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A R I T H M E T I C

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:

How can we represent a percent as a fraction, or a whole number, or a mixed number?
How can we find the Amount when we know the Base and the Percent?
What does ½% mean?
Section 2
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	=	3 5	. (Lesson 21, Question 3)

200%	=	200 100	= 2.

250%	=	200% + 50% = 2½	.

225%	=	2¼.

233

1
3

% = 2

1
3

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

3 4	× 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.

Lesson 26, Question 2.

Example 6. How much is 37½% of $40?

Answer. 37½% =

3
8

(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

1
3

% of 720? How much is 66

2
3

Answer. 33

1
3

% 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

2
3

%, 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

2
3

% of $76.27?

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

See

50.846666666

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.

(Lesson 15, Question 3.)

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

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

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

Continue on to the next Section.

First Lesson on Percent.

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