How Can I Add Repeating Decimals?

By Society & Culture & Entertainment Last updated Saturday, August/31/2024

Converting Decimals to Fractions

1). Examine the addition problem 0.56(56) ¯ + 0.333(333) ¯. The parentheses and vinculum indicate repeating digits.
2). Turn 0.56(56) ¯ into a fraction. First set the repeating decimal so that it equals x: X = 0.56(56) ¯
3). Multiply both sides by 100: 100x = 56. 56(56) ¯. Multiply both sides by a power of 10 that is equal to the number of digits in the repeating pattern. After moving the decimal over two places, you now have a whole unit and the original x factor above.
4). Simplify the equation by writing it as 100x = 56 + x.
5). Subtract x from both sides of the equation: 100x -- x = 56 + x -- x = 99x = 56
6). Divide both sides by 99 to isolate the x, thereby creating the necessary fraction, X = 56/99, which does not reduce.
7). Repeat the process for 0.333(333) ¯: X = 0.333(333) ¯
8). Multiply by 10, that is, the same number of digits in the repeating pattern: 10x = 3. (333) ¯. Simplify to 10x = 3 + x.
9). Subtract x from both sides: 9x = 3
10

Divide both sides by 9: X = 3/9, which reduces to 1/3.

1). Find the common denominator of 1/3 and 56/99. In this case, 99 is the common denominator.
2). Multiply the numerator and denominator in 1/3 by 33 to make an equivalent fraction with the denominator 99: 33/99.
3). Add 33/99 + 56/99. Add the numerators, 33 + 56 = 89. The denominator stays the same, 89/99, which does not reduce.
4). Leave the answer in this form unless the problem asks the answer be written in decimal notation --- divide 89 by 99 to find the answer 0.89 repeating.

1). Add 6.(5) ¯ + 7.(8) ¯.
2). Set the decimals to equal x: x = 0.(5) ¯ and x = 0.(8) ¯
3). Multiply by 10 and simplify: 10x = 5 + x and 10x = 8 + x
4). Subtract x from both sides: 9x = 5 and 9x = 8
5). Divide both sides by 9: X = 5/9 and x = 8/9
6). Add the fractions 6 and 5/9 + 7 and 8/9 = 13 and 13/9. Rewrite the fraction as a mixed number by dividing the numerator by the denominator: 13 ÷ 9 = 1 and 4/9.
7). Add the whole digits, 6 + 7 = 13. Add the sum, 13, and the mixed number, 1 and 4/9 for the sum 14 and 4/9. If the problem asks for a decimal answer, convert 14 and 4/9 to a mixed number by multiplying the whole number by the denominator and then adding the numerator, which equals 130/9. Divide 130 by 9 for the decimal answer 14.4 repeating.