Question

A group of college students are volunteering for Help the Homeless during their spring break. They are putting the finishing touches on a house they built. Working alone, Irina can paint a certain room in $9$ hours. Paulo can paint the same room in $8$ hours. Write an equation that can be used to find how long it will take them to work together to paint the room. How many hours will it take them to paint the room? If necessary, round your answer to the nearest hundredth.

Answer 1

Find the number of hours will it take Irina and Paulo to paint the room.

Given that,

Working alone, Irina can paint a certain room in $9$ hours.

Paulo can paint the same room in $8$ hours.

Step 1: Find the number of hours will it take Irina and Paulo to paint the room.

Let, $\mathrm{t}=$time required when they work together

Let the completed job $=1$

Irina can paint a certain room in $9$ hours.

So work of Irina in $1$ hour be$=\frac{1}{9}$

Paulo can paint a certain room in $8$ hours.

So work of Paulo in $1$ hour be$=\frac{1}{8}$

If they work together to paint the room, each will do a fraction of the job, the two fractions add up to $1$.

$\Rightarrow \frac{\mathrm{t}}{9}+\frac{\mathrm{t}}{8}=1$

Step 2: Simplify the expression.

Write each expression with a common denominator, by multiplying each by an appropriate factor of ${\displaystyle 1}$.

The common denominator is $72$, so multiply it with both terms of $\mathrm{t}$:

$\begin{array}{rcl}\Rightarrow \mathrm{}\frac{\mathrm{t}\times 8}{72}+\frac{\mathrm{t}\times 9}{72}& =& 1\\ & \Rightarrow & \frac{\mathrm{t}\times 8+\mathrm{t}\times 9}{72}=1\\ & \Rightarrow & \frac{8\mathrm{t}+9\mathrm{t}}{72}=1\\ & \Rightarrow & \frac{17\mathrm{t}}{72}=1\end{array}$

Solve for the value of $\mathrm{t}$:

$\begin{array}{rcl}& \Rightarrow & \frac{17\mathrm{t}}{72}=1\\ & \Rightarrow & 17\mathrm{t}=72\\ & \Rightarrow & \mathrm{t}=\frac{72}{17}\\ & \Rightarrow & \mathrm{t}=4.23529411..\\ & \Rightarrow & \mathrm{t}\approx 4.24\end{array}$

Hence, the number of hours that it takes Irina and Paulo to paint the room is $\mathbf{4}\mathbf{.}\mathbf{24}\mathbf{}\mathbf{hrs}.$