How do you change terminating but reapeating decimal to fraction?

Answer 1

First of all, this question is not clear. How can you have a terminating but repeating fraction? Unless you mean a fraction like 0.234234 that ends but has a pattern. In that case, see Scenario 1.

Scenario 1

For terminating decimals, simply rewrite the decimal as a fraction with a denominator of a multiple of ten (for example, 0.313 => 313/1000) and simplify it as much as possible.

Scenario 2

For repeating decimals, the process is a lot more complicated.

First, figure out the pattern in the repeating decimal. Then write only the first part of the decimal with at least two repetitions of the pattern (for example, 0.123123...) and place ... to signify that the pattern continues.

I will use the decimal number 0.123123... as an example.

Then make one "chunk" of the pattern appear before the decimal point, as in 123.123... Note what power of ten you used to make the decimal number into the number with the "chunk" before the point. In this example it is 1,000 (0.123123... x 1000 = 123.123...)

Then write that power of ten before n, the fraction you are solving for, and then just plain n with its value:

1000n = 123.123...

n = 0.123123...

Then subtract the values. The answer in this case is 999n = 122.

Then get n alone (in this example, divide both sides by 99) and you're all set!