Aptitude · Time and Work · Model 1

Everyone works throughout, so one rate covers the job

The base case: two or three people working together from start to finish. It is the only model where a single combined rate applies for the whole job, and every harder model is defined by how it breaks that.

Set the solo times and watch the rates add →

One combined rate, applied once. The moment somebody joins late or leaves early, this model no longer applies — and that is what Model 3 is for.

01 The idea

The one case where nothing changes partway

Model 1 is the version you already know: everybody starts together, everybody finishes together, nobody joins or leaves. Because the set of workers never changes, their combined rate is constant, and a constant rate means one division gives the answer.

That is worth stating explicitly, because it is exactly the assumption the later models break. Alternate days breaks it by having only one person work each day. Joining and leaving breaks it by changing the roster partway. Both then need the job split into stages, and Model 1 does not.

The model has three shapes and they all use the same machinery. Two people together. Three people together. And the reverse question — you are told the pair’s combined time and one person’s solo time, and asked for the other person’s. That last one is where students slip, because the temptation is to subtract times.

You subtract rates. If A and B together do 5 units a day and A alone does 3, then B does 2 — not the 10 minus 6 that subtracting days would suggest. Rates add and subtract; times do neither.

To find a missing worker, subtract rates. A and B together minus A alone leaves B’s rate, and B’s time is the job divided by it.

Combined rateThe sum of the individual rates. Constant throughout in this model, which is what makes a single division sufficient.

Reverse questionGiven the pair’s time and one solo time, find the other. Subtract the rates, never the times.

Sanity windowThe together time always lies between the fastest solo time divided by the head count, and the fastest solo time itself.

02 Worked example

A in 10 days, B in 15, and then B alone from a pair

Two questions on the same pair, forwards and backwards. A can do a work in 10 days and B in 15. (i) How long together? (ii) If instead you were told the pair takes 6 days and A takes 10, how long would B take alone?

Set the job to the LCMLCM of 10 and 15 is 30, so both rates come out whole.W = 30 units ⇒ A = 3/day, B = 2/day

Forwards: add and divideFive units a day between them, on a thirty-unit job.30 / (3 + 2) = 6 days

Now the reverse: what is the pair’s rate?Six days for the whole thirty units tells you the combined rate directly.30 / 6 = 5 units/day for A and B together

Subtract A’s rateA supplies 3 of those 5 units, so B supplies the rest.5 − 3 = 2 units/day for B

Turn B’s rate back into a timeTwo units a day on a thirty-unit job.30 / 2 = 15 days ✓ consistent with part (i)

Notice what the reverse question is not: 15 is not 10 plus 6, nor 10 minus 6, nor anything you can get by combining the two times arithmetically. It only appears once you move to rates. Whenever a question gives you a combined time and asks about one member, convert everything to units per day before touching it.

03 The method

The three shapes, and the shortcut worth its limits

All three shapes are the same two lines of work. The product formula is faster for exactly one of them.

Together: T = W / (E_A + E_B + …). Reverse: E_B = E_pair − E_A, then T_B = W / E_B.

Two people, both working throughout: together time = ab/(a+b). And the reverse of it: if the pair takes t days and A takes a, then B takes at/(a − t) — here 10×6/4 = 15 days. Both formulas assume the pair works together for the entire job, which is precisely what Models 2 and 3 remove.

Shape	Given	Route
Two together	both solo times	`W/(Eᴱ + Eᵇ)`
Three together	three solo times	`W/(Eᴱ + Eᵇ + Eᶜ)`
Reverse	pair time and one solo	`subtract the rates`
Two-person shortcut	both solo times	`ab/(a+b)`
Reverse shortcut	pair time t, A takes a	`at/(a−t)`
Subtracting times	`never valid`	10 − 6 = 4 is not B
Three people by product	`no such formula`	use the LCM route

05 Cheat sheet

Model 1 on one page

Five rows of method and two of the things that are never right.

