# It takes Bob twice as long as Caitlyn to clean his room. It takes Andrea 10 minutes longer than Caitlyn to clean her room. In total they work 90 minutes to clean their rooms. How long does it take Bob to clean his room?

Sep 10, 2015

It takes Bob $\text{40 minutes}$ to clean his room.

#### Explanation:

You will need to use the information provided to you to write three equations with three unknowns.

Let's say that Bob takes $b$ minutes to clean his room, Andrea takes $a$ minutes, and Caitlyn takes $c$ minutes.

The first piece of information given to you tells you that Bob needs twice as much time as Caitlyn to clean his room. This means that you can write

$b = 2 \cdot c$

Next, you were told that Andrea only takes 10 minutes longer than Caitlyn, which means that you can write

$a = c + 10$

Finally, if you add the time it took all three to clean their rooms, you get

$a + b + c = 90$

Use the value of $b$ from the first equation and the value of $a$ from the second equation to write

${\underbrace{c + 10}}_{\textcolor{b l u e}{= a}} + {\overbrace{2 c}}^{\textcolor{red}{= b}} + c = 90$

This is equivalent to

$4 c + 10 = 90$

$4 c = 80 \implies c = \frac{80}{4} = \textcolor{g r e e n}{\text{20 min}}$

To find out how long it takes Bob to clean his room, use the first equation

$b = 2 c = 2 \cdot \left(20\right) = \textcolor{g r e e n}{\text{40 min}}$

Andrea will clean her room in

$a = c + 10 = 20 + 10 = \textcolor{g r e e n}{\text{30 min}}$