# Ashley can paint a 10 ft by 14 ft wall in 20 minutes Maddie can paint the same sized wall in 14 minutes, If Ashley and Maddie are both painting at the same time, how long will it take them together to paint the same sized wall?

Apr 17, 2018

Combined, it should take them 8.2353 minutes to paint the 10'x14' wall.

#### Explanation:

What we need to do here is determine Ashley and Maddie's painting rates, before combining and evaluating. If we call Ashley's painting rate $A$ and Maddie's $M$, the equations would look like this:

$A = \frac{10 f t \cdot 14 f t}{20 \min} = \frac{140 f {t}^{2}}{20 \min}$

$A = 7 \frac{f {t}^{2}}{\min}$

$M = \frac{140 f {t}^{2}}{14 \min}$

$M = 10 \frac{f {t}^{2}}{\min}$

Now that they're working together, we can determine the amount of time for the 10'x14' wall:

$\left(A + M\right) \cdot t = 140 f {t}^{2}$

$\left(7 \frac{f {t}^{2}}{\min} + 10 \frac{f {t}^{2}}{\min}\right) \cdot t = 140 f {t}^{2}$

$17 \frac{f {t}^{2}}{\min} \cdot t = 140 f {t}^{2}$

$t = \frac{140 f {t}^{2}}{17 \frac{f {t}^{2}}{\min}} = 140 \cancel{f {t}^{2}} \cdot \frac{\min}{17 \cancel{f {t}^{2}}}$

color(green)(t=140/17 min~=8.2353 min

Apr 17, 2018

approximately $8.235 \text{ minutes}$ to 3 decimal places
$8 \frac{4}{17} \text{ minutes exactly}$

#### Explanation:

The given wall area is $10 f t \times 14 f t = 140 f {t}^{2}$

$\textcolor{b r o w n}{\text{Consider Ashley}}$

He paints it in 20 minutes so their rate of painting for each minute is: $\frac{140}{20} \frac{f {t}^{2}}{m} = 7 \frac{f {t}^{2}}{m}$

$\textcolor{b r o w n}{\text{Consider Maddie}}$
She paints it in 14 minutes so their rate of painting for each minute is: $\frac{140}{14} \frac{f {t}^{2}}{m} = 10 \frac{f {t}^{2}}{m}$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b r o w n}{\text{Consider them working together}}$

The combined rate is $7 + 10 = 17 \frac{f {t}^{2}}{m}$

Thus the working together time is:

$\textcolor{red}{140 f {t}^{2}} \textcolor{g r e e n}{\div 17 \frac{f {t}^{2}}{m}}$

color(red)(140)/color(green)(17) color(red)(cancel(ft^2))xxcolor(green)(m/(cancel(ft^2)) = 8 4/17 "minutes"

approximately $8.235 \text{ minutes}$ to 3 decimal places