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Lesson 25 Section 2 Dividing fractionsSection 1: Multiplying fractions IN DIVISION, the dividend and divisor be units of the same kind. We can only divide dollars by dollars, hours by hours, yards by yards. 15 yards ÷ 3 yards = 5 -- because 5 times 3 yards = 15 yards. (Lesson 10.) (We cannot divide 15 yards by 3 feet -- not until we change yards to feet!) When we divide pure numbers -- 6 ÷ 2 = 3 -- we mean 6 units ÷ 2 units = 3. For there is no "6" apart from 6 units, even though we do not say the word "units." With fractions, the units are named by the denominator. (Lesson 20.) Therefore:
"6 sevenths ÷ 2 sevenths = 3" -- because 3 times 2 sevenths = 6 sevenths. Here is the rule: To divide fractions, the denominators must be the same.
because
Compare Lesson 10, Example 14.
Different denominators When the denominators are not the same --
-- we can make a common denominator in the usual way:
The common denominator in this case is 8 × 3 = 24.
As in multiplication, we must change mixed numbers to improper fractions. The common denominator is then 4.
To change a whole number into a fraction, multiply the whole number by the denominator.
That product will be the numerator. (Lesson 20.) Example 7. A bottle of medicine contains 15 oz. Each dose of the medicine is 2½ oz. How many doses are there in the bottle? Solution. This is a division problem (Lesson 10) -- how many times can we subtract 2½ oz from 15 oz?
In that bottle there are 6 doses. "Invert and multiply" A method often taught is: "Invert the divisor and multiply."
As with many written methods, this is a trick that gives the right answer. But it is based on the principle of equal denominators -- because it gives the numerators, 15 and 16, if we were to make the denominators the same!
(We see that we could also obtain the numerators by cross-multiplying.) Invert and multiply is merely a rule, and therefore it is not very educational. Nevertheless, for certain problems it can be skillful, particularly when the dividend is a whole number.
Invert the divisor -- the number after the division sign ÷ . Divide 4 into 40, then multiply. When we invert a fraction, the number we obtain is called its
Reciprocals always come in pairs.
See Lesson 28, Examples 6 - 8. In general, however, the method of common denominators is to be preferred. It uses a skill the student has already learned. And what is more, it emphasizes a basic property of division, namely: The units must be the same. In summary: |
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