SOLUTION: It takes pump A 2 hours less time than pump B to empty a pool. Pump A is started at 8:00am and pump B at 10:00am. At 12:00pm 60% of the pool.is empty when pump B broke down. How mu

Question 1177723: It takes pump A 2 hours less time than pump B to empty a pool. Pump A is started at 8:00am and pump B at 10:00am. At 12:00pm 60% of the pool.is empty when pump B broke down. How much time after 12:00pm would it take pump A to empty the pool?
Found 3 solutions by Theo, greenestamps, mananth:
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
rate * time = quantity

let quantity = 1 empty pool.
let t equal the time it takes pump B to empty the pool.
let t-2 equal the time it takes pump A to empty the pool.
let a equal the rate that pump A empties the pool.
let b equal the rate that pump B empties the pool.

your equations are:
a * (t-2) = 1 for pump A.
b * t = 1 for pump B.

solve for a in the first equation to get a = 1/(t-2).
solve for b in the second equation to get b = 1/t.

pump A starts at 8:00 am and pump B starts at 10:00 am.
at 12:00 pm, they have emptied 60% of the pool.
pump B breaks down at 12:00 pm.
pump A worked from 8:00 am to 12:00 pm.
pump B worked from 10:00 am to 12:00 pm.
pump A worked for 4 hours.
pump B worked for 2 hours.

when pump A worked for 4 hours and pump B worked for 2 hours, they emptied 60% of the pool.

the equation for that is 4*a + 2*b = .6.

since a = 1/(t-2) and b = 1/t, replace a and b in the equation with their equivalent expressions in terms of t to get 4/(t-2) + 2/t = .6

multiply both sides of this equation by t(t-2) to get 4t + 2(t-2) = .6t(t-2)

simplify to get 4t + 2t - 4 = .6t^2 - 1.2t.

combine like terms to get 6t - 4 = .6t^2 - 1.2t.

subtract 6t from both sides of the equation and add 4 to both sides of the equation to get 0 = .6t^2 - 1.2t - 6t + 4.

combine like terms to get 0 = .6t^2 - 7.2t + 4

factor this quadratic equation to get t = 11.416025603091 or t = 0.58397439690936.

t = 0.58397439690936 won't work because then t - 2 would be negative.

you get t = 11.416025603091.

since a = 1/(t-2), then a = .1062019202.

since b = 1/t, then b = .0875961595.

to confirm a and b values are good, replace a and b in the equation of 4a + 2b = .6 to get .6 = .6, which is true, confirming the values are good.

you now have:

t = 11.416025603091
a = .1062019202
b = .0875961595

since .6 of the pool has been filled, then .4 of the pool remains to be filled by pump A.

let r = the time for pump A to fill .4 of the pool.
you get a * r = .4
solve for r to get r = .4/a
this becomes r = .4/.1062019202
solve for r to get r = 3.766410241 hours.

you have the final result for filling the pool as>
4 hours for pump A plus 2 hours for pump B plus another 3.766410241 hours for pump A to fill the pool.
this results in 7.766410241 for pump A and 2 hours for pump B to fill the pool.
since pump A works at a rate of .1062019202 of the pool in 1 hour and pump B works at a rate of .0875961595 of the pool in 1 hour, then:
7.766410241 * .1062019202 + 2 * .0875961595 = 1
this confirms the calculations are correct.

your answer is that it would take pump A 3.766410241 hours to finish pumping the pool after pump B broke down.

that is if my assumptions and calculations are correct.
i think that they are, so i'll go with this answer until i'm proven wrong.
let me know if you're good with this answer, or if another answer was expected.

Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!

let x = # hours A takes alone to empty the pool
then x+2 = # hours B takes alone

1/x = fraction of job A does in 1 hour
1(x+2) = fraction of job B does in 1 hour

At 12pm when pump B broke down, A has been working 4 hours and B 2 hours. The fraction of the job done at that time is

That fraction of the job done at 12pm is 60% = 3/5. Solve that equation to find out how many hours A takes to empty the pool alone.

Multiply through by the least common denominator

This does not factor; use the quadratic formula and of course use the positive root.

= 9.416 to 3 decimal places.

2/5 of the pool remains to be emptied; the number of hours it will take A to finish is

ANSWER: to a few decimal places, 3.7664 hours

Answer by mananth(16946)   (Show Source): You can put this solution on YOUR website!
Let pump B to empty a pool be x hours
It empties 1/x in 1 hour
pump A to empty a pool will be x-2 hours
It empties 1/(x-2) in 1 hour
Pump B runs for two hours and both work for two hours
They empty 40 % of the tank

factorise
Pump B takes 11 hours
Pump A takes 9 hours
Pump A has to empty remaining water (3/5) of the tank
Pump A will take 3/5 * 9 = 27/5 = 5.4 hours after 12.00
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