How to Write a Repeating Decimal As a Fraction


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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