Question

Cindy and Pam, working together, can clean the house in 10 hours. Working alone, Pam takes five times as long as Cindy. How long does it take Cindy to clean the house

Answer 1

c = Hours it take Cindy by herself

p =Hours it take Pam by herself

now, we know if they work together, they can finish in 10 hours, so let's take a peek at their rate.

whilst working together

in 1 hr Cindy has done 1/c "th" of the whole work

in 1 hr Pam has done 1/p "th" of the whole work

and since we know the whole work's time is 10 hours, in 1 hour they both have done 1/10 of whole lot, so

$\stackrel{\textit{Cindy's rate}}{\cfrac{1}{c}}+\stackrel{\textit{Pam's rate}}{\cfrac{1}{p}}=\stackrel{\textit{fraction of total work}}{\cfrac{1}{10}}~\hfill \stackrel{\textit{since Pam takes 5 times longer}}{p=5c} \\\\\\ \cfrac{1}{c}+\cfrac{1}{5c}=\cfrac{1}{10}\implies \stackrel{\textit{multiplying both sides by }\stackrel{LCD}{10c}}{10c\left( \cfrac{1}{c}+\cfrac{1}{5c} \right)=10c\left( \cfrac{1}{10} \right)}\implies 10+2=c\implies \stackrel{hours}{12=c}$