 = 4200 + 420 = 4620. (Lesson 9) Therefore, on separating two decimal digits (Lesson 10), 11% of $420 = $46.20. (Compare Lesson 10, Question 4.) Solution 2. More simply, since 11% = 10% + 1%, then
In general, if we must do a problem in writing, then we must express the percent either as a decimal or a fraction. We saw how to express a percent as a decimal in Lesson 4. As for expressing a percent as a fraction: |
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Examples 3.
In addition, the student should know
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We saw this in Lesson 10 and in Lesson 14. Example 4. How much is 75% of 108? Answer. We could write 75% as the decimal .75, and then multiply .75 × 108. However, 75% is three quarters. Therefore we could calculate Three quarters of 108. It is not difficult to do that if we decompose 108 into 100 + 8.
75% of 108 is 81. Example 5. How much is 30% of $48? Answer. The student should realize that 30% is simply three times 10%, and so will always involve multiplication by 3. Now, 10% of $48 is $4.80. (Lesson 4.) Therefore, 30% is
See Lesson 9, The distributive property of multiplication, Examples 5 and 6. Example 6. How much is 80% of $124? Answer. 10% is $12.40. Therefore, 80% is
Example 7. How much is 80% of $45? Answer. In this case, since 80% means four fifths, and 45 has an exact fifth part, we can reason as follows: One fifth of 45 is 9. Therefore four fifths are four 9's -- 36. 80% of $45 is $36. Example 8. How much is 250% of 32? Answer. 250% = 2½.
Example 9. How much is 37½% of $40?
three eighths are 3 × $5 = $15. Example 10. How much is 18.9% of $314? Answer. Use your calculator
can decompose it into multiples of 3 as follows: 720 = 600 + 120. A third of 600 is 200. A third of 120 is 40. Therefore a third of 720 is 240.
two thirds are 2 × 240 = 480.
Solution. To find two thirds, we must first find one third, and then multiply by 2. Press
See
This is approximately $50.85. The standard textbook method for finding a percent of a number, has been to change the percent to a decimal, and multiply. And so to find 24% of $412, we are taught to change 24% to the decimal .24 (Lesson 4), and multiply times 412. But is anyone with a calculator going to do that these days? And aren't there more important things to learn about percent? Like how to take 25% of $412 without writing anything! Take half of 50%, which is $206. 25% is $103. 24% of $412 will then be 25% − 1%:
Example 13. $36 is 4% of how much? Answer. Here, the Base is missing, the number that follows of. This is Example 2 in Section 3. At this point, please "turn" the page and do some Problems. or Continue on to the Section 2: Fractional percent Introduction | Home | Table of Contents Please make a donation to keep TheMathPage online. Copyright © 2012 Lawrence Spector Questions or comments? E-mail: themathpage@nyc.rr.com |
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Press 314 × 18.9%. See 59.346, which is approximately $59.35.