How do you turn a decimal to its simplest form then into a fraction?

Answer 1

To convert a decimal to a faction in its simplest form depends on the decimal:

1) If it is a terminating decimal, place the decimal over a 1 followed by the same number of 0s as digits in the decimal and then divide top and bottom by common factors until their only common factor is 1 - these divisions can be done in one step by dividing by their highest common factor. Every digit after the decimal point counts, including zeros between the decimal point and the first non-zero number.

eg 0.75 has 2 digits after decimal point → 0.75 = 75/100 = 15/20 (divide by common factor 5) = 3/4 (divide by common factor 5); it can be done in one step by dividing by 25: 75/100 = 3/4 - divide by hcf(75, 100) = 25.

eg 0.02 also has 2 digits after the decimal point → 0.02 = 02/100 = 2/100 = 1/50.

2) If it is a repeating decimal, place the repeating digits over the same number of 9s as the number of repeating digits and simplify as above.

eg 0.242424... has 2 repeating digits (24) → 0.242424... = 24/99 = 8/33

eg 0.027027027... has 3 repeating digits (027) → 0.027027027... = 027/999 = 27/999 = 1/37

3) If it has a number of non-repeating digits before repeating digits, then a combination of 1 and 2 is used: put the non-repeating digits over a 1 followed by the same number of 0s as the number of digit, add the fraction formed by the repeating digits over the same number of 9s as the repeating digits followed by the same number of 0s as in the first fraction, and simplify the result.

eg 0.08333... has 2 digits (0.08) followed by 1 repeating digit (3) → 0.08333... = 08/100 + 3/900 = 72/900 + 3/900 = 75/900 = 1/12

4) Any non-terminating, non-repeating decimal is irrational and cannot be converted into a fraction.

eg √2 (= 1.41421...) cannot be represented as a fraction;

eg π (= 3.14159...; pi - the ratio of a circle's circumference to its diameter) cannot be represented as a fraction.