SSC CGL Maths · Time and Work

Workers Joining and Leaving

When people work for different stretches of time, add each stretch's work (rate times days) and set the total equal to one whole job.

All Time and Work notes

Why this matters

Twenty-one questions in the bank, thirteen MODERATE and one HARD, eight of them from the 2025 shifts: one worker hands over to another, a helper joins, someone leaves after a few days or a few days before the end. The method never changes: one line for each stretch, then add the lines.

Concept 1 of 3: One worker starts and another finishes

Split the job into stretches. Each stretch has its own worker or team, and the work done in it is that rate times its days. The stretches together make exactly one job.

Definition

Work in a stretch $=$ daily rate of whoever is working $\times$ days of that stretch.
All stretches add to 1 (or to the whole job in LCM units).
An unknown rate: if A works first and does $\frac{t}{a}$ of the job, the rest, $1 - \frac{t}{a}$ , took B the stated days. So B's rate $=$ the rest $\div$ B's days, and B's time for the whole job follows.
Regroup, fast route: if A works $p$ days and B works $q$ days ( $p < q$ ) and you know how long A and B take together, write $pA + qB = p(A + B) + (q - p)B$ . The pair part is known, so B comes out in one step.

Work in stretches

\sum (\text{rate} \times \text{days}) = 1 \qquad pA + qB = p(A + B) + (q - p)B

$A, B$ one-day rates of the two workers
p, qdays each of them works

Worked example

A can finish a job in 24 days. He works for 6 days, and then B finishes the rest in 27 days. How long would A and B take working together?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 15 Sep 2025, Shift 3 · Q66Moderate

Example 1 · Time and Work · Workers Joining and Leaving

A and B can complete a certain task together in 30 days. After A works for 16 days, B is left to finish the remaining work by himself in 44 days. How many days would it take for B to complete the entire task on his own?

Taking the days for the rest as the whole time

If B finishes the remaining

\frac{3}{4}

of a job in 27 days, B's time for the whole job is

27 \div \frac{3}{4} = 36

days, not 27.

Drill 8 more on one worker starts and another finishes

Concept 2 of 3: A team that changes partway

The team works together for a while, then someone leaves or a helper joins and the new team finishes. That is two stretches with two different rates.

Definition

Stretch 1: team rate $\times$ days. Stretch 2: new team rate $\times$ days. The two add to 1.
When did A leave? If A left after $x$ days and B alone finished in $d$ days, $x(A + B) + dB = 1$ . Fast route: take B's $d$ days of work off first; what is left was done by the pair at the pair's rate.
A joiner's rate: work left after stretch 1 $\div$ days of stretch 2 gives the new team's rate; take away the members you know.
Efficiencies given: use them as units a day, with the whole job $=$ one worker's efficiency $\times$ that worker's days.
Pay still follows work done: add up each person's work across both stretches.
Read what is asked: only the days after the change, or the total from the start.

Team changes partway

x(A + B) + dB = 1

xdays the full team works
ddays the smaller team works after the change
$A, B$ one-day rates

Worked example

A and B can finish a job in 15 days and 30 days. They start together, but after 4 days A leaves. How many more days does B take to finish the job?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 18 Sep 2025, Shift 2 · Q55Moderate

Example 2 · Time and Work · Workers Joining and Leaving

A and B can finish a certain job in 28 and 35 days, respectively. They started working together, but A left after a while, and B managed to complete the remaining work in 17 days. How many days did A work before leaving?

Giving the days after the change when the total is asked

"How many more days" wants only the second stretch. "In how many days was the work completed" wants both stretches added. SSC prints both numbers among the options.

Forgetting the old members in a joiner's rate

If a team finishes the rest at

\frac{1}{12}

of the job a day after C joins, C's rate is

\frac{1}{12}

minus the rates of those already working, not

\frac{1}{12}

itself.

Drill 9 more on a team that changes partway

Concept 3 of 3: Leaving a few days before the end

When someone leaves k days before the job ends, you do not know when they left, but you know they missed the last k days. Call the total time T and give each person T minus the days they missed.

Definition

Let the job take $T$ days. A worker who leaves $k$ days before the end works $T - k$ days; one who stays works all $T$ .
Write $\sum \text{rate} \times (\text{days worked}) = 1$ and solve for $T$ .
LCM units, fast route: with whole-number rates, $T \times$ (all the rates added) minus each leaver's rate $\times$ the days they missed equals the job's units.

Leaving k days before the end

\sum r_i (T - k_i) = 1

Ttotal days the job takes
$r_i$ one-day rate of worker i
$k_i$ days before the end that worker i leaves (0 if they stay)

Worked example

A, B and C can finish a job in 20, 30 and 60 days. They start together. A leaves 2 days before the end and B leaves 3 days before the end, while C works to the finish. How many days does the job take?

Practice this conceptself-check · 2 quick reps

The same idea in a real exam question:

SSC CGL · 2023 · Tier I, 27 Jul 2023, Shift 2 · Q68Moderate

Example 3 · Time and Work · Workers Joining and Leaving

Benny can do a piece of work in 24 days. Chethan and David can do the same work individually in 36 and 48 days, respectively. All of them begin the work together. However, Benny leaves the work 4 days before the completion of work and Chethan leaves the work 10 days before the completion of the work. David worked till the end and completed the work. Find the number of days in which the work was completed.

Taking the leaving days off the leaver's own time

Leaving 4 days before the END means working

T - 4

of the job's

T

days. It does not turn the leaver's own time of, say, 24 days into 20.

Drill 1 more on leaving a few days before the end

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

One worker starts and another finishes
Work in stretches
$\sum (\text{rate} \times \text{days}) = 1 \qquad pA + qB = p(A + B) + (q - p)B$
A team that changes partway
Team changes partway
$x(A + B) + dB = 1$
Leaving a few days before the end
Leaving k days before the end
$\sum r_i (T - k_i) = 1$

Watch out for (4)

Taking the days for the rest as the whole time
→ One worker starts and another finishes
Giving the days after the change when the total is asked
→ A team that changes partway
Forgetting the old members in a joiner's rate
→ A team that changes partway
Taking the leaving days off the leaver's own time
→ Leaving a few days before the end

Test yourself on Time and Work

20 past SSC CGL questions from this chapter, timed at 12 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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