62 Part 1: The World of Numbers
Dividing fractions introduces only one extra piece to the multiplication rule. To divide fractions,
invert the divisor and multiply. The divisor is always the second fraction. Invert only the divisor,
never the first fraction. This inverted, or f lipped, version of the fraction is called its reciprocal. To
divide
1
83
4(^4) , realize that the fraction^3
4
is the divisor. Invert^3
4
to get^4
3
, then multiply and simplify:
1
8
4
3
4
24
1
(^556)
(^3).
DEFINITION
Two numbers are reciprocals if when they are multiplied their product is 1. The
reciprocal of the whole number 4 is the fraction 41. The reciprocal of a fraction
is another fraction with the numerator and denominator switched. 83 and^83 are
reciprocals. Each number is the reciprocal of the other.
Much of the work of multiplying and dividing fractions can be made easier by canceling before
multiplying. Canceling is dividing a numerator and a denominator by the same common factor.
You can think of it as simplifying before you multiply instead of after.
DEFINITION
Canceling is the process of simplifying a multiplication of fractions by dividing a
numerator and a denominator by a common factor.
When you multiply^893127 you multiply the numerators and multiply the denominators, so your
product, before you actually get the multiplying done, is 91287
3
3. After you do the multiplying,
it’s 10856 , and that needs to be simplified. To simplify it, you’ll factor: 108565 14 427 4
3
3. But look back to
87
912
3
3 for a moment. Can you see that you could factor this version? And because the numbers are
smaller, it might be easier. 91287555934427 27 4 934279344
3
333
33
33
33
3 3
3
. Both methods will leave you with^1427.
Canceling allows you to divide either numerator and either denominator by the same number. In
that last example, it would look like this:
You need to multiply
8
97
12(^3). You notice that both the numerator 8 and the denominator 12 have a
factor of 4.
Divide each of them by 4, and show that by crossing out the original number and writing the
quotient of that number and 4.
8
9
7
12
8
9
7
12
2
3
(^335)