M47-153

Question

Adam, Ben, and Clara are hired to weed a garden at a community center. If Adam and Ben, working simultaneously and independently at their respective constant rates, can weed the garden in 6 hours, how long would Adam and Clara, working simultaneously and independently at their respective constant rates, take to weed the garden?

(1) Adam, Ben, and Clara, working simultaneously and independently at their respective constant rates, can weed the garden in 4 hours.
 
(2) Ben, working alone at his constant rate, would take 50% more time to weed the garden than Adam, working alone at her constant rate, would take.

Bunuel · Answer

Official Solution:

Adam, Ben, and Clara are hired to weed a garden at a community center. If Adam and Ben, working simultaneously and independently at their respective constant rates, can weed the garden in 6 hours, how long would Adam and Clara, working simultaneously and independently at their respective constant rates, take to weed the garden? 
 
Let  A ,  B , and  C  be the rates of Adam, Ben, and Clara, respectively, in gardens per hour.
 
We are given that Adam and Ben together can weed the garden in 6 hours, so:
 
 A + B = \frac{1}{6} 
 
The question asks for the time Adam and Clara would take together, so we need  A + C .
 
(1) Adam, Ben, and Clara together can weed the garden in 4 hours.
 
So:
 
 A + B + C = \frac{1}{4} 
 
Since  A + B = \frac{1}{6} :
 
 C = \frac{1}{4} - \frac{1}{6} = \frac{1}{12} 
 
But we still do not know  A , so we cannot determine  A + C .
 
Not sufficient.
 
(2) Ben would take 50% more time than Adam to weed the garden.
 
So Ben’s time is  \frac{3}{2}  of Adam’s time. Since rate is the reciprocal of time, Ben’s rate is  \frac{2}{3}  of Adam’s rate:
 
 B = \frac{2A}{3} 
 
Since:
 
 A + B = \frac{1}{6} 
 
we can determine  A  and  B . However, (2) gives no information about Clara’s rate, so we cannot determine  A + C .
 
Not sufficient.
 
(1)+(2):
 
From statement (1):  C = \frac{1}{12} 
 
From statement (2):  B = \frac{2A}{3} 
 
Since  A + B = \frac{1}{6} :
 
 A + \frac{2A}{3} = \frac{1}{6} 
 
 A = \frac{1}{10} 
 
Therefore:
 
 A + C = \frac{1}{10} + \frac{1}{12} = \frac{11}{60} 
 
So Adam and Clara together would take:
 
 \frac{60}{11}  hours
 
Sufficient.
 
Answer: C

M47-153

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