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`