Time & Work • Topic 1 of 5

Individual Efficiency

Efficiency is simply the amount of work a person does in one unit of time. The CAT-fast way is to fix the total work as a convenient number — almost always the LCM of the days mentioned — so that each rate comes out as a whole number of units. If A finishes a job in 10 days, set total work = 10 (or any multiple) and A’s efficiency = 1 unit/day; if a second worker appears who takes 15 days, jump up to LCM(10,15) = 30, giving A = 3 units/day and the other = 2. The golden link is that efficiency is inversely proportional to time: if A is twice as efficient as B, A takes half the time B does, and their efficiency ratio 2:1 forces a time ratio 1:2. Reading questions as efficiency ratios (rather than day-fractions) is what separates a 30-second solution from a two-minute fraction crawl.

✅ Solved examples

1. A completes a job in 8 days and B in 12 days. Find the ratio of their efficiencies.
Efficiency ∝ 1/time, so ratio = (1/8) : (1/12) = 12 : 8 = 3 : 2. A is the faster worker.
2. A is 50% more efficient than B. If B takes 18 days, how long does A take?
Efficiency ratio A : B = 150 : 100 = 3 : 2 ⇒ time ratio A : B = 2 : 3. A’s time = (2/3) × 18 = 12 days.
3. A does a job in 15 days. After how many days of work has he completed 40% of it?
Total work = 15 units, A = 1 unit/day. 40% of 15 = 6 units ⇒ 6 days.
4. A and B together take 6 days; A alone takes 10 days. How long does B alone take?
Total work = LCM(6,10) = 30. Together = 5 units/day, A = 3 units/day ⇒ B = 2 units/day ⇒ 30/2 = 15 days.

✏️ Practice — try these, take hints as needed

1. A finishes a task in 20 days, B in 25 days. Ratio of efficiencies A : B?
Efficiency ∝ 1/time.
Invert the days: 1/20 : 1/25.
Multiply by LCM 100.
5 : 4
2. A is twice as efficient as B and takes 30 days less than B. Time taken by A?
Time ratio is inverse of efficiency: 1 : 2.
B takes 2 parts, A takes 1 part; difference 1 part = 30 days.
A = 30 days, B = 60 days.
30 days
3. A can do a job in 12 days. What fraction of the work does he finish in 5 days?
Total work = 12 units, A = 1/day.
5 days = 5 units.
5 out of 12.
5/12
4. A is 25% more efficient than B. If A takes 16 days, how long does B take?
Efficiency A : B = 5 : 4.
Time ratio = 4 : 5.
B = (5/4) × 16.
20 days
5. Together A and B finish in 8 days; B alone takes 24 days. A alone?
Total work = LCM(8,24) = 24.
Together = 3/day, B = 1/day.
A = 2 units/day ⇒ 24/2.
12 days

📝 Topic test — 8 questions

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