Question 65c13

Apr 14, 2017

$\frac{133}{99}$

Explanation:

Recall (Sum of Geometric Series Formula):

$a + a r + a {r}^{2} + a {r}^{3} + \cdots = \frac{a}{1 - r}$ if $| r | < 1$

We can view a repeated decimal as the sum of a geometric series.

$1.343434 \ldots = 1 + \left[0.34 + 0.0034 + 0.000034 + \cdots\right]$

$= 1 + \left[\frac{34}{100} + \frac{34}{100} \left(\frac{1}{100}\right) + \frac{34}{100} {\left(\frac{1}{100}\right)}^{2} + \cdots\right]$

By applying the formula above with $a = \frac{34}{100}$ and $r = \frac{1}{100}$ starting with the second term,

=1+(34/100)/(1-1/100) =1+(34/100)/(99/100) =99/99+34/99=133/99#

Apr 14, 2017

$\frac{133}{99}$

Explanation:

Obtain 2 equations with the same repeating part then subtract them.

$\text{We can represent the repeated part by } 1. \overline{34}$

$x = 1. \overline{34} \to \left(1\right) \leftarrow \textcolor{red}{\text{ multiply by 100}}$

$100 x = 134. \overline{34} \to \left(2\right)$

$\text{Subtract } \left(2\right) - \left(1\right)$

$\left(100 x - x\right) = \left(134. \overline{34} - 1. \overline{34}\right)$

$\Rightarrow 99 x = 133 \leftarrow \textcolor{red}{\text{ repeating part eliminated}}$

$\Rightarrow x = \frac{133}{99} \leftarrow \textcolor{red}{\text{in simplest form}}$