Question

What is 04.151515... into a fraction

Answer 1

`\displaystyle 4.\overline{15} = (137)/(33)`.

Explanation:

Start by separating this decimal number into its integer part and its fraction part:

`4.151515\cdots = 4 + 0.151515\cdots`

The most challenging task here is to express
`0.151515\cdots` as a proper fraction. Once that fraction is found, expressing the original number
`4.151515\cdots` will be as simple as rewriting a mixed number as an improper fraction.

Let
`x = 0.151515\cdots`.
`(x + 4)` would then represent the original number.

Note that the repeating digits appear in groups of two. Therefore, if the digits in
`x` are shifted to the left by two places, the repeating part will continue to match:

`\begin{aligned}x = 0.&151515\cdots && \\ 100\, x = 15.& 151515\cdots \end{aligned}`.

Note, that this "shifting" is as simple as multiplying the initial number by
`100` (same as
`10` raised to the power of the number of digits that needs to be shifted.)

Subtract the original number from the shifted number to eliminate the fraction part completely:

`\begin{aligned}&(100\, x) - x \\ &= 15.151515\cdots\\ & \phantom{=}- 0.151515\cdots\\&=15 \end{aligned}`.

In other words:

`99\, x = 15`.

`\displaystyle x = (15)/(99) = (5)/(33)`.

Therefore, the original number would be:

`\displaystyle x + 4 = (5)/(33) = (132 + 5)/(33) = (137)/(33)`.