Model 5: Wages and Share of Work

Time and Work · 25 min

Aptitude · Time and Work · Model 5

Pay follows work done, not days present

When everyone works the whole job, wages split in the efficiency ratio and that is the end of it. The moment their durations differ, that shortcut becomes wrong — and the fix is to count the units each person actually completed.

Set the times and the payment, then make one of them leave →
Same duration for everyone? Split by the efficiency ratio. Different durations? Split by the units actually done.

01 The idea

Two rules, and knowing which applies

A can do a job in 10 days and B in 20, and they take it on for ₹2,700. If they work together throughout, A does twice as much per day as B, so the money splits 2 : 1 — ₹1,800 and ₹900. Nothing more is needed.

That works because they worked for the same length of time. When durations are equal, the ratio of work done equals the ratio of rates, so the efficiency ratio is a legitimate shortcut for the wage ratio.

Change one thing. Suppose A and B start together, A leaves after 4 days, and B finishes alone. Now their durations differ, and the efficiency ratio is no longer the ratio of work done. Paying A two-thirds of the money would be overpaying, because A was not there for the whole job.

So fall back to the definition: wages are proportional to work done. On a 20-unit job A does 2 units a day and B does 1. In 4 days A completes 8 units, and B completes the other 12, whether alongside A or alone afterwards. The split is 8 : 12 = 2 : 3, so A gets ₹1,080 and B gets ₹1,620 — B earns more, despite being the slower worker.

Wages are proportional to work done. The efficiency ratio is only a shortcut for that, and only when everybody works for the same length of time.
Efficiency ratioThe ratio of daily rates. Equals the wage ratio only when all parties work the full duration of the job.
Work-done ratioThe ratio of units actually completed. Always the correct basis for splitting wages, in every case.
Daily wageA person’s total share divided by the days they actually worked. Useful when a question asks who was better paid per day rather than in total.

02 Worked example

₹2,700 between A and B, two ways

One pair, one payment, two scenarios. A can do a work in 10 days and B in 20 days. They undertake it for ₹2,700. (i) Split the wages if they work together throughout. (ii) Split them if A leaves after 4 days and B finishes alone.

1
Set up the unitsLCM of 10 and 20 is 20, so A does 2 units a day and B does 1.W = 20 units  ⇒  A = 2/day, B = 1/day
2
Part (i): equal durations, so use the ratioBoth work the whole job, so the work ratio equals the rate ratio.A : B = 2 : 1  ⇒  A = 2700 × 2/3 = ₹1,800, B = ₹900
3
Part (ii): count what A actually didA works only 4 days, at 2 units a day.A’s work = 4 × 2 = 8 units
4
B did the restThe remaining units are B’s, whether alongside A or alone.B’s work = 20 − 8 = 12 units  ⇒  ratio 8 : 12 = 2 : 3
5
Split the money by unitsFive parts in total, so each part is ₹540.A = 2700 × 2/5 = ₹1,080    B = ₹1,620

In part (ii) the slower worker earns more, which surprises students and is exactly right: B did 12 of the 20 units. Note too that had you used the efficiency ratio here you would have given A ₹1,800 — ₹720 too much. Compare the two parts whenever you are unsure: the only thing that changed is whether the durations matched.

03 The method

The rule, and the daily-wage variant

One principle, two shortcuts, and one further question examiners like to add.

Wage share ∝ work done. With equal durations that reduces to share ∝ efficiency. Work done by a person = their rate × the days they worked.
For a helper whose rate is not given, find it by subtraction: if A and B with a helper C finish in a stated time, C’s rate is the trio’s rate minus A’s and B’s, and C’s wage follows from that. For a daily wage, divide each person’s total share by the days they actually worked — a person can have a smaller total share and a higher daily wage, which is a favourite question.
SituationSplit byOn the ₹2,700 job
Both work throughoutefficiency ratio2 : 1 → 1,800 / 900
A leaves after 4 daysunits done8 : 12 → 1,080 / 1,620
Three workers, all throughoutefficiency ratio—
A helper of unknown ratesubtract rates first—
Daily wage askedshare / days workedA 270/day, B 202.50/day
Efficiency ratio with unequal dayswrongwould overpay A by 720
Splitting by days presentwrongignores their rates

05 Cheat sheet

Model 5 on one page

The principle, the shortcut and its precondition, plus the variants.

CaseRouteOn A 10, B 20, ₹2,700
The principleshare ∝ work donealways valid
Equal durationsshare ∝ efficiency2 : 1 → 1,800 / 900
Unequal durationsrate × days worked8 : 12 → 1,080 / 1,620
Unknown helper ratetrio rate minus the known two—
Daily wageshare / days workedA 270, B 202.50
Efficiency ratio, unequal dayswrongoverpays A by 720
Splitting by days presentwrongignores efficiency entirely
The slower worker can earn moreIf B does 12 of 20 units, B is paid for 12 of them. Total pay tracks work, not speed, so a slow worker who stays longer out-earns a fast one who leaves.
Check the shares total the paymentThey must add back to the stated amount. It is a one-second check and it catches a mis-split immediately.
Total share and daily wage are different questionsA person can take a smaller total and still have the higher daily wage. Read which one the question wants.

06 Where & why

Where Model 5 shows up

Wages questions are set because they punish using a shortcut outside its precondition.

Bank PO · SSC CGL
Contract split between two or three workers

The straightforward version when everyone works throughout, and the trap version when one leaves. Both appear.

TCS Digital · Infosys
The unknown helper

“A and B with the help of C finish in 4 days — find C’s share.” Subtract rates to price C, then split by work.

