Twin brothers, bully and bobby

Question

Twin brothers, Billy and Bobby, can mow their grandparents lawn together in 56 minutes. Billy could mow the lawn by himself in 15 minutes less time that it would take Bobby. How long would it take Bobby to mow the lawn himself?

Answer 1

Bobby can mow the yard in x minutes.

Billy can mow the yard in x-15 minutes

1/x + 1/(x-15) = 1/56

Multiplies by 56x(x-15)

56(x-15) + 56x = x(x-15)

56x - 840 + 56x = x^2 - 15x

112x - 840 = x^2 - 15x

x^2 -15x -112x + 840 = 0

x^2 -127x + 840 = 0

( x - 120 )(x - 7 ) = 0

x = 120 or x = 7

x=7 is disqualified because it leads to negative measures

x=120 --> Bobby mows the lawn in 120 minutes

Billy mows the lawn in 105 minutes

Check: adding their workrates...

1/120 + 1/105 = 7/840 + 8/840 = 15/840 is the combined workrate

Working together they finish is 840/15 = 168/3 = 56 minutes