Word Problem (Help Setting Up)

[desihatcher]

If it takes one pump 12 minutes to empty a pool, but only 8 minutes when a second pump is also used, how long would it take to empty the pool using only the second pump?

Im having trouble setting up the equation. We're using a "box" method in class, where you put the fraction of the job completed in the first box, "x" in the second box, and add the two while making it equal to one. (ie, 5/x + 7/x = 1) However, this format doesnt work here because were not adding the two. Any help would be greatly appreciated!!

 

[soroban]

Hello, desihatcher!

If it takes one pump 12 minutes to empty a pool,
but only 8 minutes when a second pump is also used,
how long would it take to empty the pool using only the second pump?

You can "talk" your way through this one.

Pump A takes 12 minutes to do the job.
In one minute, it does \(\displaystyle \tfrac{1}{12}\) of the job.
In 8 minutes, it does \(\displaystyle \tfrac{8}{12} \,=\,\tfrac{2}{3}\) of the job.

Pump B takes \(\displaystyle x\) minutes to do the job.
In one minute, it does \(\displaystyle \tfrac{1}{x}\) of the job.
In 8 minutes, it does \(\displaystyle \tfrac{8}{x}\) of the job.

Working together for 8 minutes, the two pumps will do the entire job.

There is our equation! . . \(\displaystyle \dfrac{2}{3} + \dfrac{8}{x} \:=\:1\)

 

[HallsofIvy]

If it takes one pump 12 minutes to empty a pool, but only 8 minutes when a second pump is also used, how long would it take to empty the pool using only the second pump?

Im having trouble setting up the equation. We're using a "box" method in class, where you put the fraction of the job completed in the first box, "x" in the second box, and add the two while making it equal to one. (ie, 5/x + 7/x = 1) However, this format doesnt work here because were not adding the two. Any help would be greatly appreciated!!

You are adding, just not the data you are given. If one pump take 12 minutes to empty the pool, then its rate is 1/12 "pool per minute". We are not told how long it would take the second pump, that is what we are asked- so call that "T" minutes. Then its rate is 1/T "pool per minute. The sum of those is the rate of the two pumps together and since we are told it takes "8 minutes when a second pump is also used" that rate is 1/8 "pool per minute". The equation is \(\displaystyle \frac{1}{12}+ \frac{1}{T}= \frac{1}{8}\). Solve that for T.

(I'm going to have to type faster!)

 

[HallsofIvy]

A nonsensical question- Mrs Halls can type much faster than Mr Halls!