Question

Some please help
Harry can rake the leaves in the yard 8 hours faster than his little brother Jimmy can. If they work together, they can complete the job in 3 hours. Using complete sentences, explain each step in figuring out how to determine the time it would take Jimmy to complete the job on his own.

Answer 1

use rates, the rate a person is working is the reciprocal of the time

working together, they work at a rate of 1/3 job per hour

when two people work together their rates add together
---> 1/h + 1/j = 1/3
---> h = j - 8 (harry takes 8 hrs less than jimmy)

substitute and solve for j
\[\frac{1}{j-8} + \frac{1}{j} = \frac{1}{3}\]
combine fractions
\[\frac{j+(j-8)}{j(j-8)} = \frac{1}{3}\]
cross-multiply
\[j^2 -8j = 6j -24\]
\[j^2 -14j +24 = 0\]
Factor
\[(j-12)(j-2) = 0\]
j = 12

It takes jimmy 12 hours to rake the leaves