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.

Start with this

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

Work per day, and the LCM total: the rules

Worked example

A 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.
   Why: 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.

Answer: 8

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).

Worked example

A 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.
   Why: Efficiency is work per day.
2. Total = 3 × 14 = 42 units.
3. A alone: 42 / 2 = 21 days.

Answer: 21

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.

Worked example

12 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.
   Why: Same job, so the same total man-hours.
2. 960 = 96 D, so D = 10.

Answer: 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).

Worked example

A 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.
   Why: Check whose turn it is after the blocks.
4. 12 days.

Answer: 12

Shortcut: 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.

LCM method: try it

A 10 days, B 15, C 30 together?

1. 30 units, 6 per day.

Answer: 5 days

Shortcut: 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.

Alternate days: try it

A 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.

Answer: 14 1/3 days

Common mistakes

Your turn 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

Your turn 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

Your turn 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

Your turn 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

Recap

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

Now practise

11 questions with full solutions.

Open practice Back to the lesson