8  Time and Work

8.1 Introduction

The time and work concept builds on the fundamental principle of proportionality between work done, rate of work, and time. It forms the basis for problems on individual and group efficiency, pipes and cisterns, work with wages, alternate working days, partial work, and efficiency ratios.

This chapter provides formulas, shortcuts, solved examples, and graded practice problems.

8.2 1) Fundamental Relationship

If \(W\) = total work, \(R\) = rate of work, \(T\) = time taken:

\[ W = R \times T \quad \text{or} \quad R = \frac{W}{T} \quad \text{or} \quad T = \frac{W}{R} \]

In aptitude problems, “1 unit of work” is often assumed, so \(R\) = fraction of work done per day/person.

8.3 2) Work Rate Basics

  • If A completes work in \(n\) days, then A’s 1-day work = \(1/n\).
  • If A and B work together: 1-day work = \(1/n + 1/m\).
  • Time taken together = \(\frac{nm}{n+m}\).

8.4 3) Efficiency Ratio

If A completes work in \(n\) days, B in \(m\) days:
- Efficiency ratio = \(m:n\).
- Time ratio = inverse = \(n:m\).

8.5 4) Pipes and Cisterns (Inlet & Outlet)

  • Inlet pipe fills at \(1/n\) per hour.
  • Outlet pipe empties at \(1/m\) per hour.
  • Together: rate = \(1/n - 1/m\) (if \(m\) > \(n\), tank still fills).

8.6 5) Work and Wages

  • Wages ∝ work done.
  • If A:B efficiency = \(m:n\), then wages shared = \(m:n\).

8.7 6) Partial Work and Remaining Work

If part of work is completed, leftover can be calculated by subtracting fractions.

Example: A alone 10 days, B alone 15 days. Together = \(1/10+1/15=1/6\). Work in 5 days = 5/6. Remaining = 1/6.

8.8 7) Alternate Days / Cyclic Work

If A and B work alternately:
- Compute 2-day work = (A’s 1-day + B’s 1-day).
- Extend for multiple cycles, then add last partial day.

8.9 8) Solved Examples

  1. A alone can finish in 20 days. Work done in 1 day = 1/20. B in 30 days = 1/30. Together = 1/12. ⇒ Finish in 12 days.
  2. A can do a piece in 12 days, B in 15. They work together for 5 days. Work done = 5(1/12+1/15)=5(3/20)=15/20=3/4. Remaining = 1/4. A alone → 3 days.
  3. A=10 days, B=15 days. Efficiency ratio=15:10=3:2.
  4. A fills in 6 h, B fills in 8 h, C empties in 12 h. Effective = 1/6+1/8−1/12=7/24. ⇒ Time=24/7=3.43 h.
  5. A, B, C complete work in 10, 15, 30 days. Paid ₹900. Work ratio = 1/10:1/15:1/30=6:4:2. Wages=₹450:₹300:₹150.
  6. Tank filled in 5 h by 3 pipes of equal capacity. Each pipe = 1/15 per h.
  7. A works alone 2 days, then B joins. Work=2×1/12=1/6. Remaining=5/6. Together rate=1/12+1/15=3/20. Time= (5/6)/(3/20)=5.55 days.

8.10 9) Common Traps

  • Forgetting base = “1 unit work”.
  • Misreading inlet/outlet signs.
  • Confusing wages ratio with time ratio (inverse needed).
  • Partial work leftover miscalculated.
  • Alternate day cycles miscounted.

8.11 Practice Set – Level 1

  1. A can finish work in 12 days. Find his 1-day work.
  2. A=12 days, B=18 days. Together time?
  3. A fills a tank in 8 h, B in 12 h. Both together time?
  4. A completes work in 10 days. B is 25% more efficient. Find B’s days.
  5. A alone 15 days. B alone 20 days. Together work for 4 days. What fraction remains?

8.12 Practice Set – Level 2

  1. A=10 d, B=15 d. They work together 5 d, then A leaves. Remaining work days for B?
  2. A fills in 3 h, B fills in 4 h, C empties in 6 h. Find fill time.
  3. Wages: A=12 d, B=15 d, C=20 d. Paid ₹3,600. Share each?
  4. Tank has 2 inlets (6 h, 8 h) and 1 outlet (12 h). Time to fill?
  5. A alone 8 d, B alone 12 d. Efficiency ratio? Wages ratio?

8.13 Practice Set – Level 3 (Challengers)

  1. A and B together 12 d. B and C together 15 d. A and C together 20 d. All three together?
  2. A completes work in 15 d. B is 50% more efficient. Together time?
  3. A and B fill tank in 10, 12 h. Outlet empties in 15 h. If all open, time?
  4. A works 2 d, B works 3 d, alternating. Total work in 30 d?
  5. A=5 d, B=6 d, C=10 d. Work together but C leaves after 2 days. Remaining time?

8.14 Answer Key (Outline)

Level 1: 1) 1/12; 2) 7.2 d; 3) 4.8 h; 4) 8 d; 5) 2/3.
Level 2: 6) 7.5 d; 7) 4 h; 8) ₹1,200, ₹1,440, ₹960; 9) 4 h; 10) 3:2.
Level 3: 11) 10 d; 12) 6 d; 13) 20 h; 14) 24 d; 15) 2.5 d. 

8.15 Summary

  • Work concept: \(W=R\times T\).
  • Efficiency ∝ 1/time.
  • Pipes: add for inlets, subtract for outlets.
  • Wages shared ∝ work done.
  • Alternate/cyclic work handled with cycle totals.
  • Care with partial work and remaining fractions.