# Two friends are painting a living room. Ken can paint it in 6 hours working alone. If Barbie works alone, it will take her 8 hours. How long will it take working together?

Mar 17, 2018

Let,the total work is of $x$ amount.

So,ken does $x$ amount of work in $6 h r s$

So,in $1 h r$ he will do $\frac{x}{6}$ amount of work.

Now,Barbie does $x$ amount of work in $8 h r s$

So,in $1 h r$ she does $\frac{x}{8}$ amount of the work.

Let,after working $t h r s$ together the work will be finished.

So, in $t h r s$ Ken does $\frac{x t}{6}$ amount of work and Barbie does $\frac{x t}{8}$ amount of work.

Clearly, $\frac{x t}{6} + \frac{x t}{8} = x$

Or, $\frac{t}{6} + \frac{t}{8} = 1$

So, $t = 3.43 h r s$

Mar 17, 2018

Detailed solution given so that you can see where everything comes from.

$3 \text{ hours and "25 5/7" minutes"larr" Exact value}$

$3 \text{ hours and "26" minutes}$ to the nearest minute

#### Explanation:

People work at different rates. So the time taken by different people to complete a set amount of work will also be different. This is what we need to model

Let the total amount of work required to complete the task be $W$

Let Ken's work rate per hour be ${w}_{k}$
Let Barbie's work rate per hour be ${w}_{b}$
Let the total time working together be $t$

If Ken works on his own he can complete the whole task in 6 yours

$\text{work rate "xx" time = work done.} \ldots \ldots \ldots \ldots . E q u a t i o n \left(1\right)$

$\textcolor{w h i t e}{\text{ddd")w_kcolor(white)("dddd")xxcolor(white)("ddd")6color(white)("d")=color(white)("ddd}} W$

So ${w}_{k} = \frac{W}{6} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots E q u a t i o n \left(2\right)$

If Barbie works on her own she can complete the whole task in 8 hours

Using the above method

${w}_{b} = \frac{W}{8.} \ldots \ldots \ldots \ldots \ldots \ldots \ldots . . E q u a t i o n \left(3\right)$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Consider $E q n \left(1\right)$ but combine the two work rates $E q n \left(2\right) + E q n \left(3\right)$

$\textcolor{w h i t e}{\text{d") (w_bxxt) color(white)("d")+ color(white)("d}} \left({w}_{k} \times t\right) = W$

$\left(\frac{W}{8} \times t\right) + \left(\frac{W}{6} \times t\right) = W$

Factor out the $t$

$t \left(\frac{W}{8} + \frac{W}{6}\right) = W$

$t \left(\frac{3 W}{24} + \frac{4 W}{24}\right) = W$

$t \frac{7 W}{24} = W$

$t = \frac{24 \cancel{W}}{7 \cancel{W}}$

$t = \frac{24}{7} \text{ hours}$

$t = 3 \frac{3}{7} \text{ hours} \leftarrow$ Exact value

$t = 3 \text{ hours and } \left(\frac{3}{7} \times 60\right)$

$t = 3 \text{ hours and "25 5/7" minutes"larr" Exact value}$

Mar 17, 2018

$3 \frac{3}{7}$ hours or $3$ hours and $26$ minutes

#### Explanation:

First find out what fraction of the task they would each complete in $1$ hour.

Ken will finish $\frac{1}{6}$ of the task in $1$ hour.

Barbie will finish $\frac{1}{8}$ of the task in $1$ hour.

If they work together, in one hour they will finish:

$\frac{1}{6} + \frac{1}{8}$ of the painting task.

$= \frac{4 + 3}{24} = \frac{7}{24}$ is the fraction completed in one hour.

So to complete the whole task $\left(\frac{24}{24}\right)$ we need to divide:

$\frac{24}{24} \div \frac{7}{24}$

$= \frac{24}{24} \times \frac{24}{7}$

$= \frac{24}{7}$ hours.

This simplifies to $3 \frac{3}{7}$ hours

Which is easier given as $3$ hours and $\frac{3}{7} \times 60$ minutes

$= 3 \text{ hours } \mathmr{and} 25 \frac{5}{7}$ minutes

or $3 \text{ hours } \mathmr{and} 26$ minutes