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, or one quarter, 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 16, Example 4.)
Now, each eighth will be another 12½%.
2 8 
= 
1 4 
= 12½% + 12½% = 25%. 
See Problem 15.
times larger?
Answer. Five times larger. Because 25 is 5 × 5.
Example 9. The following problem appeared in a recent textbook:
The writer no doubt intend it to be translated as
1 2 
times what number is 
1 5 
? 
 thus making it a division problem:
However, if we make verbal sense of the problem 
1fifth is half of what number?
 then the answer is obvious. Just as 1 apple is half of 2 apples, so
1fifth is half of 2fifths.
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,
The green dye consists of three parts: 2 parts blue and 1 part yellow.
To add equal amounts of O and G, we may simply add half of each. Therefore, half of O is
And half of G is
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 is 30. (Lesson 23.) The fractions of Y 
1 5 
Y + 
1 6 
Y 


 become 
6 30 
Y + 
5 30 
Y = 
11 30 
Y. 
(Lesson 22, Question 3.)

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
We now see why we multiply the numerators and multiply the denominators:
It follows, as it must, from what the symbols mean.
Similarly, Half of 
6 8 
becomes 
"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,
"Two thirds of 
5 7 
is 
10 21 
." 
And so we have arrived where we began (Lesson 26, Question 3), at the formal rule for multiplying fractions:
Multiply the numerators and multiply the denominators.
Please "turn" the page and do some
Problems.
or
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