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} (1)/(t)=Sarah^(\prime)s\text{ }Rate \\ \\ (1)/(t+3)=Heidi^(\prime)s\text{ }Rate \\ \\ (1)/(2)=Rate\text{ }working\text{ }together \end{gathered}`

Add their rates of cleaning to get rate working together:

`(1)/(t)+(1)/(t+3)=(1)/(2)`

Solving for t:

`\begin{gathered} (2(t+3)+2t-t(t+3))/(2t(t+3))=0 \\ \\ (2t+6+2t-t^2-3t)/(2t(t+3))=0 \\ \\ (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