# Sue did a job for $120. It took her 2 hours longer than she expected, and therefore she earned$2 per hour less than she anticipated.. How long did she expect that it would take to do the job?

Oct 22, 2015

Expected time to complete job $= 10$ hours

#### Explanation:

Let
$\textcolor{w h i t e}{\text{XXX}} {t}_{x} =$ expected time required
$\textcolor{w h i t e}{\text{XXX}} {t}_{a} =$ actual time required
$\textcolor{w h i t e}{\text{XXX}} {r}_{x} =$ expected rate of income
$\textcolor{w h i t e}{\text{XXX}} {r}_{a} =$ actual rate of income

We are told
$\textcolor{w h i t e}{\text{XXX}} {t}_{a} = {t}_{x} + 2$
$\textcolor{w h i t e}{\text{XXX}} {r}_{a} = {r}_{x} - 2$

${r}_{x} = \frac{120}{t} _ x$ and ${r}_{a} = \frac{120}{t} _ a = \frac{120}{{t}_{x} + 2}$

therefore
$\textcolor{w h i t e}{\text{XXX}} \frac{120}{{t}_{x} + 2} = \frac{120}{t} _ x - 2$

simplifying
$\textcolor{w h i t e}{\text{XXX}} 120 = \frac{120 \left({t}_{x} + 2\right)}{{t}_{x}} - 2 \left({t}_{x} + 2\right)$

$\textcolor{w h i t e}{\text{XXX}} \cancel{120 {t}_{x}} = \cancel{120 {t}_{x}} + 240 - 2 {t}_{x}^{2} - 2 {t}_{x}$

$\textcolor{w h i t e}{\text{XXX}} {t}_{x}^{2} + 2 {t}_{x} - 120 = 0$

this factors as
$\textcolor{w h i t e}{\text{XXX}} \left({t}_{x} + 12\right) \left({t}_{x} - 10\right) = 0$

and since ${t}_{x} > 0$
$\Rightarrow {t}_{x} = 10$