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A R I T H M E T I C

Lesson 27

THE MEANING OF MULTIPLYING FRACTIONS

FRACTIONS INTO PERCENTS

PARTS OF FRACTIONS

In this Lesson, we will answer the following:

What does it mean to multiply a number by a fraction?
How do we multiply a whole number by a mixed number?
How can we express a fraction as a percent?
Section 2
How can we take a part of a fraction?

In the previous Lesson we simply stated the rule for multiplying one fraction by another. In this Lesson we want to understand where that rule comes from. It comes from what multiplying by a fraction means.

What does it mean to multiply a number by a fraction?

1 2	× 8, .15 × 20

It means to take that fractional part of the number.

1 2	× 8 means One half of 8, which is 4.

1 4	× 20 means One fourth of 20, which is 5.

3 4	× 20 means Three fourths of 20, which is 15.

.15 × 20 means Fifteen hundredths of 20, which is 3.

For, according to the meaning of multiplication, we are to repeatedly add the multiplicand as many times as there are 1's in the multiplier. In the multiplier ½ there is one half of 1. Therefore we are to add the multiplicand 8 one half a time. We are to take one half of 8.

Also, although "½ × 8" looks like multiplication, there is nothing to multiply! "½ × 8" is a symbolic abbreviation for "One half of 8." And to calculate it, we have to divide. (Lesson 16.) We can now begin to see why we have the cancelation rules.

In fact, the symbolic statement, "4 = ½ × 8," expresses the ratio of 4 to 8: "4 is one half of 8."

This is another use for fractions: Multiplication by a fraction expresses a ratio.

For the most general definition of multiplication, see below.

Example 1. Calculate

2
3

× 21 "Two thirds of 21." (We may read

2
3

× 21" as "Two thirds of 21" rather than "Two thirds times 21.")

One third of 21 is 7 -- "3 goes into 21 seven (7) times." 2 × 7 = 14.

(Lesson 15, Question 6.)

If the problem were just to evaluate Two thirds of 21,

the student should not have to resort to writing

2
3

× 21.

Simply say, "One third of 21 is 7. So two thirds are 14." (Lesson 15.) The point of this Lesson is to explain what it means to multiply by a fraction.

Problem.

5
8

× 32. What does that mean?

To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").
Do the problem yourself first!

Calculate it.

Compare Lesson 15, Example 5.

Example 2. Calculate

3
4

× 5. "Three fourths of 5."

Solution. Although 5 is not exactly divisible by 4, we can still take its fourth part -- by dividing by 4:

"4 goes into 5 one (1) time with 1 left over."

One fourth of 5 is 1

1
4

; therefore, three fourths are 3 × 1

1
4

= 3

3
4

(Lesson 26.)

Alternatively, we can multiply first:

3
4

× 5

15
4

= 3

3
4

"4 goes into 15 three (3 ) times (12) with 3 left over."

We see:

We may take a part first or multiply first.

See below. Compare Lesson 11.

Example 3. You are going on a a trip of four miles, and you have gone two thirds of the way. How far have you gone?

Solution. We must take two thirds of 4.

"One third of 4 is 1

1
3

4
3

= 1

1
3

"Therefore two thirds are 2 × 1

1
3

= 2

2
3

You have gone 2

2
3

miles.

Example 4. How much is a fifth of 3?

Solution. While we could write

1
5

× 3 =

3
5

, we know that to find a fifth

of a number, we divide by 5. And 3 ÷ 5 is

3
5

-- Lesson 11, Example 15.

Therefore, we could know immediately:

A fifth of 3 is

3
5

Example 5. How much is a fourth of 9 gallons?

Answer.

9
4

= 2

1
4

gallons.

That should be a simple mental calculation.

Example 6. How much money is 64 quarters?

Answer. 64 quarters would be 64 × $.25. But according to the order property of multiplication,

64 × .25 = .25 × 64.

But .25 is the decimal for

1
4

. Therefore we can evaluate 64 quarters

by taking one quarter of 64 And we can do that by taking half of half. (Lesson 16.)

Half of 64 is 32. Half of 32 is 16. Therefore 64 quarters are $16.

Example 7. A slot machine at a casino paid 93 quarters. How much money is that?

Answer. To find a quarter of 93, divide 93 by 4. We can easily do that mentally by decomposing 93 into multiples of 4. For example:

93 = 80 + 12 + 1.

On dividing each term by 4, we have

20 + 3 + ¼ = 23¼.

93 quarters, then, are $23.25.

Example 8. A recipe calls for 3 cups of flour and 4 cups of milk. Proportionally, how much milk should you use if

a) you use 1½ cups of flour? b) you use 2 cups of flour?

c) you use 2½ cups of flour?

Answers.

a) 1½ cups flour are half of 3 cups. Therefore you should use half as
a) much milk -- you should use 2 cups of milk.

b) 2 cups flour are two thirds of 3 cups. That is the ratio of 2 cups to 3. b) Therefore you should use two thirds as much milk.

2
3

× 4 =

8
3

= 2

2
3

cups milk.

c) What ratio has 2½ cups of flour to the original 3 cups?

On expressing 2½ as the improper fraction

5
2

, then on cross-

b) multiplying:

5
2

is to 3 as 5 is to 6.

Lesson 23, Question 4.

2½ cups are five sixths of 3 cups.

Therefore, you should use five sixths of 4 cups of milk.

5
6

× 4

20
6

= 3

2
6

= 3

1
3

cups of milk.

2.	How do we multiply a whole number by a mixed number?
2½ × 8
	Multiply by the whole number of the mixed number, then multiply by the fraction. It is not necessary to change to an improper fraction.