Aptitude · Time and Work · Model 3

Start from the stage you actually know

Somebody leaves before the job is done, or joins after it has started. The combined rate changes partway, so a single division cannot work. Split the job into stages — and begin with the stage the question has fully specified.

Set the times and how early A leaves, and watch the split →

Find the fixed stage first, subtract its work from the job, and solve the rest. Setting up one equation for the whole job is the slow route.

01 The idea

Two stages, and one of them is already known

A and B can finish a job in 8 and 16 days respectively. They start together, but A leaves 4 days before the work is completed. How long does the whole job take? The combined rate applies for part of the time and B’s rate alone for the rest, so there is no single rate to divide by.

You can set up an equation. Let the total be x days: then A and B work together for (x − 4) days and B works alone for 4, giving 3(x − 4) + 4 = 16 on a 16-unit job, so x = 8. That works and it is fine.

But there is a faster reading. One of the two stages is completely specified: the last 4 days are B alone, at 1 unit a day, which is 4 units. Subtract that from the 16-unit job and 12 units were done together, at 3 a day, which is 4 days. Total 8 days — no algebra at all.

That is the habit worth building: identify the stage the question has pinned down, compute its work, subtract, and solve what remains. In a leaving question the fixed stage is at the end; in a joining question it is at the beginning. Either way, starting from the known stage turns an equation into two subtractions.

The rate changes when the roster changes, so split at that moment. One of the two stages is always fully specified — do that one first.

StageA span during which the set of workers is constant, so the combined rate is constant. A joining or leaving question has two; a question with both has three.

Fixed stageThe stage whose duration the question tells you outright — the last 4 days, or the first 6. Compute its work first and subtract.

Remaining workThe job minus the fixed stage’s work. Divided by the other stage’s rate, it gives the unknown duration.

02 Worked example

A leaves 4 days before the end

The source question, worked the short way. A and B alone can complete a work in 8 days and 16 days respectively. They start working together, but A leaves 4 days before the completion of the work. In how many days is the entire work completed?

Set up the units and name the stagesLCM of 8 and 16 is 16. Stage one is A and B together; stage two is the last 4 days, B alone.W = 16 units ⇒ A = 2/day, B = 1/day, together = 3/day

Work the fixed stageThe last 4 days are fully specified: B alone at 1 unit a day.4 × 1 = 4 units done in the final stage

Subtract to get the joint stage’s workEverything else was done with both of them working.16 − 4 = 12 units done together

Divide by the joint rateTwelve units at three a day.12 / 3 = 4 days working together

Add the stagesFour days together, then four days of B alone.4 + 4 = 8 days in total

Verify by adding up who did what: A worked 4 days at 2 units and B worked all 8 days at 1 unit, giving 8 + 8 = 16 units, the whole job. That check is worth doing every time in this model, because the commonest error is a stage boundary off by a day — and a boundary error always shows up as the contributions failing to total the job.

03 The method

The variants, and where the fixed stage sits in each

Every version of this model is the same two moves. What changes is which end of the job is pinned down.

Leaving: last-stage work = d × E_remaining, then joint time = (W − that) / E_joint. Joining: first-stage work = d × E_starter, then the rest at the joint rate.

Read where the question pins the job down. “A leaves 4 days before completion” fixes the end. “A works alone for 4 days, then B joins” fixes the beginning. “A leaves after 4 days” also fixes the beginning — note that this is a different sentence from the first one and a different question. Mis-reading which end is fixed is the single biggest source of error here.

The question says	Fixed stage	Then find
A leaves d days before the end	The last d days	The joint stage
A leaves after d days	The first d days	The stage after A goes
A works alone d days, then B joins	The first d days	The joint stage
B joins d days before the end	The last d days	The stage before B arrives
Both join and leave	Two of the three stages	The middle one
One equation for the whole job	—	Works, but slower
Confusing the two 'leaves' phrasings	`different questions`	Re-read before calculating

05 Cheat sheet

Model 3 on one page

Two moves and the readings that decide which end to start from.

Step	Do this	On A 8, B 16, leaves 4 early
1. Units and rates	`W = LCM`	16 units, A = 2, B = 1
2. Fixed stage work	`days × that stage’s rate`	4 × 1 = 4 units
3. Remaining work	`W − fixed`	16 − 4 = 12 units
4. Unknown duration	`remaining / other rate`	12 / 3 = 4 days
5. Add the stages	`sum the durations`	4 + 4 = 8 days
Verify	`Σ (days × rate) = W`	4×2 + 8×1 = 16 ✓
One equation instead	`slower but valid`	3(x−4) + 4 = 16

Start from the stage you knowOne stage is always fully specified. Computing its work and subtracting turns an equation into two subtractions.

