algebra word problems

Question

Andy and Brian can mow their grandparents lawn in 18 minutes Brian can mow the lawn by himself in 15 minutes more than it would take Andy How long would it take each person to mow the lawn alone

Answer 1

Let x = time it takes Andy to mow the lawn alone. Then x + 15 = time it takes Brian to mow the lawn alone.

 

Since it takes Andy x minutes to mow the lawn, he mows at a rate of 1/x of the lawn per minute. Likewise, Brian mows at a rate of 1/(x+15) of the lawn per minute. Together, their rate is 1/18 of the lawn per minute (since they take 18 minutes to mow the entire lawn). So, their separate rates per minute, when combined, should equal 1/18 of the lawn per minute:

 

1/x + 1/(x+15) = 1/18

(x+15)/((x)(x+15)) + x/((x)(x+15)) = 1/18     (find a common denominator for the left-hand side)

(x + 15 + x)/((x)(x+15)) = 1/18     (combine the fractions on the left-hand side)

(2x+15)/(x²+15x) = 1/18     (combine like terms and distribute)

18(2x+15) = x² + 15x     (cross-multiply)

36x + 270 = x² + 15x

0 = x² - 21x - 270

0 = (x + 9)(x - 30)

x + 9 = 0    OR    x - 30 = 0

x = -9    OR    x = 30

 

Since Andy can't mow a negative amount of the lawn, he mows the lawn in x = 30 minutes. Brian therefore mows the lawn in x+15 = 30+15 = 45 minutes.