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Lesson 26  Section 2

Parts of Fractions

Back to Section 1

  Example 1.   A recipe calls for 6 3
4		  cups of flour, and you are going to

make a third of the recipe.  How much flour will you use?

  Answer.   A third of 6 cups is 2 cups.  But how much is a third of   3
4		 ?

A third of   3
4		   is  1
4		 .  Simply take a third of the numerator.

(See Lesson 19.)

A third of  6 3
4		  cups is 2 1
4		  cups.
But say that the recipe calls for 6 1
2		  cups of flour.  Since the
  numerator 1 does not have a third part, how will you take a third of  1
2		  ?

If we divide each  1
2		  into thirds, then the whole 1 will be in six equal
  parts.  A third of   1
2		  is  1
6		 .

To take a third of a fraction, we can multiply the denominator by 3.

(Again, see Lesson 19.)

A third of  6 1
2		  cups is 2 1
6		  cups.

There are two ways, then, to take a part of a fraction.
 4.   How can we take a part of a fraction?
 
 
  Take that part of the numerator -- if the numerator has that part.
 
  If the numerator does not have that part,
 
  multiply the denominator by the cardinal number that corresponds to the part. That is, to take half, multiply the denominator by 2; to take a third, multiply by 3; and so on.
 
 

Example 2.   

Half of   6
7		  is  3
7		 .    Take half of the numerator.
 
A third of   6
7		  is  2
7		 .    Take a third of the numerator.
 
A third of   4
7		  is   4 
21		 .  4 does not have a third part.  Therefore multiply the denominator by 3.
  Example 3.   How much is a fifth of   1
2		  ?
  Answer.     1 
10		 .   Multiply 2 by 5.
  Example 4.   How much is half of    1 
10		  ?
  Answer.     1 
20		 .  Multiply 10 by 2.
  Example 5.    1
8		   is which part of   1
2		  ?
  Answer.   Since 8 is 4 × 2,   1
8		   is the fourth part of  1
2		 .
  Example 6.     1 
16		  is which part of   1
8		  ?   What ratio has   1 
32		  to  1
8		  ?
  Answer.     1 
16		   is half of   1
8		  .  16 = 2 × 8.     1 
32		  is a fourth of   1
8		  .  32 = 4 × 8.
  Example 7.  Percent.   Since  1
8		  is half of   1
4		   (8 = 2 × 4),
  and since   1
4		   is equal to 25%, then what percent is  1
8		 ?
  Answer.   Half of 25%, which is 12½%.  (Lesson 15, Question 9.)
 
  See Problem 15.
  Example 8.      2
5		  is larger than   2 
25		 . (Lesson 19, Question 11.)  How many

times larger?

Answer.  Five times larger.  Because 25 is 5 × 5.

Example 9.   The following problem appeared in a recent textbook:

1
5		  is  1
2		  of what number?

The writer no doubt intend it to be translated as

1
2		  times what number is  1
5		 ?

-- thus making it a division problem:

1
5		  ÷  1
2		 .

However, if we make verbal sense of the problem --

1-fifth is half of what number?

-- then the answer is obvious.  Just as 1 apple is half of 2 apples, so

1-fifth is half of 2-fifths.

  Example 10.    3
7		  is half of what number?
  Answer.    3
7		  is half of   6
7		 .
  Example 11.    3
7		  is a third of what number?
  Answer.    3
7		  is a third of   9
7		 , which is 1 2
7		 .

Example 12.   To make an orange dye, 3 parts of red dye are mixed with 2 parts of yellow dye.  To make a green dye, 2 parts of blue dye are mixed with 1 part of yellow dye.  If equal amounts of orange and green are mixed, what fraction of the new mixture is yellow dye?

Solution.   For convenience, let

O = Orange dye,  R = Red dye,  Y = Yellow dye,

G = Green dye,  B = Blue dye.

Now, the orange dye consists of a total of five parts:  3 parts red and 2 parts yellow.  That is,

O  =  3
5		  R +  2
5		  Y.

The green dye consists of three parts:  2 parts blue and 1 part yellow.

G  =  2
3		  B +  1
3		  Y.

To add equal amounts of O and G, we may simply add half of each.   Therefore, half of O is

O  =   3 
10		  R +  1
5		  Y.

And half of G is

O  =  1
3		  B +  1
6		  Y.

Upon adding those equal parts of O and G, the new mixture will consist of

 3 
10		  R +  1
5		  Y +  1
3		  B +  1
6		  Y.

The LCM of those fractions (Lesson 22) is 30.  The fractions of Y then become

 6 
30		  Y +   5 
30		  Y =  11
30		  Y.

(Lesson 21.)

   11
30		  is the fraction of yellow dye in the new mixture.

*

  Consider "Half of  5
8		 " -- which we know is   5 
16		  -- and let us write it in

symbols as

1
2		  ×  5
8		 .

We now see why we multiply the numerators and multiply the denominators:

1
2		  ×  5
8 		 =   5 
16		 .

It follows, as it must, from what the symbols mean.

Similarly,  Half of  6
8		  becomes
1
2		  ×  6
8		  =  3
8		 .

"2 goes into 6 three (3) times."


  Example 13.     How much is a third of  5
7		 ?  How much is two thirds?
  Answer.     A third of  5
7		  is   5 
21.		   (Multiply the denominator by 3).
Two thirds of  5
7		  is twice as much as one third:  2 ×   5 
21		  =  10
21		 .

In symbols,

2
3		  ×  5
7		  =  10
21 		.
"Two thirds of   5
7		  is  10
21		 ."

And so we have arrived where we began (Lesson 25, Question 2), at the formal rule for multiplying fractions:

Multiply the numerators and multiply the denominators.


Please "turn" the page and do some Problems.

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