Question

How do you convert $ \dfrac{3}{8} $ to a decimal?

Answer 1

Hint: There is more than one way to do this. The simplest would be to divide the number 3 by 8 and place the decimal point at the appropriate place. A slightly complicated, but faster, way to do it would be to multiply both the numerator and the denominator with a number so that the denominator becomes a power of 10, and then simply move the decimal places in the numerator according to the number of 0s in the denominator.

Complete step-by-step answer:
Let us convert the denominator of $ \dfrac{3}{8} $ into a power of 10 by multiplying it with some number.
We observe that 8 = 2 × 2 × 2 and 10 = 2 × 5. Therefore, if we multiply each 2 with a 5, we will get a power of number 10. The multiplier is, therefore, 5 × 5 × 5 = 125.
Now, $ \dfrac{3}{8} $ = $ \dfrac{3}{8} $ × $ \dfrac{125}{125} $ = $ \dfrac{375}{1000} $ = 0.375, which is the required answer.

Note: Dividing by $ {{10}^{n}} $ moves the decimal point n places towards the left and multiplying by $ {{10}^{n}} $ moves the decimal point n places to the right.
In the decimal (base 10) system, the numbers whose denominators have only 2s and 5s as their factors in the lowest forms, are terminating, or else they repeat. e.g. $ \dfrac{7}{20} $ = 0.35, $ \dfrac{5}{11} $ = $ 0.\overline{45} $ etc. This is because 10 is a multiple of 2 and 5 and thus it is easy to convert any number with 2s and 5s as its factors into a power of 10 by multiplying some more 2s and 5s.
For numbers which don't have a denominator which is a multiple of 10, the pattern is: $ \dfrac{x}{9} $ = $ 0.\overline{x} $ , $ \dfrac{xy}{99} $ = $ 0.\overline{xy} $ and so on. So, every such denominator can be multiplied with a sum number to give a string of 9s. For example: $ \dfrac{4}{7} $ = $ \dfrac{4}{7} $ × $ \dfrac{142857}{142857} $ = $ \dfrac{571428}{999999} $ = $ 0.\overline{571428} $ .