Case	Route	On A = 10, B = 15
Together, two	`W/(Eᴱ+Eᵇ)` or `ab/(a+b)`	6 days
Together, three	`add all three rates`	with C = 30 → 5 days
Find B from the pair	`Eᵇ = Eᴅᶜⁱᵣ − Eᴱ`	5 − 3 = 2 → 15 days
Reverse shortcut	`at/(a−t)`	10×6/4 = 15
Sanity window	`fastest/n to fastest`	between 5 and 10 days
Subtracting times	`wrong`	gives 4, not 15
Averaging times	`wrong`	12.5 is slower than A alone

One rate for the whole jobThis is the only model where that holds. If anybody joins or leaves, the rate changes and you need stages instead.

Subtract rates to find a personThe combined rate minus a known rate is the unknown rate. Times cannot be subtracted this way and doing so is the model’s standard error.

Check the windowThe together time sits between the fastest solo time divided by the head count and the fastest solo time. Two comparisons, and they catch nearly every slip.

06 Where & why

Where Model 1 shows up

The most-set time-and-work question, and the base every other model builds on.

TCS NQT · Wipro · Cognizant

Two or three workers together

A speed item. Fifteen seconds via the LCM route, and the trap options are the sum and the average of the times.

Bank PO · SSC CGL

The reverse question

“A and B together take 6 days, A alone 10 — find B.” Set because subtracting times gives 4, which is in the options.

Multi-part questions

Model 1 as the opening step

Harder questions often start by asking for the combined rate and then complicate it. Getting this part automatic frees attention for the rest.

Pipes and cisterns

Two inlet pipes

Identical arithmetic. An inlet is a worker; the whole of Model 1 transfers directly.

The value of learning this model precisely is that it tells you what the other six are doing. Each of them takes the single constant rate assumed here and breaks it in one specific way.

07 Interview questions

What gets asked

Ten, and the reverse question in the middle is the one that separates rate thinking from time thinking.

A takes 10 days, B takes 15. How long together?

Six days. On a job of 30 units A does 3 a day and B does 2, so together 5 a day and 30/5 = 6. Or use the two-person formula: 10 × 15 / 25 = 6.

A and B together take 6 days, and A alone takes 10. How long does B take alone?

Fifteen days. The pair does 30/6 = 5 units a day and A does 3, so B does 2, giving 30/2 = 15 days. Subtracting the times to get 4 is the standard error — rates subtract, times do not.

Why can’t you subtract the times in that question?

Because time is not additive over workers. Rate is work per day and rates combine linearly; time is the reciprocal of rate, and reciprocals do not add. That is the same reason the together time is not the sum of the individual times.

A in 10, B in 15, C in 30. How long together?

Five days. On 30 units the rates are 3, 2 and 1, totalling 6 a day, and 30/6 = 5. The two-person product formula does not extend to three workers, which is why the LCM route should be the default.

Give the reverse shortcut formula.

If the pair takes t days and A alone takes a days, B alone takes at/(a − t). For a = 10 and t = 6 that is 60/4 = 15 days. Note it requires a to be larger than t, which it always is if the numbers are consistent.

How do you sanity-check a together time?

It must be less than the fastest worker’s solo time, since help cannot slow a job, and with n workers it cannot be below the fastest time divided by n. For A = 10 and B = 15 the answer must lie between 5 and 10 days.

What makes Model 1 different from the rest of the module?

The set of workers never changes, so the combined rate is constant for the whole job and one division suffices. Every other model changes who is working partway through, which forces the job to be split into stages.

A and B together finish in 6 days. If A is twice as efficient as B, how long does B alone take?

Eighteen days. Take the job as 6 units so the pair does 1 unit a day; splitting that 2:1 gives A two-thirds and B one-third of a unit a day, so B needs 18 days. This is the bridge into Model 6, where the ratio is given rather than the times.

Three equally efficient workers each take 12 days alone. How long together?

Hard

(a) Derive the two-person together-time formula ab/(a+b) from rates. (b) Rearrange it to show that if the pair takes t days and A takes a days, then B takes at/(a−t). (c) State what happens to your formula in part (b) when t equals a, and explain what that means physically. (d) Explain why a question stating that the pair takes 12 days while A alone takes 10 is self-contradictory.

Follow-up

Parts (c) and (d) are the point. As t approaches a the formula diverges, which says that if the pair is no faster than A alone then B contributes nothing; and a pair slower than one of its members is impossible, which the formula signals by returning a negative time. A formula that misbehaves exactly where the physics does is a formula you can trust.

Show the hint

For (c), look at the denominator. For (d), work out what B’s rate would have to be.