Question 1

The work efficiency of A is 3 times that of B. If A and B together complete a project in 9 days, how many days will B alone take to complete the project?

Accepted Answer

Step 1: Let B's efficiency = x work per day. Then A's efficiency = 3x work per day. Step 2: Combined efficiency = 3x + x = 4x work per day. Step 3: Since they complete work in 9 days: 4x × 9 = 1 (total work), so 4x = 1/9, thus x = 1/36. Step 4: B's efficiency = 1/36 work per day. Step 5: Time for B alone = 1 ÷ (1/36) = 36 days. Therefore, answer is 36 days.

Question 2

A's efficiency is twice that of B. If A and B together can complete a project in 9 days, how many days will B alone take to complete it?

Accepted Answer

Step 1: Let B's rate = 1/x per day. Then A's rate = 2/x per day (twice efficient). Step 2: Combined rate = 1/x + 2/x = 3/x per day. Step 3: They complete in 9 days, so 3/x = 1/9. Step 4: Solving: 3 × 9 = x, so x = 27 days.

Question 3

A contractor hires workers to complete a bridge in 20 days. After 5 days, he realizes the work is only 15% complete. He then increases the workforce by 60%. How many additional days beyond the original 20 days will be needed to finish the project?

Accepted Answer

Step 1: After 5 days, 15% complete → original rate = 15/5 = 3% per day. Step 2: Remaining work = 85%. Step 3: New workforce = 1.6 × original → new rate = 1.6 × 3% = 4.8% per day. Step 4: Days for remaining 85% = 85 ÷ 4.8 = 17.708 days ≈ 17.71 days. Step 5: Total time = 5 + 17.71 = 22.71 ≈ 23 days. Step 6: Additional days = 23 - 20 = 3 days.

Question 4

If 8 workers can build a wall in 15 days, how many days will 12 workers take to build the same wall, assuming all workers have equal efficiency?

Accepted Answer

Step 1: Total work = 8 workers × 15 days = 120 worker-days. Step 2: With 12 workers, time required = 120 ÷ 12 = 10 days.

Question 5

A's work efficiency is twice that of B. If A alone can complete a task in 9 days, how many days will B take to complete the same task?

Accepted Answer

Step 1: A's efficiency = 1/9 of work per day. Step 2: Since A's efficiency is twice B's, B's efficiency = (1/9) ÷ 2 = 1/18 of work per day. Step 3: Time for B = 1 ÷ (1/18) = 18 days.

Question 6

The efficiency of X is twice the efficiency of Y. If X alone can complete a job in 9 days, how many days will Y take to complete the same job?

Accepted Answer

Step 1: Let Y's efficiency = e. Then X's efficiency = 2e. Step 2: X completes work in 9 days, so X's efficiency = 1/9 per day. Step 3: Therefore, 2e = 1/9, which gives e = 1/18. Step 4: Y's efficiency = 1/18 per day, so Y takes 18 days to complete the work. Therefore, answer is 18 days.

Question 7

P can do 1/4 of a work in 5 days. At this rate, how many days will P take to complete the entire work?

Accepted Answer

Step 1: P completes 1/4 of work in 5 days. Step 2: To complete the entire work (which is 4 times 1/4), P needs 4 × 5 = 20 days. Therefore, answer is 20 days.

Question 8

The efficiency of worker M is 25% more than worker N. If M and N together can complete a job in 12 days, how many days will N alone take to complete the job?

Accepted Answer

Step 1: Let N's rate = 1/x per day. M's rate = 1.25 × (1/x) = 1.25/x per day (25% more efficient). Step 2: Combined rate = 1/x + 1.25/x = 2.25/x per day. Step 3: Combined time = 12 days, so 2.25/x = 1/12. Step 4: Solve: x = 2.25 × 12 = 27 days. Step 5: N alone takes 27 days.

Question 9

A contractor hires workers such that the work efficiency increases by 20% every day. On day 1, the workers complete 10 units of work. If the total work required is 132 units, on which day will the work be completed?

Accepted Answer

Step 1: Day 1 work = 10 units. Day 2 work = 10 × 1.2 = 12 units. Day 3 work = 12 × 1.2 = 14.4 units. Day 4 work = 14.4 × 1.2 = 17.28 units. Day 5 work = 17.28 × 1.2 = 20.736 units. Step 2: Cumulative work = 10 + 12 + 14.4 + 17.28 + 20.736 = 74.416 units (after 5 days). Step 3: Day 6 work = 20.736 × 1.2 = 24.8832 units. Cumulative = 74.416 + 24.8832 = 99.2992 units. Step 4: Day 7 work = 24.8832 × 1.2 = 29.85984 units. Cumulative = 99.2992 + 29.85984 = 129.15904 units. Step 5: Day 8 work = 29.85984 × 1.2 = 35.831808 units. Cumulative = 129.15904 + 35.831808 = 165.0 units (exceeds 132). Therefore, work completes on day 8.

Question 10

A work is completed by 8 men in 10 days. If the work is to be completed in 5 days, how many additional men are needed, assuming all men work at the same efficiency?

Accepted Answer

Step 1: Total work = 8 men × 10 days = 80 man-days. Step 2: To complete in 5 days, men needed = 80 ÷ 5 = 16 men. Step 3: Additional men = 16 - 8 = 8 men.

Question 11

Machine A produces 240 units per hour. Machine B produces 30% more units per hour than A. If both machines run together for 5 hours, then only B runs for 3 more hours, what is the total production?

Accepted Answer

Step 1: Machine A's rate = 240 units/hour. Step 2: Machine B's rate = 240 × 1.30 = 312 units/hour. Step 3: Combined rate = 240 + 312 = 552 units/hour. Step 4: Production in first 5 hours = 552 × 5 = 2760 units. Step 5: Production by B alone in next 3 hours = 312 × 3 = 936 units. Step 6: Total production = 2760 + 936 = 3696 units.

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