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Lesson 26

# HOW TO MULTIPLY FRACTIONSHOW TO DIVIDE FRACTIONS

 3 × 27 ("3 times 27 ").  What does that mean?

According to the meaning of multiplication,

 3 × 27 = 27 + 27 + 27 .

That is,

 3 × 27 = 2 + 2 + 2       7 = 67 .

We can therefore answer the following:

 1. How do we multiply a fraction by a whole number? Multiply the numerator by the whole number. Do not change the denominator. If the fraction becomes improper, extract the whole number.

Example 1.    2 × = Example 2.    It takes yards of material to make a shirt.

How many yards will it take to make 6 shirts?

 Answer.   6 × = 12 5 = 2 25 yards.

Example 3.  Dividing first.

18 × = 6 × 2 = 12.

"3 goes into (is contained in) 18 six (6) times. 6 times 2 is 12."

The order of multiplying and dividing does not matter. (Property 3 of division.)  If we had multiplied first --

 18 × = 18 × 2    3

-- we would divide by 3 eventually.  But dividing first results in multiplying smaller numbers.

 Example 4. 20 × 11 5 = 4 × 11 = 44

"5 goes into 20 four (4) times."

 Example 5. 3 × 11 3 = 11

In this case, we may simply say that the 3's "cancel."

When the multiplier is a whole number, as 3 in this case, it is never necessay to write it as .

 2. How do we multiply a mixed number by a whole number? Multiply the whole number of the mixed number, multiply the fraction.
 2 × 4 = 8. 2 × 13 = 23 .

When multiplying by a whole number, it is never necessary to change to an improper fraction.

This is exactly what we did to multiply dollars and cents (Lesson 9).

4 × \$6.20 = \$24 + \$.80 = \$24 .80

 Example 6. 5 × 3 2 11 = 15 1011 .
 Example 7. 7 × 6 3 5 = 42 21 5 = 42 + 4 15 = 46 15 .

Always, if the fraction becomes improper, extract the whole number.

 3. How do we multiply a fraction by a fraction? Multiply the numerators and multiply the denominators.

In the next Lesson, we will see what multiplying by a fraction means.

Example 8. × = When multiplying fractions, do not change to a common denominator.

Example 9. × = If any denominator has a divisor in common with any numerator, then, just as in reducing a single fraction, we can divide them by that common divisor.

 Example 10. 89 × 1516

8 and 16 have a common divisor, which is 8 itself.  9 and 15 have a common divisor 3. "8 goes into 8 once (1);  8 goes into 16 two (2) times."

"3 goes into 9 three (3) times;  3 goes into 15 five (5) times."

"1 × 5 = 5.   3 × 2 = 6."

Whenever possible it is more skillful to cancel -- reduce -- before multipliying. The fraction will then immediately be in its lowest terms.
If we had multiplied first --

 89 × 1516 = 120144

-- the fraction would be much more difficult to reduce.

 Example 11. 12 × 34 × 57 = 1556

Multiply all the numerators:   1 × 3 × 5 = 15

Multiply all the denominators:   2 × 4 × 7 = 8 × 7 = 56

The time to cancel is before multiplying when the numbers are smaller.  If nothing cancels before , as in this example, then the answer is already in its lowest terms.

 Example 12. 43 × 9 2

2 is a common divisor of 4 and 2. 3 is a common divisor of 3 and 9. "2 goes into 4 two (2) times;  2 goes into 2 one (1) time."

"3 goes into 3 one (1) time;  3 goes into 9 three (3) times."

"2 × 3 = 6.   1 × 1 = 1."

 A fraction with denominator 1  ( 61 )  is simply the

numerator. It is not necessary to say, "1 goes into 6 six times."

Here is another example:

 51 = 5.
 Example 13. 4 × 56

We may cancel before multiplying 4 × 5: "2 goes into 4 two (2) times."

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

"2 × 5 = 10; over 3."

Since we would find the divisor 2 after multiplying 4 × 5, we can find it even more easily before.  Again, the advantage of reducing first is that we work with smaller numbers.

 4. How do we multiply a mixed number by a mixed number, or a fraction by a mixed number? Change the mixed numbers to improper fractions (Lesson 20). This is the only place in arithmetic where it is necessary to change to an improper fraction.

 Example 15. 2 × 4 13 = 8 23

It is not necessary to change to an improper fraction.  When multiplying by a whole number, multiply the whole number times the whole number, and multiply the whole number times the numerator.

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

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Continue on to the Section 2:  Dividing fractions