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52 Mastering Technical Mathematics
7/17 ?
20/21 ?
21/22 ?
26/27 ?
97/99 ?
851/999 ?
6,845/9,999 ?
25,611/99,999 ?
Do you notice something in common about the last four examples above?
CONVERTING RECURRING DECIMALS TO FRACTIONS
Although recurring decimals are easier to handle when you know what they mean,
using old-fashioned fractions is often easier. How can we convert a recurring decimal
to a fraction? There’s a neat little trick that works for decimal numbers less than 1—
that is, numbers that you write down as a zero followed by a decimal point and then a
group of digits that keeps recurring. We can call this general rule the “law of the
nines.”
First, find the pattern that repeats, and write it down. For example, you might see the
following decimal:
0.673867386738 . . .
Here, the sequence of digits that repeats is 6738. Now put this in the numerator of a
fraction, and then put an equal number of 9s in the denominator. In this case, you get
6,738/9,999. This is the fractional equivalent of the above decimal number. Check it
out with your calculator. Punch in the digits 6, 7, 3, and 8; then press the “divide by”
key, then the digits 9, 9, 9, and 9, and finally the “equals” key. Here are some more
examples:
0.979797 . . . 97/99
0.851851851 . . . 851/999
0.684568456845 . . . 6,845/9,999
0.256112561125611 . . . 25,611/99,999
You will recognize these as the last four numbers from the previous section.