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A R I T H M E T I C

Lesson 26  Section 2

Parts of Fractions

This Lesson depends on Lesson 14: Parts of Natural Numbers.

 How much is half of 25 ?
 Half of 25 is 15 .
 How much is half of 15 ?

If the whole "pie" is in 5 pieces, and we take half of each piece, then the whole pie will be in 10 pieces.

 Half of 15 is 1 10 .
 How much is half of 35 ?

Since

 35 = 15 + 15 + 15 (Lesson 20),
 then on taking half of each 15 , we get
 1 10 + 1 10 + 1 10 .
 Half of 35 is 3 10 .

To take half, the denominator 5 has been multiplied by 2

 5. 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 1.

 Half of 67 is 37 . Take half of the numerator. A third of 67 is 27 . Take a third of the numerator. A third of 47 is 4 21 . 4 does not have a third part.  Therefore multiply the denominator by 3.
 Example 2.   How much is a fifth of 12 ?
 Answer. 1 10 .   Multiply 2 by 5.
 Example 3.   How much is half of 1 10 ?
 Answer. 1 20 .  Multiply 10 by 2.
 Example 4. 18 is which part of 12 ?
 Answer.   Since 8 is 4 × 2, 18 is the fourth part of 12 .
 Example 5. 1 16 is which part of 18 ?   What ratio has 1 32 to 18 ?
 Answer. 1 16 is half of 18 .  16 = 2 × 8. 1 32 is a fourth of 18 .  32 = 4 × 8.
 Example 6. 25 is larger than 2 25 . (Lesson 22, Question1.)  How many

times larger?

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

 Example 7.  Percent.   Since 18 is half of 14 (8 = 2 × 4),
 and since 14 is equal to 25%, then what percent is 18 ?

Answer.  Half of 25%, which is 12½%.  (Lesson 15, Question 7.)

Example 8.   A recipe calls for 6¼ cups of flour.  You are going to make half the recipe.  How much flour will you use?

 Answer.   Half of 6 is 3.  Half of 14 is 18 .  You will use 3 18 cups.

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

 15 is 12 of what number?

The writer no doubt intend it to be translated as

 12 times what number is 15 ?

-- thus making it a division problem:

 15 ÷ 12 .

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. 37 are half of what number?
 Answer. 37 are half of 67 .
 Example 11. 37 are a third of what number?
 Answer. 37 are a third of 97 , which is 1 27 .

*

 Consider "Half of 58 " -- which we know is 5 16 -- and let us write it in

symbols as

 12 × 58 .

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

 12 × 58 = 5 16 .

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

 Similarly,  Half of 68 becomes
 12 × 68 = 38 .

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

 Example 12.     How much is a third of 57 ?  How much is two thirds?
 Answer. A third of 57 is 5 21. (Multiply the denominator by 3).
 Two thirds of 57 is twice as much as one third:  2 × 5 21 = 1021 .

In symbols,

 23 × 57 = 1021 .
 "Two thirds of 57 is 1021 ."

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.

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