Time and Work — Study Notes

Overview

Time and Work is a high-yield topic in the UP Police Constable exam, typically contributing 2–4 questions in the Numerical & Mental Ability section. This chapter tests your ability to convert real-world productivity scenarios into mathematical equations and solve them efficiently.

The core principle is simple: **Work = Rate × Time**. Every person, machine or pipe has a rate of work (how much they complete per unit time), and questions revolve around finding time taken, work done or efficiency when multiple entities work together or in shifts. Mastery requires understanding three sub-areas: **basic work problems** (individuals working alone or together), **pipes and cisterns** (filling/emptying tanks), and **wages distribution** (payment proportional to work done). These problems appear frequently and are often solved faster using the "efficiency method" rather than algebraic equations.

Strong command over LCM, fractions and unitary method is essential. Practice converting "days to complete work" into "work done per day" and vice versa. With systematic practice, you can solve most Time and Work questions in under 90 seconds.

Key Concepts

• **Work is constant**: Total work is typically assumed as 1 unit or taken as LCM of all given days for easier calculation. This allows you to focus on rates rather than absolute quantities.

• **Rate (Efficiency) = 1/Time**: If A completes work in 10 days, A's one-day work = 1/10. If B does it in 15 days, B's rate = 1/15. Combined rate when working together = 1/10 + 1/15.

• **Time taken together = Total Work / Combined Rate**: When multiple workers collaborate, add their individual rates to get the combined efficiency, then divide total work by this sum.

• **Negative work for emptying**: In pipes and cisterns problems, outlet pipes (emptying) have negative rates. If inlet fills in 6 hours and outlet empties in 8 hours, net rate = 1/6 - 1/8.

• **Work done = Efficiency × Time**: If A works for 5 days at rate 1/10 per day, work done = 5 × (1/10) = 1/2. Remaining work = 1 - 1/2 = 1/2.

• **Wages ∝ Work done**: Payment is distributed in the ratio of work done. If A works 10 days at efficiency 2 units/day and B works 15 days at 3 units/day, ratio = (10×2):(15×3) = 20:45 = 4:9.

• **Man-days concept**: 10 men in 6 days = 60 man-days of work. Same work by 15 men = 60/15 = 4 days. Total work capacity (M₁D₁) = (M₂D₂) when work is equal.

• **Efficiency and time are inversely proportional**: If A is twice as efficient as B, A takes half the time B takes. More efficient workers complete work faster.