Question 1177723
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.