← All topics🔄 Three, Leaving & AlternateMaths
Time & Work · Topic 4 of 6

🔄 Three, Leaving & Alternate

3 exam question types, fully solved

Once you treat each worker as “so many units per day”, three workers, someone joining or leaving partway, and alternate-day working are all the same single skill: add up the units actually done, and compare with the total.

Count the units done, day by day

Always start by setting total work = LCM of the times and writing each person's units/day. Then:

• Three together: just add all three rates. If A, B, C take 12, 15, 20 days, total = LCM = 60; rates 5, 4, 3; together units/day, so days.

• Someone joins or leaves: work out the units done while everyone is present, subtract from the total, then see who finishes the leftover. “A leaves 3 days before the end” means add back the units A would have done in those 3 days to find the real total time.

• Alternate days: A and B work on alternate days, so handle them as a 2-day block. Add the units of one A-day and one B-day, see how many full blocks fit in the total, then finish the remainder. Who starts matters — the leftover after the last full block is done by whoever's turn is next.

Key habit: never average the days. Always go back to units/day, because only rates can be added or subtracted safely.

🧱Time & Work LabTotal work = LCM units — no fractions
A finishes in (days)
B finishes in (days)
Schedule:
Total = LCM(10, 15) = 30 units
A: 3 units/dayB: 2 units/day
DAY 1
A
+33/30
DAY 2
B
+25/30
DAY 3
A
+38/30
DAY 4
B
+210/30
DAY 5
A
+313/30
DAY 6
B
+215/30
DAY 7
A
+318/30
DAY 8
B
+220/30
DAY 9
A
+323/30
DAY 10
B
+225/30
DAY 11
A
+328/30
DAY 12
B
+230/30
Finishes exactly at the end of day 12 days
Alternate days → work in 2-day blocks (A, B take turns). One full block = 3 + 2 = 5 units. Count blocks, then finish the leftover — whoever's turn is next does it.

A, B, C do 6, 4, 2 units/day on a 24-unit job. Together that is 12 units/day, so they finish in days.

📝Practice Questions

Q1A, B and C can finish a task alone in 42, 84 and 28 days. Working together, how long do they take?

Q2A alone takes 8 days and B alone takes 12 days. A starts on day 1 and B joins on day 3. On which day is the work finished?

Q3P alone takes 12 days and Q alone takes 18 days. They work on alternate days with P starting first. On which day is the work completed?

📚

Real exam questions — Three, Leaving & Alternate

3 question types · 12 solved examples from real SSC papers

Three-worker teams, mid-job joiners/leavers, and alternate-day turns — all cracked with one tool: total work = LCM of the times, each rate = total ÷ time.

How to solve this type
Take total work = LCM of the given times, then each rate = total ÷ time (units/day). To find the team, add all three rates and divide total work by the sum. If two pair-times are given (A+B, B+C, A+C), add the three pair-rates: that sum equals twice A+B+C, so halve it — then subtract a pair to isolate the third worker.

A, B and C can finish a job alone in 20, 30 and 60 days respectively. Working together, how long do they take?

A8 daysB10 daysC12 daysD15 days

A, B and C can do a piece of work alone in 6, 8 and 12 days. Together they finish it in:

A3 daysB83\dfrac{8}{3} daysC2.5 daysD4 days

A and B together do a work in 12 days, B and C in 15 days, and A and C in 20 days. How long does B alone take?

A40 daysB30 daysC24 daysD20 days

A, B and C together finish a work in 10 days. A alone takes 30 days and B alone takes 20 days. How long does C alone take?

A30 daysB45 daysC20 daysD60 days