Question

Working together, Sarah and Heidi can clean the garage in 2 hours. If they work alone, it takes Heidi 3 hours longer than it takes Sarah. How long would it take Heidi to clean the garage alone?

Answer 1

Given the rates:

$\begin{gathered} \frac{1}{t}=Sarah^{\prime}s\text{ }Rate \\ \\ \frac{1}{t+3}=Heidi^{\prime}s\text{ }Rate \\ \\ \frac{1}{2}=Rate\text{ }working\text{ }together \end{gathered}$

Add their rates of cleaning to get rate working together:

$\frac{1}{t}+\frac{1}{t+3}=\frac{1}{2}$

Solving for t:

$\begin{gathered} \frac{2(t+3)+2t-t(t+3)}{2t(t+3)}=0 \\ \\ \frac{2t+6+2t-t^2-3t}{2t(t+3)}=0 \\ \\ \frac{t+6-t^2}{2t(t+3)}=0 \\ \\ -t^2+t+6=0 \\ \\ (t+2)(t-3)=0 \end{gathered}$

Hence:

t = -2

t = 3

Time can't be negative; then:

Heidi's time: t + 3

3 + 3 = 9

ANSWER

It will take Heidi 9 hrs to clean garage working alone