Question

Working alone, Henry takes 3 hours less than Mary to clean the carpets in the entire office. Working together, they can clean the carpets in 2 hours. How long would it take Henry to clean the office carpets if Mary were not there to help?

Answer 1

It would take Henry 3 hours to clean the office carpets if Mary were not there to help

Explanation:

Assume that it takes Henry t hours to clean the carpets

∵ Henry takes 3 hours less than Mary to clean the carpets

∵ Henry takes t hours

- That means add 3 to t to find Mary time

∴ Mary takes t + 3 hours

∵ The rate of Henry is
`(1)/(t)`

∵ The rate of Mary is
`(1)/(t+3)`

∵ Working together, they can clean the carpets in 2 hours

- Add the two rates and equate the answer by
`(1)/(2)` (their rate together)

∴
`(1)/(t)+(1)/(t+3)=(1)/(2)`

Multiply the two denominators and multiply each numerator by the other denominator to add the two fractions of the left hand side

∵
`((1)(t+3))/(t(t+3))+((1)t)/(t(t+3))=(1)/(2)`

∴
`((t+3))/(t(t+3))+(t)/(t(t+3))=(1)/(2)`

- Add the two fractions in the left hand side

∴
`(t+3+t)/(t(t+3))=(1)/(2)`

∴
`(2t+3)/(t^(2)+3t)=(1)/(2)`

Use the cross multiplication

∵ (t² + 3t)(1) = 2(2t + 3)

∴ t² + 3t = 4t + 6

- Subtract 4t from both sides

∴ t² - t = 6

- Subtract 6 from both sides

∴ t² - t - 6 = 0

- Factorize it into two factors

∴ (t - 3)(t + 2) = 0

Equate each factor by 0 to find t

∵ t - 3 = 0

- Add 3 to both sides

∴ t = 3

OR

∵ t + 2 = 0

- Subtract 2 from both sides

∴ t = -2 ⇒ reject this answer because there is no - ve time

∴ t = 3

∴ Henry would take 3 hours to clean the carpets alone

It would take Henry 3 hours to clean the office carpets if Mary were not there to help