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Lesson 29 PERCENT OF A NUMBER
Every statement of percent involves three numbers. For example, 8 is called the Amount. 50% is the Percent. 16 is called the Base. The Base always follows "of." In any percent problem, we are given two of those numbers, and we are asked to find the third. We have already seen how to solve that problem with a calculator. The same procedures apply in a written calculation, where we would typically change the percent to a decimal. After learning how to divide by a power of 10, we saw how to take 1% and 10% of a number simply by placing the decimal point. They 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. In this Lesson we will see how to find the Amount and the Base with a minimum amount of writing. 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.) 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. In general, since percent is how many for each 100, then we may think
of 100 as being divided into one hundred equal parts, that is, into hundredths. A percent is a number of hundredths. 32 of those 100 little squares have been shaded, which is 32%, or 32 hundredths, of them. Example 2. How much is 11% of $420? Solution. 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.) More simply, however, since 11% = 10% + 1%, then
In general, if we are required to do a problem in writing, then we must express the percent either as a decimal or a fraction, and then multiply that with the Base. We saw how to express a percent as a decimal in Lesson 4, Question 8. 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 14. Example 4. How much is 75% of 104? Answer. We could write 75% as the decimal .75. Therefore we could multiply .75 × 104. However, 75% is three quarters. Therefore we could calculate
"Three quarters of 104." First, we must take one quarter by dividing by 4. It is easy to do that by decomposing 104 into 100 + 4. One quarter of (100 + 4) = 25 + 1 = 26. Therefore, Three quarters = 3 × 26 = 60 + 18 = 78. (Lesson 9) 75% of 104 is 78. 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, which is 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?
eighths are 3 × $5 = $15. Example 10. How much is 18.9% of $314? Answer. Use your calculator
of 720, we 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. Thus, 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%! It comes out to $103. 24% of $412 will then be 25% − 1%:
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.