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Lesson 26

THE MEANING OF MULTIPLYING FRACTIONS

PARTS OF FRACTIONS


In this Lesson, we will answer the following:

  1. What does it mean to multiply a number by a fraction?
  2. How do we multiply a whole number by a mixed number?
  3. How can we express a fraction as a percent?

    Section 2

  4. 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.
 1.   What does it mean to multiply a number by a fraction?
1
2		  × 8
 
  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.

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.

Furthermore, although ½ × 8 looks like multiplication, there is nothing to multiply! To take half of 8, we divide by 2 (Lesson 15).  And we begin to see why we have the cancellation rules.

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 14, Question 5.)

  Example 2.      5
8		 × 32 -- "Five eighths of 32."

"One eighth of 32 is 4 .  5 × 4 = 20."

  Example 3.   Calculate  3
4		 × 5.  "Three fourths of 5."

Solution.  Although 5 is not exactly divisible by 4, we can still take its fourth part.  We can divide 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		 .

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.

The order in which we multiply or divide does not matter.  (Lesson 10)

Example 4.   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:

2
3		  × 4.

Since 4 is not exactly divisible by 3, we will multiply first:

2
3		  × 4  =  8
3		  =  2 2
3

"3 goes into 8 two (2) times (6) with 2 left over."

But again, we could take a third of 4 first:

"3 goes into 4 one (1) time with 1 left over.   2 × 1 1
3		  = 2 2
3		 .
You have gone 2 2
3		  miles.

Example 5.   How much is a third of 2?

  Solution.   While we could write   1
3		  × 2 =  2
3		 , we know that to find a third
  of a number, we divide by 3. (Lesson 15.)  And 2 ÷ 3 is   2
3		   --  the division

bar.  Therefore, we could write immediately:

A third of 2  =   2
3		 .

Example 6.   How much is a fifth of 9?

  Answer.    9
5		  = 1 4
5		 .

Example 7.   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		 .  (Lesson 23.)  Therefore we can

evaluate 64 quarters by taking one quarter of 64!  And we can do that by taking half of half.  (Lesson 15, Question 8.)

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

Example 8.   A slot machine at a casino paid 93 quarters.  How much money is that?

 Answer.   To find a quarter of 93 is equivalent to dividing 93 by 4. (Lesson 15.)  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 9.   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.  Therefore you should use two
b)  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-

multiplying:

5
2		  is to 3  as  5 is to 6.    Lesson 22, Question 6.

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.

We saw this in Lesson 15, Question 1 as a mixed number of times.

Example 10.   2½ × 8.

   Answer.   2½ × 8   =  2 × 8  +  ½ × 8  -- "Two times 8 + Half of 8"
   Answer.   2½ × 8   =  16 + 4
   Answer.   2½ × 8   =  20.

In multiplication, when one of the numbers is a whole number, it is not necessary to change to an improper fraction.

(See also the previous Lesson, Question 1.)

Example 11.  Mental calculation.   What is the price of 12 items at $3.25 each?

Answer.  12 × $3.25 is the same as $3.25 × 12,  or, 3¼ × 12:

3¼ × 12  = Three times 12 + A quarter of 12
3¼ × 12  = 36 + 3
3¼ × 12  = $39.

Example 12.  Multiplying by numbers ending in 5.   Calculate mentally:  75 × 6.

Answer.  Rather than 75 × 6, let us do

7.5 × 6

that is,

7½ × 6.

7½ × 6 = 42 + 3 = 45.

Now, by replacing 75 with 7.5, we divided by 10.  (Lesson 3,
Question 5.)  Therefore, to compensate we must multiply by 10:

75 × 6 =450.

Example 13.     35 × 16

Answer.   3.5 × 16 = 48 + 8 = 56.  Therefore,

35 × 16 = 560.
 3.   How can we express a fraction as a percent?
  Multiply it times 100%.

That is how to change any number to a percent (Lesson 3).


  Example 14.    Express  1 
11		 as a percent.  Express  5 
11		 as a percent.

Solution.  100% is the whole.  Therefore, take one eleventh of 100%:

 1 
11		 × 100%  =  100% ÷ 11
100%
  11		  =  9  1 
11		 %.

"11 goes into 100 nine (9) times (99) with 1 left over."

 1 
11		   =  9  1 
11		 %.
As for   5 
11		 , it is five times   1 
11		 :
5 × 9  1 
11		 %  =  45  5 
11		 %.

See the previous Lesson, Example 3.

Here is the most general definition of multiplication:

Whatever ratio the multiplier has to 1
      the product shall have to the multiplicand.

Consider this multiplication of whole numbers:

3 × 8 = 24

The multiplier 3 is three times 1; therefore the product will be three times the multiplicand; it will be three times 8.

Similarly, to make sense out of

½ × 8,

the multiplier ½ is half of 1 (Lesson 15, Question 6); therefore the product will be half of 8.

Proportionally,

As the Multiplier is to 1, so the Product is
to the Multiplicand.

Or inversely,

As 1 is to the Multiplier, so the Multiplicand is
to the Product.

The Product, then, is the fourth proportional to 1, the Multiplier, and the Multiplicand.


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

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