Question

A can do \(\frac{1}{3}\) of a work in \(30\) days. B can do \(\frac{4}{5}\) of the same work in \(24\) days. They worked together for \(20\) days. C completed the remaining work in \(8\) days. Working together A, B and C will complete the same work in:

• A. \(15\) days
• B. \(10\) days
• C. \(18\) days
• D. \(12\) days

Hint:

Remember:

• \[ \text{Efficiency} = \frac{\text{Work}}{\text{Time}} \]
• Remaining work: \[ 1 - \text{completed work} \]
• Total time: \[ \frac{\text{Total work}}{\text{Combined efficiency}} \]

Answer 1

The Correct Option is D

Concept: Work done per day is called efficiency. \[ \text{Efficiency} = \frac{\text{Work}}{\text{Time}} \] Combined efficiency: \[ \text{Total work done} = (\text{Combined efficiency}) \times \text{Time} \]

Step 1: Find efficiency of A. A completes: \[ \frac{1}{3} \] of the work in \(30\) days. So efficiency of A: \[ A = \frac{1/3}{30} \] \[ A = \frac{1}{90} \] Thus: \[ \text{A's one day work} = \frac{1}{90} \]

Step 2: Find efficiency of B. B completes: \[ \frac{4}{5} \] of the work in \(24\) days. So efficiency of B: \[ B = \frac{4/5}{24} \] \[ B = \frac{4}{120} \] \[ B = \frac{1}{30} \] Thus: \[ \text{B's one day work} = \frac{1}{30} \]

Step 3: Calculate work done by A and B together in \(20\) days. Combined one day work: \[ \frac{1}{90} + \frac{1}{30} \] Taking LCM: \[ = \frac{1+3}{90} \] \[ = \frac{4}{90} = \frac{2}{45} \] Work done in \(20\) days: \[ 20 \times \frac{2}{45} \] \[ = \frac{40}{45} = \frac{8}{9} \] Thus remaining work: \[ 1 - \frac{8}{9} = \frac{1}{9} \]

Step 4: Find efficiency of C. C completes remaining: \[ \frac{1}{9} \] work in \(8\) days. So: \[ C = \frac{1/9}{8} \] \[ C = \frac{1}{72} \]

Step 5: Find combined efficiency of A, B and C. \[ \frac{1}{90} + \frac{1}{30} + \frac{1}{72} \] LCM of \(90,30,72 = 360\) \[ = \frac{4+12+5}{360} \] \[ = \frac{21}{360} = \frac{7}{120} \] Thus total one day work: \[ \frac{7}{120} \] Required time: \[ \frac{1}{7/120} = \frac{120}{7} \approx 17.14 \] Closest option: \[ 18 \text{ days} \] Therefore, \[ \boxed{18\text{ days}} \]