How to Convert a Repeating Decimal into a Fraction - The Math District
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How to Convert a Repeating Decimal into a Fraction
Example 1
Convert the decimal 0.47474747 ... into a fraction.
1. Use a letter to represent the decimal 0.47474747 ...
= 0.4747474747 ... ( 1)
2. Multiply both sides of the equation by 100, because the decimal has 2 repeating digits. When multiplying the right side by 100, move the decimal point two places to the right.
100() = 100(0.4747474747 ... )
100 = 47.4747474747 ... ( 2)
3. Subtract equation 1 from equation 2.
100 = 47.4747474747 ... = 0.4747474747 ...
99 = 47
4. Solve the equation for .
99 = 47
99 47 99 = 99
47 = 99 So, we converted the repeating decimal into a fraction.
47 0.47474747474747 ... = 99
Example 2
Convert the decimal 0.222222 ... into a fraction.
1. Use a letter to represent the decimal 0.222222 ...
= 0.222222 ... ( 1)
2. Multiply both sides of the equation by 10, because the decimal has 1 repeating digit. When multiplying the right side by 10, move the decimal point one place to the right.
10() = 10(0.222222 ... )
10 = 2.22222 ... ( 2)
3. Subtract equation 1 from equation 2.
10 = 2.22222 ... = 0.222222 ...
9 = 2
4. Solve the equation for .
9 = 2
9 2 9 =9
2 = 9 So, we converted the repeating decimal into a fraction.
2 0.222222 ... = 9
Example 3
Convert the decimal 0.285714285714 ... into a fraction.
1. Use a letter to represent the decimal 0.285714285714 ...
= 0.285714285714 ... ( 1)
2. Multiply both sides of the equation by 1,000,000, because the decimal has 6 repeating digits. When multiplying the right side by 1,000,000, move the decimal point six places to the right.
1,000,000() = 1,000,000(0.285714285714 ... )
1,000,000 = 285,714.285714 ... ( 2)
3. Subtract equation 1 from equation 2.
1,000,000 = 285,714.285714 ... = 0.285714285714 ...
999,999 = 285,714
4. Solve the equation for .
999,999 = 285714
999,999 285,714 999,999 = 999,999
285,714 = 999,999
Reduce the fraction.
285,714 ? 142,857 2 = 999,999 ? 142,857 = 7
So, we converted the repeating decimal into a fraction.
2 0.285714285714 ... = 7
Example 4
Convert the decimal 0.45454545 ... into a fraction.
1. Use a letter to represent the decimal 0.4545454545 ...
= 0.4545454545 ... ( 1)
2. Multiply both sides of the equation by 100, because the decimal has 2 repeating digits. When multiplying the right side by 100, move the decimal point two places to the right.
100() = 100(0.45454545 ... )
100 = 45.4545454545 ... ( 2)
3. Subtract equation 1 from equation 2.
100 = 45.4545454545 ... = 0.4545454545 ...
99 = 45
4. Solve the equation for .
99 = 45
99 45 99 = 99
45 = 99 Reduce the fraction.
45 ? 9 5 = 99 ? 9 = 11 So, we converted the repeating decimal into a fraction.
5 0.4545454545 ... = 11
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