Turning a repeating decimal back into a fraction
We turned fractions into decimals. Can we turn a repeating decimal back into a fraction?
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- d = 0.444…
- 10d = 4.444…
- Both have the same tail. It cancels.
- Take one from the other: 9d = 4.
- So d = 4/9.
Two lines with the same tail
Yes. Write two lines with the same endless tail. Subtract one from the other, and the tail cancels.
Call the number d. Multiply by 10.
- d = 0.444…
- 10d = 4.444…
Both lines have the same tail, the endless 4s after the point.
Take d from 10d. On the left, 10d − d = 9d. On the right, 4.444… − 0.444… = 4. The endless 4s cancel. So 9d = 4, and d = 4/9.
Sometimes a digit comes once before the repeating part. Multiply by 10 to put that digit before the point. For 0.8333…, use 10d and 100d.
- 10d = 8.333…
- 100d = 83.333…
- Take away: 100d − 10d = 90d.
- On the right, 83.333… − 8.333… = 75.
- So 90d = 75, and d = 75/90 = 5/6.
| Decimal | Lines used | After taking away | Cut down |
|---|---|---|---|
| 0.444… | d, 10d | 9d = 4 | 4/9 |
| 0.3636… | d, 100d | 99d = 36 | 4/11 |
| 0.8333… | 10d, 100d | 90d = 75 | 5/6 |
| 0.41666… | 100d, 1000d | 900d = 375 | 5/12 |
The last row gives 5/12. Dividing 5 by 12 gives 0.41666… again.
Watch out!
0.8333… is not 83/99. 83/99 is 0.8383…, a different number. Writing the digits over 9s works only when all digits repeat. For example, 0.3636… is 36/99.