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Lesson 28

# PERCENT OF A NUMBER

In this Lesson we will learn how to find the Amount and the Base. We have already seen how to find them with a calculator (Lesson 13). And in Lesson 27 we saw how to find the most frequent examples. Here, we present examples that require a minimum amount of writing.

In this Lesson, we will answer the following:

1. How can we represent a percent as a fraction, or a whole number, or a mixed number?
2. How can we find the Amount when we know the Base and the Percent?
3. What does ½% mean?

Section 2

4. How can we find the Base when we know the Amount and the Percent?

 1. How can we represent a percent as a fraction, or a whole number, or a mixed number? 29% = ? We can represent a percent as a fraction whose denominator is 100.

Examples .

 29% = 29  100 . 60% = 60  100 = 6 10 = 35 .   (Lesson 21, Question 3) 200% = 200100 = 2.
 250% = 200% + 50% = 2½ . 225% = 2¼.
 233 13 % =  2 13 .

In addition, the student should know

 12.5% = 1 8 .  ( 1 8 is half of 1 4 , which is 25%. ) Lesson 23
 37.5% = 3 8 ,    62.5% = 5 8 ,    87.5% = 7 8 .

A percent is a ratio (Lesson 27).  Nevertheless, for purposes of calculation we can represent a percent as a fraction (or a decimal), because a fraction in turn indicates the ratio of the numerator to the denominator. The meaning of multiplying by a fraction depends upon that ratio. (Lesson 26.)

 2. How can we find the Amount when we know the Base and the Percent? Amount = Base × Percent

We saw this in Lesson 13.

Example 1.   How much is 75% of 104?

Answer.  We saw in Lesson 3 that we could write 75% as the decimal .75.  Therefore we could multiply .75 × 104.

However, 75% is three quarters.  While we could write

 34 × 104  (Lesson 26)

and calculate that, we can easily decompose 104 into 100 + 4 (Lesson 15, Question 3) -- and so it is easy take three quarters of it.

One quarter of (100 + 4) = 25 + 1 = 26.

Therefore,

Three quarters = 3 × 26 = 60 + 18 = 78.  (Lesson 8)

75% of 104 is 78.

Example 2.   How much is 30% of \$48?

Answer.  The student should anticipate that finding 30% of a number will involve multiplication by 3 -- because 10% is \$4.80.  (Lesson 3)  20% will be 2 times that.  30% will be 3 times.  And so on.

Now, 3 × 48 = 120 + 24 = 144.

Where does the decimal point go?

Since 10% is \$4.80, then 30% -- 3 times that -- is \$14.40.

Example 3.   How much is 80% of \$124?

Answer.   8 × 124 = 800 + 160 + 32 = 992.

Where does the decimal point go?

Since 10% is \$12.40, then 80% is 8 times that:  \$99.20.

Example 4.   How much is 80% of \$45?

Answer.   In this case, since 45 has an exact fifth part, reason that 80% is four fifths.

One fifth of 45 is 9.  Therefore four fifths are four 9's, which is 36.

80% of \$45 is \$36.

Example 5.   How much is 250% of 32?

Answer.   250% = 2½.

 2½ × 32 =  2 × 32  +  ½ × 32  -- "Two times 32 + Half of 32"
 2½ × 32 =  64 + 16
 2½ × 32 =  80.

Example 6.   How much is 37½% of \$40?

 Answer.  37½% = 38 (Lesson 23).  One eighth of \$40 is \$5.  Therefore, three

eighths are 3 × \$5 = \$15.

Example 7.   How much is 18.9% of \$314?

Answer.   Use your calculator! Lesson 13.

 Example 8.  Thirds.  How much is 33 13 % of 720?  How much is 66 23 %?
 Answer.   33 13 % means a third.  (Lesson 27, Question 1.)  To take a third

of 720, we can decompose it as follows:

720 = 600 + 120.

A third of 600 is 200.

A third of 120 is 40.

Therefore a third of 720 is 240.

 As for 66 23 %, it means two thirds.  One third of 720 is 240.  Therefore,

two thirds are 2 × 240 = 480.

 Example 9.  Calculator problem.   How much is 66 23 % of \$76.27?

Solution.  To find two thirds, we must first find one third, and then multiply by 2.  Press

 7 6 . 2 7 ÷ 3 × 2 =

See

 50.8467

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 (see Lesson 3, Question 10), 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 much is 25% of \$412 -- without writing anything!  Take half of half.

Fractional percent

 2. What does ½% mean? Half of 1%.  Half of the hundredth part.

Example 10.   Distinguish the following:

a)  Half of \$600        b)  ½% of \$600

Answers.

a)  Half of \$600 is \$300.

b)  ½% of \$600  means  Half of 1% of \$600.

1% of \$600 is \$6. 00.

(Lesson 3, Question 8.) Therefore,

½% of \$600 is \$3. 00.

Example 11.   How much is ½% of \$824?

Answer.  \$4.12.

Again, ½% means Half of 1%.

1% of \$824 is \$8.24.

½% of \$824 is \$4.12.

Notice that Half of 1% is the same as 1% of Half  (1% of \$412).  It does not matter which operation we do first.

Note, also, that ½% of \$824  and half of \$824  have the same digits:  4, 1, 2.

Example 12.   How much is ¼% of \$800?

Answer.   ¼% means a quarter of 1%.

1% of \$800 is \$8.00 .

¼% of \$800 is \$2.00.

Example 13.   Jacqueline deposited \$2000 in the bank, where the interest is 3½% annually.  In one year, how much interest will she earn?

Answer.  We have to calculate 3½% of \$2000.  We can get it from 1%:

1% of \$2000 is \$20.00.

Therefore, 3½% is

 3½ × 20 = 60 + 10 = 70. "Three times 20 + Half of 20."

She will earn \$70 in interest.

Example 14.  Calculator problem.   How much is 6½% of \$104.16?

Solution.   The Amount is missing.  Press

 1 0 4 . 1 6 × 6 . 5 %

See:

 6.7704

This is approximately, \$6.77.

If your calculator does not have a percent key, then change 6.5% to a decimal (Lesson 3), and press = .   Press

 1 0 4 . 1 6 × . 0 6 5 =

At this point, please "turn" the page and do some Problems.

or

Continue on to the next Section.

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