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Recurring decimals to fractions
Every recurring decimal is really a fraction. A neat bit of algebra reveals which one.
Notation
0.4̇ means 0.444...
0.2̇7̇ means 0.272727...
A dot over the first and last digits shows the repeating block.
0.2̇7̇ means 0.272727...
A dot over the first and last digits shows the repeating block.
The method
Let x = the decimal. Multiply by 10, 100 or 1000 so the repeating parts line up. Subtract to cancel the repeating tail, then solve.
x = 0.272727...
100x = 27.272727...
99x = 27, so x = 2799 = 311
x = 0.272727...
100x = 27.272727...
99x = 27, so x = 2799 = 311
A non-repeating start
x = 0.16666...
10x = 1.6666... and 100x = 16.666...
90x = 15, so x = 1590 = 16
10x = 1.6666... and 100x = 16.666...
90x = 15, so x = 1590 = 16
Write 0.5̇ as a fraction.
- x = 0.555...
- 10x = 5.555...
- 9x = 5
- x = 59
Answer: 59
Set x equal to the decimal, multiply by a power of 10 that matches the repeat length, subtract, and solve. Simplify the fraction.
Check you have got it
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