# Charles can shovel the snow off his driveway in 40 minutes. He shovels for 20 minutes and then is joined by Joaney. If they shovel the remaining snow in 10 minutes, how long would it have taken Joaney to shovel the driveway alone?

May 30, 2018

So Joaney would take 40 minutes to complete the task on their own.

Very full explanation given.

#### Explanation:

Let the amount of 'work' to complete the job be $W$ effort

Let the work rate of Charles be ${w}_{c}$
Let the work rate of Joaney be ${w}_{j}$
Let generic designation of time be $t$

Let time Charles works be ${t}_{c}$
Let the time Joaney works be ${t}_{j}$

$\textcolor{b l u e}{\text{Initial condition:}}$

We have the relationship

$\underbrace{\text{work efort") xx ubrace("time") = ubrace("total work done}}$
$\textcolor{w h i t e}{\text{ddd")w_c color(white)("ddd")xxcolor(white)("d.")t_c color(white)("d")=color(white)("ddddd}} W$

Set ${t}_{c} = 40 \text{ minutes}$

${w}_{c} \times {t}_{c} = W$
${w}_{c} \times 40 = W$
${w}_{c} = \frac{W}{40}$ effort per minute $\text{ } \ldots E q u a t i o n \left(1\right)$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$\textcolor{b l u e}{\text{Considering joint effort}}$

Charles works on his own for the initial time of 20 minutes giving:

${w}_{c} \times 20$

He is then joined by Joaney and they work together for 10 minutes more. So the above becomes

$\left[{w}_{c} \times \left(20 + 10\right)\right] + \left[{w}_{j} \times 10\right] = W \text{ } \ldots \ldots \ldots \ldots \ldots . E q u a t i o n \left(2\right)$

${t}_{c} = 30 \mathmr{and} {t}_{j} = 10$

Using $E q n \left(1\right)$ substitute for ${w}_{c}$ in $E q n \left(2\right)$

$\left[\frac{W}{40} \times 30\right] + \left[{w}_{j} \times 10\right] = W$

$\frac{{\cancel{30}}^{3} W}{\cancel{40}} ^ 4 + 10 {w}_{j} = W$

Make everything have a common denominator of 4

$\frac{3 W}{4} + \frac{40 {w}_{j}}{4} = \frac{4 W}{4}$

$3 W + 40 {w}_{j} = 4 W$

$40 {w}_{j} = W$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
So Joaney would take 40 minutes to complete the task on their own.