Basic (1-7)
▶1. What is the basic formula for work in Time & Work problems?
Answer: Work = Rate × Time
Step-by-step Explanation:
1. The amount of work done is the product of the rate (work per unit time) and the time taken.
2. If a person does 1/5 of the work in 1 day, in 5 days, the work done is 1 (complete work).
▶2. If A can do a job in 10 days, what part of the job does A do in 1 day?
Answer: 1/10
Step-by-step Explanation:
1. In 10 days, A does 1 job.
2. In 1 day, A does 1/10 of the job.
▶3. If A can do a job in 8 days and B can do it in 12 days, how long will they take together?
Answer: 4.8 days
Step-by-step Explanation:
1. A's rate = 1/8 per day.
2. B's rate = 1/12 per day.
3. Combined rate = 1/8 + 1/12 = (3/24 + 2/24) = 5/24 per day.
4. Time = 1 ÷ (5/24) = 24/5 = 4.8 days.
▶4. If A can do a work in 6 days, how much work does A do in 2 days?
Answer: 1/3
Step-by-step Explanation:
1. In 1 day, A does 1/6 of the work.
2. In 2 days, A does 2/6 = 1/3 of the work.
▶5. If A and B together can do a work in 5 days, and A alone can do it in 10 days, how long will B alone take?
Answer: 10 days
Step-by-step Explanation:
1. A + B's rate = 1/5 per day.
2. A's rate = 1/10 per day.
3. B's rate = 1/5 - 1/10 = 1/10 per day.
4. So, B alone takes 10 days.
▶6. If A is twice as fast as B and can finish a job in 8 days, how long will B take?
Answer: 16 days
Step-by-step Explanation:
1. A's rate = 1/8 per day.
2. B is half as fast, so B's rate = 1/16 per day.
3. B takes 16 days to finish the job.
▶7. If A can do a work in 12 days and B in 18 days, how long will they take together?
Answer: 7.2 days
Step-by-step Explanation:
1. A's rate = 1/12 per day.
2. B's rate = 1/18 per day.
3. Combined rate = 1/12 + 1/18 = (3/36 + 2/36) = 5/36 per day.
4. Time = 1 ÷ (5/36) = 36/5 = 7.2 days.
Moderate (8-14)
▶8. If A, B, and C can do a work in 10, 12, and 15 days respectively, how long will they take together?
Answer: 4.21 days
Step-by-step Explanation:
1. A's rate = 1/10, B's = 1/12, C's = 1/15.
2. Combined rate = 1/10 + 1/12 + 1/15 = (6/60 + 5/60 + 4/60) = 15/60 = 1/4 per day.
3. Time = 1 ÷ (1/4) = 4 days.
▶9. If A can do a work in 8 days and B in 16 days, but B leaves after 4 days, how long will the work take?
Answer: 6 days
Step-by-step Explanation:
1. A's rate = 1/8, B's = 1/16.
2. Together for 4 days: (1/8 + 1/16) × 4 = (3/16) × 4 = 3/4 of work done.
3. Remaining = 1 - 3/4 = 1/4.
4. A alone does 1/8 per day, so 1/4 ÷ 1/8 = 2 days.
5. Total = 4 + 2 = 6 days.
▶10. If A and B can do a work in 6 days, B and C in 8 days, and A and C in 12 days, how long will A, B, and C take together?
Answer: 4 days
Step-by-step Explanation:
1. Let A+B = 1/6, B+C = 1/8, A+C = 1/12.
2. Add: (A+B) + (B+C) + (A+C) = 1/6 + 1/8 + 1/12 = (4+3+2)/24 = 9/24 = 3/8.
3. But this is 2(A+B+C), so (A+B+C) = 3/16 per day.
4. Time = 1 ÷ (3/16) = 16/3 ≈ 5.33 days.
▶11. If A can do a work in 20 days and B in 30 days, but A works for 4 days and leaves, how long will B take to finish the rest?
Answer: 18 days
Step-by-step Explanation:
1. A's rate = 1/20, B's = 1/30.
2. A works 4 days: 4 × 1/20 = 1/5.
3. Remaining = 1 - 1/5 = 4/5.
4. B does 1/30 per day, so 4/5 ÷ 1/30 = 24 days.
5. Total = 4 + 24 = 28 days.
▶12. If A is 50% more efficient than B and can finish a job in 12 days, how long will B take?
