Time and Work

Overview

Time and Work is a staple arithmetic topic in IBPS PO Prelims, typically contributing 1–3 questions either as standalone problems or embedded within Data Interpretation sets. The core concept is deceptively simple—understanding the inverse relationship between time taken and work done per day—but examiners test this through multiple persons working together, persons leaving midway, and the popular pipes and cisterns variant.

Mastery of this topic offers reliable marks because the question types are predictable and the calculations, once you adopt the LCM method, become fast and error-free. Students who rely solely on the traditional "fraction of work" approach often struggle with time pressure; the efficiency-based LCM technique is the modern competitive exam standard and should be your default.

Key Concepts

• **Work-Efficiency Relationship**: If A completes a job in 'n' days, A's one-day work (efficiency) = 1/n of the total work. More days means lower efficiency.

• **LCM Method**: Assume total work = LCM of all given times. This converts fractional efficiencies into whole numbers, making addition and comparison instant.

• **Combined Work**: When A and B work together, their combined efficiency = sum of individual efficiencies. Combined time = Total Work ÷ Combined Efficiency.

• **Work Equivalence**: Work = Efficiency × Time. If one person's efficiency is double another's, they complete the same work in half the time.

• **Pipes and Cisterns Analogy**: An inlet pipe "does positive work" (fills), an outlet pipe/leak "does negative work" (empties). Net effect = Inlet efficiency − Outlet efficiency.

• **Alternate Day / Part-Work Problems**: Track work done in cycles. If A works on odd days and B on even days, calculate work done per 2-day cycle, then handle the remainder.

• **Wages Distribution**: Wages are divided in the ratio of total work done (Efficiency × Days worked), not just in ratio of efficiencies or days alone.

Formulas / Key Facts

| Scenario | Formula | |----------|---------| | A's 1-day work | 1/A (if A finishes in A days) | | A and B together | 1/A + 1/B = (A+B)/(A×B); Combined time = (A×B)/(A+B) | | LCM-based efficiency | Total Work = LCM; Efficiency of A = LCM/A | | Time when working together | Total Work ÷ Sum of Efficiencies | | Pipes: Net filling rate | (1/Inlet) − (1/Outlet) | | Part work given, find time | Remaining Work ÷ Efficiency | | Wages ratio | Efficiency of A × Days worked by A : Efficiency of B × Days worked by B | | M₁D₁H₁/W₁ = M₂D₂H₂/W₂ | Relates Men, Days, Hours, and Work across two scenarios |

Worked Examples

### Example 1: Basic Combined Work (LCM Method) **Problem**: A can do a work in 12 days, B can do it in 18 days. In how many days will they finish working together?

**Solution**:

• Total Work = LCM(12, 18) = 36 units
• A's efficiency = 36/12 = 3 units/day
• B's efficiency = 36/18 = 2 units/day
• Combined efficiency = 3 + 2 = 5 units/day
• Time = 36/5 = **7.2 days or 7 days 4 hours**

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### Example 2: Pipes and Cisterns **Problem**: Pipe A fills a tank in 20 minutes, Pipe B fills it in 30 minutes, and Pipe C empties it in 15 minutes. If all three are opened, how long to fill the tank?

**Solution**:

• Total Capacity = LCM(20, 30, 15) = 60 units
• A's rate = +60/20 = +3 units/min (fills)
• B's rate = +60/30 = +2 units/min (fills)
• C's rate = −60/15 = −4 units/min (empties)
• Net rate = 3 + 2 − 4 = 1 unit/min
• Time = 60/1 = **60 minutes**

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### Example 3: Work with Person Leaving **Problem**: A and B together can finish work in 12 days. A alone can do it in 20 days. After working together for 4 days, A leaves. In how many more days will B finish the remaining work?

**Solution**:

• Total Work = LCM(12, 20) = 60 units
• (A+B)'s efficiency = 60/12 = 5 units/day
• A's efficiency = 60/20 = 3 units/day
• B's efficiency = 5 − 3 = 2 units/day
• Work done in 4 days together = 5 × 4 = 20 units
• Remaining work = 60 − 20 = 40 units
• Time for B alone = 40/2 = **20 days**

Common Mistakes

• **Adding times instead of efficiencies**: Students write "A takes 10 days, B takes 15 days, so together 25 days." → **Correct fix**: Add work rates (1/10 + 1/15), not times.

• **Forgetting to subtract for outlet pipes**: When a leak or drain is present, its efficiency must be negative. → **Correct fix**: Assign negative efficiency to anything that removes work/water.

• **Applying wages ratio incorrectly**: Dividing wages only by efficiency or only by days worked. → **Correct fix**: Wages ∝ (Efficiency × Days worked by that person).

• **Ignoring fractional days in alternate-day problems**: When work doesn't complete exactly at a cycle end, students round incorrectly. → **Correct fix**: After full cycles, check who works on the next day and compute partial day if needed.

• **Misreading "A is twice as efficient as B"**: This means A's efficiency = 2 × B's efficiency, so A takes half the time, not double. → **Correct fix**: Higher efficiency = less time required.

Quick Reference

• **LCM method is faster**: Always set Total Work = LCM of given times.
• **Together time formula**: (A × B)/(A + B) for two workers.
• **Inlet = positive, Outlet = negative** in pipes problems.
• **Efficiency ratio = Inverse of time ratio**: If A:B time = 3:4, then A:B efficiency = 4:3.
• **Remaining work problems**: First calculate work done, subtract from total, then divide by new efficiency.
• **MDH/W formula**: Use when men, days, or hours change across scenarios.