Time and Work — SSC CHSL Study Notes

Overview

Time and Work is a cornerstone of the Arithmetic section in SSC CHSL Tier-1, appearing in 2–4 questions per paper. The topic tests your ability to relate the amount of work done to the time taken and the efficiency of workers. Understanding this chapter is critical because the same logic extends to Pipes and Cisterns problems, which are essentially Time and Work dressed in different language.

Most questions revolve around calculating how long multiple workers take to finish a job together, determining the efficiency of individuals, or finding the fraction of work completed in a given time. The beauty of this topic is its formulaic consistency — once you master the core relationship between work, time and efficiency, variants become mechanical. The typical SSC CHSL difficulty is moderate, emphasizing quick mental calculation and formula application rather than multi-layered reasoning.

Students must develop fluency in converting verbal statements ("A can do a work in 10 days") into algebraic expressions, working with fractions of work done per day, and handling negative work scenarios (Pipes and Cisterns). This chapter pairs naturally with Ratio and Proportion, so clarity here strengthens your foundation across multiple Arithmetic topics.

Key Concepts

• **Work is constant**: If no quantity is mentioned, assume total work = 1 unit. Alternatively, use the LCM of all given times as total work to avoid fractions.
• **Efficiency = Work done per unit time**: If A completes work in 10 days, A's one-day work = 1/10. Efficiency is inversely proportional to time taken.
• **Joint work formula**: If A does work in x days and B in y days, working together they complete 1/(1/x + 1/y) days, which simplifies to (xy)/(x+y) days.
• **Man-Day equivalence**: Total Work = Number of persons × Number of days. If 5 men finish in 6 days, then 1 man finishes in 30 days (or 30 man-days of work exist).
• **Negative work in Pipes**: A filling pipe adds work (positive efficiency), while an emptying/leak pipe removes work (negative efficiency). Combine them algebraically.
• **Part of work done**: If someone works for t days at efficiency e, work done = e × t. Always express as a fraction of total work.
• **Work left after partial completion**: If A completes x fraction of work, remaining work = 1 − x. Then calculate time for the remaining portion separately.
• **Efficiency ratio ↔ Time ratio**: If efficiencies are in ratio a:b, times taken are in ratio b:a (inverse relationship).

Formulas / Key Facts

1. **One-day work**: If A completes work in n days, A's 1 day work = 1/n. 2. **Combined work (two persons)**: Time taken together = (t₁ × t₂)/(t₁ + t₂), where t₁, t₂ are individual times. 3. **Combined work (multiple persons)**: Combined one-day work = 1/t₁ + 1/t₂ + 1/t₃ + … Then time = 1/(combined one-day work). 4. **Total Work using LCM method**: Total work = LCM(t₁, t₂, t₃, …). Then efficiency of each person = Total work / their time. 5. **Man-Day formula**: M₁D₁ = M₂D₂ (if same work). With varying work or hours: M₁D₁H₁/W₁ = M₂D₂H₂/W₂. 6. **Pipes formula**: Net filling rate = (Sum of filling rates) − (Sum of emptying rates). Time to fill = Capacity / Net rate. 7. **Alternate day work**: Calculate work done in one full cycle (all persons working their turns), then find number of complete cycles and remaining work. 8. **Efficiency from ratio**: If A:B efficiency = 3:2, and total work = 5k, A does 3k and B does 2k in the same time.

Worked Examples

**Example 1: Basic Joint Work** *A can do a piece of work in 12 days, B in 15 days. How long will they take together?*

**Solution**:

• A's 1-day work = 1/12
• B's 1-day work = 1/15
• Combined 1-day work = 1/12 + 1/15 = (5 + 4)/60 = 9/60 = 3/20
• Time together = 1 ÷ (3/20) = 20/3 = 6⅔ days

**Shortcut**: Time = (12 × 15)/(12 + 15) = 180/27 = 20/3 = 6⅔ days.

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**Example 2: Pipes and Cisterns** *Pipe A fills a tank in 6 hours. Pipe B empties it in 8 hours. If both are opened, how long to fill the empty tank?*

**Solution**:

• A's rate = 1/6 tank/hour (positive)
• B's rate = −1/8 tank/hour (negative, empties)
• Net rate = 1/6 − 1/8 = (4 − 3)/24 = 1/24 tank/hour
• Time to fill = 1 ÷ (1/24) = 24 hours

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**Example 3: Man-Day Variation** *10 men can finish a work in 8 days. How many men are needed to finish the same work in 5 days?*

**Solution**: Using M₁D₁ = M₂D₂: 10 × 8 = M₂ × 5 M₂ = 80/5 = 16 men

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**Example 4: Partial Work + Remaining Work** *A can do work in 10 days, B in 15 days. A works for 4 days, then B joins. How many more days to finish?*

**Solution**:

• A's 1-day work = 1/10
• A's 4 days work = 4/10 = 2/5
• Remaining work = 1 − 2/5 = 3/5
• (A+B)'s 1-day work = 1/10 + 1/15 = (3+2)/30 = 1/6
• Days to finish remaining = (3/5) ÷ (1/6) = (3/5) × 6 = 18/5 = 3.6 days

Common Mistakes

1. **Adding times instead of rates**: Students often add 12 days + 15 days = 27 days when combining two workers. **Fix**: Always add *rates* (1/12 + 1/15), never times.

2. **Forgetting negative sign for emptying pipes**: Treating an emptying pipe as positive work leads to absurd answers (tank fills faster with a leak!). **Fix**: Subtract the emptying rate explicitly.

3. **Confusing efficiency ratio with time ratio**: If A is twice as efficient as B, students incorrectly assume A takes twice the time. **Fix**: Efficiency and time are *inversely* proportional — if A:B efficiency = 2:1, then A:B time = 1:2.

4. **Not using LCM to avoid messy fractions**: Calculating 1/12 + 1/18 + 1/24 directly creates ugly arithmetic. **Fix**: Set total work = LCM(12,18,24) = 72 units, then efficiencies become integers.

5. **Ignoring the part already completed**: When A works alone for x days then B joins, students recalculate the whole work instead of just the remainder. **Fix**: Subtract work done by A, then apply joint efficiency only to what's left.

Quick Reference

• **A in x days, B in y days → Together in (xy)/(x+y) days**
• **Total Work = LCM of all given days → Avoid fractions**
• **Efficiency ∝ 1/Time → Double efficiency = Half time**
• **Man-Day constant: M₁D₁ = M₂D₂ for same work**
• **Pipes: Add filling rates, subtract emptying rates**
• **Alternate days: Calculate one full cycle, count cycles + remainder**