VJesus12 wrote: ↑Fri Dec 10, 2021 8:06 am
Hoses \(A, B,\) and \(C\) pump a swimming pool full of water. Hoses \(A\) and \(B\) working simultaneously can pump the pool full of water in \(4\) hours, and Pumps \(B\) and \(C\) working simultaneously can pump the pool full of water in \(6\) hours. How long does it take the pump \(A\) to work alone to fill the pool?
(1) All three hoses working simultaneously can fill the pool in \(3\) hours and \(36\) minutes.
(2) Hose \(A\) and Hose \(C\) working simultaneously can fill the swimming pool in twice the time it would take all three hoses together to fill the swimming pool.
Answer:
D
Source: Princeton Review
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Rule #1: If a person can complete an entire job in k hours, then in one hour, the person can complete 1/k of the job
Example: If it takes Sue 5 hours to complete a job, then in one hour, she can complete 1/5 of the job. In other words, her work rate is 1/5 of the job per hour
Rule #2: If a person completes a/b of the job in one hour, then it will take b/a hours to complete the entire job
Example: If Sam can complete 1/8 of the job
in one hour, then it will take him 8/1 hours to complete the job.
Likewise, if Joe can complete 2/3 of the job in one hour, then it will take him 3/2 hours to complete the job.
Let’s use these rules to solve the question. . . .
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Target question: How long does it take pump A working alone to fill the pool?
Given: Hoses A and B working simultaneously can pump the pool full of water in 4 hours, and Pumps B and C working simultaneously can pump the pool full of water in 6 hours.
Let A = the RATE at which hose A can fill the pool alone
Let B = the RATE at which hose B can fill the pool alone
Let C = the RATE at which hose C can fill the pool alone
Hoses A and B working simultaneously can pump the pool full of water in 4 hours
From
rule #1. the
combined RATE of hoses A and B is 1/4 of the pool PER HOUR
In other words,
A + B = 1/4
B and C working simultaneously can pump the pool full of water in 6 hours.
From
rule #1. the
combined RATE of hoses B and C is 1/6 of the pool PER HOUR
In other words,
B + C = 1/6
Statement 1: All three hoses working simultaneously can fill the pool in 3 hours and 36 minutes.
In other words, working together hoses A, B and C can fill the pool in 3.6 hours
From
rule #1. the
combined RATE of hoses A, B and C is 1/3.6 of the pool PER HOUR
In other words,
A + B + C = 1/3.6
At this point, we have the following system:
A + B = 1/4
B + C = 1/6
A + B + C = 1/3.6
Since we have 3 different equations with 3 variables, we can definitely solve the system to determine the value of A.
Of course, we're not going to waste our time and actually solve the system. We need only recognize that we COULD determine the value of A, which means we COULD determine
the time it would take pump A working alone to fill the pool
Since we can answer the
target question with certainty, statement 1 is SUFFICIENT
Statement 2: Hose A and Hose C working simultaneously can fill the swimming pool in twice the time it would take all three hoses together to fill the swimming pool.
This tells us that the combined RATE of hoses A and C is HALF the combined RATE of hoses A, B and C
So, we can write:
A + C = 0.5(A + B + C)
At this point, we have the following system:
A + B = 1/4
B + C = 1/6
A + C = 0.5(A + B + C)
Once again, we have 3 different equations with 3 variables, which means we can definitely solve the system to determine the value of A.
So, we COULD determine
the time it would take pump A working alone to fill the pool
Since we can answer the
target question with certainty, statement 2 is SUFFICIENT
Answer: D
