A repeating decimal is a decimal that has a repeating pattern. A simple example is 0.33333.... where the ... means continue like this. Many fractions, when expressed as decimals, are repeating. For instance, 0.33333.... is 1/3. But sometimes the repeating portion is longer. For instance, 1/7 = 0.142857142857. However, any repeating decimal can be converted into a fraction. Repeating decimals are often represented with a bar, over the repeating portion.
Identify the repeating portion. For instance, in 0.33333..... the 3 is the repeating portion. In 0.1428571428, it is 142857
Count the number of digits in the repeating portion. In 0.3333 the number of digits is one. In 0.142857 it is six. Call this "d."
Multiply the repeating decimal by 10^d, that is, one with "d" zeroes after it. So, multiply 0.3333.... by 10^1 = 10 to get 3.3333...... Or multiply 0.142857142857 by 10^6 = 1,000,000 to get 142857.142857.....
Note that the result of this multiplication is a whole number plus the original decimal. For instance 3.33333...... = 3 + 0.33333..... Or, in other words, 10x = 3 + x. With 0.142857, you would get 1,000,000x = 142,857 + x.
Subtract x from each side of the equation. For example, if 10x = 3 + x, then subtract x from each side to get 9x = 3 or 3x = 1 or x = 1/3 In the other example, 1,000,000x = 142,857 + x, so 999,999x = 142,857 or 7x = 1 or x = 1/7
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