Question

write the following number as ratios of integers

Answer 1

Our goal here is to somehow "surgically remove" the repeating part of the number, so let's start by putting the original value in a variable and messing around with it a bit.

We'll let $x=7.\overline{6}$. We want to cut the $0.\overline{6}$ bit off completely, so let's create the scalpel that'll let us do that. If $x=7.\overline{6}$, then we can also say that $10x=76.\overline{6}$. Maybe I was lying a bit: the $x$ is our real scalpel here, and $10x$ is where we'll be making the cut. Mathematically, a "cut" is almost always shorthand for subtraction, so let's see what our operation (cutting $x$ off of $10x$) leaves us with:

$10x-x=76.\overline{6}-7.\overline{6}\\9x=69$

The operation was a success! We can now simply divide either side by 9 to find $x = 69/9$, which, when reduced by dividing the numerator and denominator by 3, gives us $x=23/3$