Answer:
a) 81/500
b) 7/90
c) 3/11
Explanation:
You want to write the fractions that correspond with the decimals ...
a) 0.162
b) 0.0777...
c) 0.272727...
Terminating decimals
The fraction corresponding to a terminating decimal can be written based on how you pronounce the fraction.
a) 0.162
0.162 = "one hundred sixty-two thousandths" = 162/1000
This can be reduced by removing a common factor of 2.
= 81/500
Repeating decimals
The fraction equivalent to a repeating decimal can be found by a multiply and subtract process. In general, this makes the denominator be a number that has as many 9s as the decimal has repeating digits.
b) 0.0777...
This has one repeating digit, 7. This means we want to convert to a fraction by multiplying by 10 to the power of 1:
x = 0.0777...
10x = 0.7777...
Now, we can subtract the original number:
10x -x = 0.7777... -0.0777... = 0.7000...
Notice that the repeating digits cancel. Now we have ...
9x = 0.7 = 7/10
Dividing by 9 gives ...
x = 7/90
c) 0.272727...
Using the above method with 10 to the power of 2 (with a 2-digit repeat), we get ...
100x = 27.272727...
100x -x = 27.272727... -0.272727... = 27
99x = 27
x = 27/99 = 3/11
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Additional comment
In general, the repeating part of a repeating decimal can be expressed over an equivalent number of 9s. For example, the 6-digit repeat 0.142857142857... is equivalent to 142857/999999. As it happens, this one reduces to 1/7.
If there are intervening digits, as in 0.4333... for example, then they can be treated as a terminating decimal, and the repeating part can be multiplied by an appropriate power of 10 to account for the shift from the decimal point:
0.4333... = 0.4 + 0.1 × 0.333...
= 4/10 + 1/10 × 3/9 = 4/10 + 1/30
= 12/30 +1/30 = 13/30