Two 'leaves' phrasings are different questions“Leaves 4 days before completion” fixes the end; “leaves after 4 days” fixes the beginning. Read which before calculating.

Check the contributions total the jobAdd each person’s days times their rate. A stage boundary that is off by a day always shows up here.

06 Where & why

Where Model 3 shows up

The most-set hard time-and-work question, because it rewards reading the sentence carefully rather than remembering a formula.

Bank PO · SSC CGL

“A leaves d days before completion”

The standard form. Fast via the fixed stage, slow via an equation, and wrong if you read it as “leaves after d days”.

TCS Digital · Infosys

Joining questions

“A works alone for 4 days, then B joins.” The fixed stage is at the front instead, and the method is unchanged.

Three-stage questions

Somebody joins and somebody leaves

Two of the three stages are usually specified, leaving one to solve. The same subtract-and-divide move, applied twice.

Pipes and cisterns

A tap closed partway through

One of the four question sets in that module is exactly this model with a tank instead of a wall.

The reading is the work here. Underline the phrase that pins down a duration, decide which end of the job it fixes, and the arithmetic that follows is two lines.

07 Interview questions

What gets asked

Ten, and the two “leaves” phrasings appear next to each other on purpose.

A and B take 8 and 16 days. They start together and A leaves 4 days before completion. How long in total?

Eight days. On 16 units A does 2 a day and B does 1. The last 4 days are B alone, contributing 4 units, so 12 units were done together at 3 a day — that is 4 days. Total 4 + 4 = 8.

What is the fastest route through this model?

Start with the stage the question has fully specified. Compute its work, subtract it from the job, and divide what remains by the other stage’s rate. That replaces setting up and solving an equation with two subtractions and a division.

What is the difference between “A leaves 4 days before completion” and “A leaves after 4 days”?

They fix opposite ends of the job. The first tells you about the final stage — B works alone for the last 4 days. The second tells you about the first stage — both work for the first 4 days. They are different questions with different answers, and confusing them is the commonest error here.

Why can’t you use a single combined rate for the whole job?

Because the set of workers changes partway, so the rate is not constant. A single division assumes a constant rate, which is exactly what Model 1 has and this model does not. Hence the split into stages.

How do you check a Model 3 answer?

Add up each person’s contribution: days worked times their rate, summed over everybody, should equal the whole job. For the worked example, A did 4 × 2 = 8 and B did 8 × 1 = 8, totalling 16. A stage boundary off by one day always fails this check.

A and B take 14 and 21 days. They begin together and A leaves after 4 days. How long in total?

About 15⅓ days. On 42 units A does 3 and B does 2. The first 4 days move 4 × 5 = 20 units, leaving 22 for B alone at 2 a day, which is 11 days. Total 15 days. Note the fixed stage is at the front here, because of the phrasing.

How would you handle a question where somebody joins and somebody leaves?

Three stages. Usually two of the three durations are given, so compute their work, subtract both from the job, and divide the remainder by the middle stage’s rate. It is the same move done twice rather than a new method.

Can you still use the equation method?

Yes, and it is perfectly valid — let the total be x days and write the work done in each stage in terms of x. It is slower and gives more chances to slip, but it handles awkward phrasings where identifying the fixed stage is not obvious, so it is worth being able to fall back on.

What if the question says the work was finished 3 days earlier than expected?

Hard

A, B and C can do a job alone in 10, 15 and 30 days. A and B start together. C joins after 2 days. A leaves 3 days before the job is completed. (a) Find the total time taken. (b) State each person’s number of working days and verify the contributions sum to the whole job. (c) A student solves it by averaging the three possible combined rates over the whole duration. Explain why that is wrong even though it produces a number close to the right one.

Follow-up

Part (c) is the reason stages exist. Averaging the rates weights each roster equally regardless of how long it actually applied, so it only coincides with the truth when the stages happen to be equal in length. Seeing that the near-miss is luck rather than method is what makes the stage split feel necessary instead of fussy.

Show the hint

Three stages: A+B, then A+B+C, then B+C. Two of the three durations are pinned by the question.