The core idea
Work = rate × time. If A finishes a job in a days, A's rate is 1/a of the job per day. Rates add when people work together. The LCM method makes it easy: take the total work as the LCM of the days given, so each person's rate is a whole number of units per day.
Worked examples
Example 1. A can do a job in 12 days and B in 18 days. Together? LCM = 36 units; A = 3 per day; B = 2 per day; together 5 per day; 36 ÷ 5 = 7.2 days. Shortcut: ab ÷ (a + b) = 216 ÷ 30 = 7.2.
Example 2 (one leaves). A and B can do a job in 10 and 15 days. They work together for 3 days and then A leaves. Total = 30 units; A = 3, B = 2; in 3 days they do 15 units; 15 remain; B alone takes 15 ÷ 2 = 7.5 days.
Example 3 (efficiency). A is twice as efficient as B and together they finish in 12 days. Rates are in the ratio 2 : 1, so together 3 parts in 12 days: 36 part-days. A alone: 36 ÷ 2 = 18 days; B alone: 36 days.
Example 4 (pipes). A pipe fills a tank in 6 hours and another empties it in 10 hours. Both are open. LCM = 30; fill 5 per hour; empty 3 per hour; net 2 per hour; 30 ÷ 2 = 15 hours.
Example 5 (wages). A and B together earn 1,500 for a job, A working 6 days and B working 4 days at the same daily wage. Wages are shared in the ratio of work done: 6 : 4 = 3 : 2, so A gets 900 and B gets 600.
Example 6 (men and days). 15 men can build a wall in 24 days. How many men in 18 days? Man-days = 360; 360 ÷ 18 = 20 men.
Common traps
- Adding days instead of rates.
- Forgetting that a person who leaves stops contributing.
- Treating pipes that empty as adding to the fill rate.
- Assuming equal efficiency when the question gives ratios.
Speed tips
- Use the LCM method for three or more workers.
- For two workers, use ab ÷ (a + b).
- Keep man-days constant for inverse proportion problems.
A practice routine
- Day 1: LCM method for two workers, ten questions.
- Day 2: leaving and joining, and efficiency.
- Day 3: pipes and cisterns, and wages.
- Day 4: a timed set of 12 at 60 seconds each.
Where it appears
Time and work appears in SSC CGL, banking, railway mathematics, MAT and GATE General Aptitude. Related: HCF and LCM, ratio and proportion and time, speed and distance.
Practice tests below are published by institutes and educators on TestMint.
