Question

When working individually, A and B take 6 days and 8 days to paint a fence. They started working together. After 2 days, C also joined them. If the work was completed after 1 more day (3 days in total), in how many days can C paint the entire fence alone?

• A. 3 days
• B. \( \frac{10}{3} \) days
• C. 4 days
• D. \( \frac{13}{3} \) days
• E. 8 days

Hint:

When multiple workers are involved, their combined work rate can be found by adding their individual rates.

Answer 1

The Correct Option is D

Let the total work required to paint the fence be 1 unit. Step 1: Work rates: - A takes 6 days to paint the fence, so A's rate is \( \frac{1}{6} \) of the work per day. - B takes 8 days to paint the fence, so B's rate is \( \frac{1}{8} \) of the work per day. Step 2: In the first 2 days, A and B work together: \[ \text{Work done by A and B in 2 days} = 2 \times \left( \frac{1}{6} + \frac{1}{8} \right) = 2 \times \frac{7}{24} = \frac{7}{12} \] Step 3: After 2 days, the remaining work is \( 1 - \frac{7}{12} = \frac{5}{12} \). Step 4: C joins A and B. Let C's rate be \( \frac{1}{c} \), where \( c \) is the number of days it takes C to paint the entire fence alone. Together, A, B, and C's rate is \( \frac{1}{6} + \frac{1}{8} + \frac{1}{c} \). Step 5: In the next 1 day, A, B, and C complete the remaining \( \frac{5}{12} \) of the work: \[ \left( \frac{1}{6} + \frac{1}{8} + \frac{1}{c} \right) \times 1 = \frac{5}{12} \] \[ \frac{7}{24} + \frac{1}{c} = \frac{5}{12} \] \[ \frac{1}{c} = \frac{5}{12} - \frac{7}{24} = \frac{10}{24} - \frac{7}{24} = \frac{3}{24} = \frac{1}{8} \] Step 6: Thus, \( c = 8 \). Therefore, C can paint the entire fence in 8 days.