Rational Aplication

Question

Two pumps of different sizes are available to empty a flooded basement. Working together, the two pumps could drain the basement in 3 hours. Working alone, the small pump would take 8 hours longer to drain than the large pump. How long would it take to drain the basement if only the large pump was used?



Answer 1

The small pump does 3 / 8 of the job

The large pump does 5 / 8 of the job in 3 hours

The large pump would nee (8 / 5)(3) or 4.8 hours working alone

Another way

(3 / 8) + (3 / t) = 1, t represents time for large pump

3t + 24 = 8t

24 = 5t

4.8 = t

Answer 2

write what happens in 1 hour

if together they do it all in 3 hours, then in one hour they do 1/3 of the draining

alone the Larger does it in L hours, and in one hour does 1/L of the draining

alone the Smaller does it in S hours, and in one hour does 1/S = 1/(8+L) of the draining.

the Smaller takes 8 hours longer than the Larger pipe: S=8+L

1/S + 1/L = 1/3

1/(8+L) + 1/L = 1/3

multiply by 3(8+L)L to eliminate the fractions

3L + 3(8+L) = (8+L)L

3L + 24+3L = 8L + L^2

L^2 +2L -24 = 0

(L+6)(L-4) = 0

L = 4 (ignore the negative solution)

L = 4 hours for the Larger to drain alone

S=4+8 = 12 hours for the Smaller to drain alone

check the answers

1/12 + 1/4 = 1/3

1/12+3/12=4/12

(1+3)/12 = 4/12

the negative solution L=-6, S=-6+8=2

also works, but makes no sense, as a practical matter

1/2 + 1/-6 = 3/6-1/3 = 2/6 = 1/3