Daily-wage questions
Who was paid better per day

Divide each share by days actually worked. The answer often reverses the total-pay ordering, which is the point of asking.

Partnership
The same proportional-share idea

There the product is capital × time rather than rate × days, but the reasoning is identical — a share follows a contribution.

Before splitting any payment, ask one question: did everybody work for the same length of time? If yes, use the efficiency ratio. If no, count units. That single check is the whole model.

07 Interview questions

What gets asked

Nine, and the third is the precondition that makes the shortcut legal.

A takes 10 days, B takes 20, and they share ₹2,700 working together throughout. How is it split?
₹1,800 to A and ₹900 to B. On a 20-unit job A does 2 units a day and B does 1, so the efficiency ratio is 2 : 1. Both worked the whole job, so that ratio is also the ratio of work done and therefore of pay.
Now A leaves after 4 days and B finishes alone. How is the ₹2,700 split?
₹1,080 to A and ₹1,620 to B. A completed 4 × 2 = 8 units and B completed the remaining 12, so the ratio is 8 : 12 = 2 : 3. Note B earns more despite being slower, because B did more of the work.
Why can’t you use the efficiency ratio in the second case?
Because the efficiency ratio equals the work ratio only when the durations are equal. A worked 4 days and B worked the whole job, so their work is not in the ratio of their rates. Using 2 : 1 would pay A ₹1,800, overpaying by ₹720.
State the principle wages actually follow.
Wages are proportional to work done. Everything else in the model is a shortcut for that, valid only under a specific condition. Falling back to units completed always works.
How do you find the share of a helper whose rate isn’t given?
By subtraction. If A and B alone are known and the three of them together finish in a stated time, the trio’s rate minus A’s and B’s gives the helper’s. Then split the money by the units each completed as usual.
What is a daily wage and why does it matter?
A person’s total share divided by the days they actually worked. It matters because it can reverse the ordering: in the second scenario A takes ₹1,080 over 4 days, which is ₹270 a day, while B takes ₹1,620 over 8 days, which is ₹202.50 a day. A is better paid per day and worse paid in total.
Can the slower worker ever earn more than the faster one?
Yes, whenever the slower worker does more of the job — typically by staying longer. Pay tracks work, not speed. It is a good sanity check that your answer is about units rather than about who is quicker.
Would splitting the wages by days worked ever be right?
Only if the workers are equally efficient, in which case days worked and units done are proportional anyway. In general it ignores efficiency entirely and is wrong — it would pay A and B the same per day despite A doing twice the work.
How is this related to partnership questions?
Closely. Partnership splits profit in the ratio of capital × time; this splits wages in the ratio of rate × days. Both are a share following a contribution, and the product form is what makes them the same idea. Recognising that makes the partnership module noticeably shorter.

08 Practice problems

Six splits

For each one, first answer: did everybody work for the same length of time? That decides which rule you may use.

Equal durations

Easy
A and B can do a job in 12 and 18 days respectively. They complete it together and are paid ₹3,000. How should the money be divided?
Follow-up
Both work throughout, so the efficiency ratio is legitimate. Check the two shares add back to ₹3,000.
Show the hint
On 36 units A does 3 a day and B does 2.

Three workers, all throughout

Easy
A, B and C can individually complete a job in 10, 15 and 30 days. They work together throughout and are paid ₹5,400. Find each share.
Follow-up
The same rule with three parties. The slowest worker’s share should be noticeably small, which is a useful plausibility check.
Show the hint
On 30 units the rates are 3, 2 and 1.

One leaves

Medium
A and B can do a work in 8 and 16 days. They start together, A leaves after 3 days, and B finishes alone. They are paid ₹4,800. Find each share, and state what A would wrongly have received on the efficiency ratio.
Follow-up
Computing the wrong answer deliberately fixes the precondition in place. The gap between the two is the whole point of the model.
Show the hint
On 16 units A does 2 a day — find A’s units in 3 days, and give B the rest.

The unknown helper

Medium
A and B can do a job in 12 and 20 days respectively. With the help of C they finish it in 5 days, working together throughout, and are paid ₹6,000. Find C’s share.
Follow-up
C’s rate is not given and must be recovered by subtracting the two known rates from the trio’s. Everything after that is an ordinary equal-duration split.
Show the hint
On 60 units the trio does 12 a day; work out how much of that is not A or B.

Daily wage reverses the order

Medium
A and B can do a work in 9 and 18 days. They start together, A leaves after 3 days and B finishes alone. They are paid ₹3,600. Find (a) each total share, (b) each daily wage, and (c) state who is better off on each measure.
Follow-up
The two measures disagree, and part (c) makes you say so explicitly. Reading which one a question wants is the practical skill.
Show the hint
Find the units each did for (a), then divide by the days each actually worked for (b).

Derive the precondition

Hard
Two workers with rates rᴱ and rᵇ work for dᴱ and dᵇ days respectively on a job paying P. (a) Write down the correct wage ratio. (b) Write down what the efficiency-ratio shortcut would give. (c) Show algebraically that the two agree exactly when dᴱ = dᵇ, and only then. (d) Hence explain why the shortcut is safe for the standard 'both work throughout' question and unsafe everywhere else.
Follow-up
This turns a rule you were told into a rule you can derive, and the derivation names its own precondition. That matters because the shortcut is genuinely useful — you want to keep using it, and you want to know exactly when you may.
Show the hint
The correct ratio is rᴱdᴱ : rᵇdᵇ and the shortcut is rᴱ : rᵇ. Ask when those two are equal.