Lessons / Quantitative aptitude
Quantitative aptitude · Day 4

Time and Work

Never add days. Make the total work the LCM of the days, turn each person into units-per-day, then add those.

Practise (11 questions) Present as slides

Why this topic matters

If A paints a wall in 10 days, then in one day A paints one tenth of it. Time and work is about per-day work, because per-day work can be added.

2 or 3 questions in most papers. Days cannot be added; work per day can. The LCM method from Easy Maths turns every question into whole numbers.

Work per day, and the LCM total

If A takes a days, A does 1/a of the job per day. Working together, rates add: 1/a + 1/b.

Faster: let the total work be the LCM of the days. Then each person does a whole number of units per day.

A3/dayB2/dayA+B5/day
Job = 30 units (LCM of 10 and 15). Together 5 a day: 6 days
Together: T = ab/(a + b)
Two people only.
Total work = LCM of the days
Units per day = total / days.
M1 D1 H1 / W1 = M2 D2 H2 / W2
Men × days × hours per unit of work stays the same.
Example 1A can do a job in 12 days, B in 18 days. They work together for 4 days, then A leaves. How long does B take to finish?
  1. Total = LCM(12, 18) = 36 units. A does 3/day, B does 2/day.Whole numbers, no fractions.
  2. 4 days together: 4 × 5 = 20 units done.
  3. Left: 16 units. B alone: 16 / 2 = 8 days.
  4. 8 more days.

Efficiency

'A is twice as efficient as B' means A does twice the work per day, so A takes half the time.

If A is x% more efficient than B, A's rate = B's rate × (1 + x/100), so A's time = B's time / (1 + x/100).

Example 1A is twice as efficient as B. Together they finish a job in 14 days. How long would A take alone?
  1. Rates 2 : 1. Together 3 units a day.Efficiency is work per day.
  2. Total = 3 × 14 = 42 units.
  3. A alone: 42 / 2 = 21 days.

Men, days and hours

The amount of work = number of workers × days × hours per day. If the job is the same, that product is the same.

If the job changes, divide by the work: M1 D1 H1 / W1 = M2 D2 H2 / W2.

Example 112 men working 8 hours a day finish a job in 10 days. How many days will 16 men working 6 hours a day take?
  1. 12 × 8 × 10 = 16 × 6 × D.Same job, so the same total man-hours.
  2. 960 = 96 D, so D = 10.

Alternate days

If A and B work on alternate days, treat one pair of days as a block. Work in a block = A's rate + B's rate.

Fill whole blocks, then finish the remainder carefully day by day (whoever's turn it is).

Example 1A can do a job in 10 days and B in 15. They work on alternate days, starting with A. In how many days is the job done?
  1. Total 30 units. A 3/day, B 2/day. One 2-day block = 5 units.
  2. 5 blocks = 10 days = 25 units. Left 5.
  3. Day 11 (A): 3 units, left 2. Day 12 (B): 2 units, done.Check whose turn it is after the blocks.
  4. 12 days.

Formula summary

Together
1/a + 1/b = 1/T → T = ab/(a + b).
LCM method
Total work = LCM of days. Each person's rate = LCM / their days.
Pairs given (A+B, B+C, C+A)
Add all three rates and halve to get A+B+C.
Men-days-hours
M1 D1 H1 / W1 = M2 D2 H2 / W2.
One leaves × days before the end
Let total be T; the leaver worked T - x days.

Shortcuts and tricks

LCM method

Use it when: Three or more people, or pairs.

  1. A 10, B 15, C 30 → total 30 units.
  2. Rates 3, 2, 1 = 6 a day.
  3. 30/6 = 5 days.
ExampleA 10 days, B 15, C 30 together?
  1. 30 units, 6 per day.

Alternate days

Use it when: A and B work on alternate days.

  1. Units per 2-day cycle.
  2. Run full cycles, then finish the leftover with whoever is next.
ExampleA 12 days, B 18 days, alternate, A starts.
  1. 36 units. A 3, B 2 → 5 per cycle.
  2. 7 cycles = 14 days, 35 units.
  3. Day 15: A needs 1 unit = 1/3 day.

Common mistakes

A takes 10 days and B 15, so together 25 (or 12.5).
Add rates: 1/10 + 1/15 = 1/6, so 6 days.

Before you move on, you should be able to...

  • add rates, not days
  • set the job to the LCM of the days
  • use M × D × H = constant
  • solve alternate-day questions with blocks

Practice questions (11)

Try each one before opening the solution. Or practise them one by one so your score is saved.

easy A can do a job in 12 days, B in 18. Together?
  1. 36/5.

Answer: 7.2 days

easy 12 men working 8 h a day finish in 18 days. How many days for 16 men at 9 h a day?
  1. 12 × 18 × 8 / (16 × 9).

Answer: 12 days

medium A+B do a job in 12 days, B+C in 15, C+A in 20. How long for A alone?
  1. Work 60. Rates 5, 4, 3. A+B+C = 6.
  2. A = 6 - 4 = 2 → 30 days.

Answer: 30 days

medium A takes 20 days, B 30. They start together, but A leaves 5 days before the end. Total days?
  1. (T - 5)/20 + T/30 = 1 → 5T = 75.

Answer: 15 days

medium A is twice as good as B; together they finish in 14 days. A alone?
  1. Rates 2 and 1 = 3 per day, work 42.
  2. 42/2 = 21.

Answer: 21 days

hard 10 men finish a job in 15 days; 15 women finish it in 12. How many days for 5 men and 6 women?
  1. Work = 150 man-days = 180 woman-days, so 1 man = 1.2 women.
  2. 5 men + 6 women = 12 women → 180/12 = 15 days.

Answer: 15 days

hard A is 50% more efficient than B. Together they finish in 18 days. A alone?
  1. Rates 3:2, 5 units a day, work 90.
  2. 90/3.

Answer: 30 days

hard 20 men can finish a job in 30 days. After 10 days, 5 men leave. How many more days to finish?
  1. Work 600 man-days, 200 done.
  2. 400 / 15 men.

Answer: 26 2/3 days

hard 2 men and 3 women finish a job in 10 days; 3 men and 2 women in 8 days. How long for 2 men and 1 woman?
  1. Man = 7/200, woman = 1/100 per day (solve the two equations).
  2. 2 men + 1 woman = 16/200 = 2/25 → 12.5 days.

Answer: 12 1/2 days

hard A takes 6 days, B 8 days; with C they finish in 3 days. The fee is Rs 3,200. C's share?
  1. In 3 days A does 1/2, B does 3/8, so C does 1/8.
  2. 1/8 of 3200.

Answer: Rs 400

hard A takes 10 days, B 15. They work together, but A leaves after 2 days. How many more days does B need?
  1. 2 days together = 2 × 1/6 = 1/3.
  2. B does 2/3 at 1/15 a day.

Answer: 10 days

Class plan (2 hours, for trainers)

A tested order for teaching this topic in one 2-hour session. Present mode follows the same order.

10 minWarm-up.
25 minRates and the LCM total; together; one leaves.
20 minEfficiency.
20 minMen-days-hours.
15 minAlternate days.
25 minPractice set.
5 minRecap.
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