Question

Let $x=(0.\overline{6})(0.\overline{60})$. When $x$ is written out as a decimal, what is the sum of the first $15$ digits after the decimal point?

Answer 1

`32\text{ for }x=0.\overline{40}`

Explanation:

It looks like you want the sum of digits of the product ...

`x = (0.\overline{6})(0.\overline{60})=(2)/(3)\cdot(20)/(33)=(40)/(99)=0.\overline{40}`

Since the decimal fraction starts off with 4 and has an even number of repeating digits, there will be 8 4's and 7 0's in the sum of the first 15 decimal digits:

sum = 8×4 +7×0 = 32

The sum of the first 15 digits after the decimal point is 32.