Answer:
It would take Henry 3 hours to clean the office carpets if Mary were not there to help
Step-by-step 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 [tex]\frac{1}{t}[/tex]
∵ The rate of Mary is [tex]\frac{1}{t+3}[/tex]
∵ Working together, they can clean the carpets in 2 hours
- Add the two rates and equate the answer by [tex]\frac{1}{2}[/tex] (their rate together)
∴ [tex]\frac{1}{t}+\frac{1}{t+3}=\frac{1}{2}[/tex]
Multiply the two denominators and multiply each numerator by the other denominator to add the two fractions of the left hand side
∵ [tex]\frac{(1)(t+3)}{t(t+3)}+\frac{(1)t}{t(t+3)}=\frac{1}{2}[/tex]
∴ [tex]\frac{(t+3)}{t(t+3)}+\frac{t}{t(t+3)}=\frac{1}{2}[/tex]
- Add the two fractions in the left hand side
∴ [tex]\frac{t+3+t}{t(t+3)}=\frac{1}{2}[/tex]
∴ [tex]\frac{2t+3}{t^{2}+3t}=\frac{1}{2}[/tex]
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
