Question

What is 04.151515... into a fraction

Answer 1

Answer:

$\displaystyle 4.\overline{15} = \frac{137}{33}$.

Step-by-step 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:

$99\, x = 15$.

$\displaystyle x = \frac{15}{99} = \frac{5}{33}$.

Therefore, the original number would be:

$\displaystyle x + 4 = \frac{5}{33} = \frac{132 + 5}{33} = \frac{137}{33}$.