How do you calculate decimals to fractions?

Answer 1

First, the decimal must be a representation of a rational number. If it is a non-repeating decimal, then multiply the decimal by the smallest power of ten to make it an integer - that number is the numerator. The power of ten that you used is the denominator. Then simplify.

Example: 0.372 --> 0.372 * 1000 = 372, so 0.372 = 372/1000 = 93/250

Repeating decimals are a little more involved. I'll show by example.

1/7 = 0.142857142857142857.... (I bolded the repeating 6 digits)

Multiply the number by a power of ten equal to number of digits that repeat (6 digits repeat, so we multiply by 106). Subtract the original decimal from the 'multiplied' number, and the repeating digits all cancel out:

142857.142857142857 - 0.142857142857 = 142857.00000 = 142857, this is the numerator of the fraction.

The denominator is the [power of ten minus one]. In this case 106 - 1 = 999999

So you have 142857/999999 which simplifies to 1/7.

Here's how this works: Let T be the power of ten that you're using, and x be the repeating decimal that you want to convert.

So you have T*x - x --> x*(T - 1). Then if we divide this by (T - 1) we are back to the original x [x*(T -1)] / (T-1) = x. The (T -1) in numerator/denominator cancel.

T*x - x

--------- = x {multiplications, subtractions, & division = the original number}

T - 1