Answer: 18 days
Step-by-step Explanation:
1. A's rate = 1/12.
2. B is less efficient, so B's rate = 1/18.
3. B takes 18 days to finish the job.
▶13. If A and B together can do a work in 10 days, and A alone can do it in 15 days, how long will B alone take?
Answer: 30 days
Step-by-step Explanation:
1. A+B's rate = 1/10, A's = 1/15.
2. B's rate = 1/10 - 1/15 = 1/30.
3. B takes 30 days to finish the job.
▶14. If A can do a work in 9 days and B in 18 days, but they work alternately starting with A, how long will the work take?
Answer: 13 days
Step-by-step Explanation:
1. A's rate = 1/9, B's = 1/18.
2. In 2 days: A+B = 1/9 + 1/18 = 1/6.
3. In 12 days: 6 × 1/6 = 1 (work done).
4. So, 12 days to finish the work.
Advanced (15-20)
▶15. If A, B, and C can do a work in 8, 12, and 16 days respectively, but C leaves after 2 days, how long will the work take?
Answer: 4.8 days
Step-by-step Explanation:
1. A's rate = 1/8, B's = 1/12, C's = 1/16.
2. Together for 2 days: (1/8 + 1/12 + 1/16) × 2 = (6/48 + 4/48 + 3/48) × 2 = (13/48) × 2 = 13/24.
3. Remaining = 1 - 13/24 = 11/24.
4. A+B's rate = 1/8 + 1/12 = 5/24.
5. Time = 11/24 ÷ 5/24 = 2.2 days.
6. Total = 2 + 2.2 = 4.2 days.
▶16. If A can do a work in 10 days, B in 15 days, and C in 30 days, how long will they take together?
Answer: 5 days
Step-by-step Explanation:
1. A's rate = 1/10, B's = 1/15, C's = 1/30.
2. Combined rate = 1/10 + 1/15 + 1/30 = (3/30 + 2/30 + 1/30) = 6/30 = 1/5.
3. Time = 1 ÷ (1/5) = 5 days.
▶17. If A and B can do a work in 12 days, B and C in 15 days, and A and C in 20 days, how long will A, B, and C take together?
Answer: 10 days
Step-by-step Explanation:
1. Let A+B = 1/12, B+C = 1/15, A+C = 1/20.
2. Add: (A+B) + (B+C) + (A+C) = 1/12 + 1/15 + 1/20 = (5+4+3)/60 = 12/60 = 1/5.
3. But this is 2(A+B+C), so (A+B+C) = 1/10.
4. Time = 1 ÷ (1/10) = 10 days.
▶18. If A can do a work in 5 days, B in 10 days, and C in 20 days, but A leaves after 2 days, how long will the work take?
Answer: 4.5 days
Step-by-step Explanation:
1. A's rate = 1/5, B's = 1/10, C's = 1/20.
2. Together for 2 days: (1/5 + 1/10 + 1/20) × 2 = (4/20 + 2/20 + 1/20) × 2 = (7/20) × 2 = 7/10.
3. Remaining = 1 - 7/10 = 3/10.
4. B+C's rate = 1/10 + 1/20 = 3/20.
5. Time = 3/10 ÷ 3/20 = 2 days.
6. Total = 2 + 2 = 4 days.
▶19. If A is 25% more efficient than B and B can do a work in 16 days, how long will A take?
Answer: 12.8 days
Step-by-step Explanation:
1. B's rate = 1/16.
2. A is 25% more efficient, so A's rate = 1/16 × 1.25 = 1/12.8.
3. A takes 12.8 days to finish the job.
▶20. If A and B can do a work in 15 days, B and C in 20 days, and A and C in 30 days, how long will A, B, and C take together?
Answer: 12 days
Step-by-step Explanation:
1. Let A+B = 1/15, B+C = 1/20, A+C = 1/30.
2. Add: (A+B) + (B+C) + (A+C) = 1/15 + 1/20 + 1/30 = (4+3+2)/60 = 9/60 = 3/20.
3. But this is 2(A+B+C), so (A+B+C) = 3/40.
4. Time = 1 ÷ (3/40) = 40/3 ≈ 13.33 days.