Model 4: The Chain Rule

Time and Work · 30 min

Aptitude · Time and Work · Model 4

Four things moving at once, and only the directions to get right

Men, days, hours a day and the size of the job all change between the two halves of the question. There is no new formula — only a decision, made four times, about whether each quantity pushes the answer up or down.

Change any of the four and watch the headcount follow →
More work means more men. More days means fewer. More hours a day means fewer. Say all four aloud before you write anything.

01 The idea

One invariant, four directions

If 28 men can build 10 houses in 5 days working 4 hours a day, how many men are needed for 15 houses in 6 days working 10 hours a day? Four quantities change at once, which is what makes this look hard and is the only thing that is.

The way through is an invariant. Multiply men by days by hours and you get the total labour poured in: 28 × 5 × 4 = 560 man-hours, and that built 10 houses. So a house costs 56 man-hours, and that figure does not care how you staff it.

Now run it forwards. Fifteen houses need 15 × 56 = 840 man-hours. Each worker supplies 6 × 10 = 60 hours. So 840/60 = 14 men. No proportion was set up, and nothing could be inverted by accident, because every step was a multiplication or a division with an obvious meaning.

The proportion form is equivalent and faster once you trust it, but it is where errors live: each of the four ratios has to be the right way up. Work is direct — more houses, more men. Days and hours are inverse — more of either, fewer men. Efficiency is inverse too. Say those aloud and the placement follows.

Man-hours per unit of work is the invariant. Compute it once, then scale up for the new work and divide by the new time available.
Man-hoursWorkers × days × hours per day. The labour content of a job, unchanged by how you distribute it. Man-days is the same idea when the hours are fixed.
Direct proportionWork against men, and work against days. More of one needs more of the other, so in the proportion it sits the same way up as the answer.
Inverse proportionMen against days, hours against days, efficiency against days. More of one means less of the other, so it sits the opposite way up.

02 Worked example

28 men, 10 houses, 5 days, 4 hours → 15 houses, 6 days, 10 hours

The source question, worked through the invariant rather than a proportion. If 28 men can build 10 houses in 5 days working 4 hours a day, how many men are required to build 15 houses in 6 days working 10 hours a day?

1
Total the labour in the first jobTwenty-eight men, five days, four hours each day.28 × 5 × 4 = 560 man-hours for 10 houses
2
Cost one houseDivide by the work, giving a figure that is independent of staffing.560 / 10 = 56 man-hours per house
3
Scale up to fifteen housesWork is direct, so fifteen houses need fifteen times the labour of one.15 × 56 = 840 man-hours needed
4
See how much each worker suppliesSix days at ten hours a day.6 × 10 = 60 hours per worker
5
Divide to get the headcountEight hundred and forty hours of labour, sixty hours per worker.840 / 60 = 14 men

Note the answer is fewer men despite half again as much work — because the hours per day went from 4 to 10, which more than compensates. That is worth checking as a habit: predict the direction of the answer before computing it. Here work pushed up, days pushed down and hours pushed down hard, so a fall from 28 was expected.

03 The method

The proportion form, placed by direction

Once the directions are automatic, the single-line proportion is quicker than the invariant. Until then, use the invariant.

M₁D₁H₁ / W₁ = M₂D₂H₂ / W₂. Solve for whichever term is unknown. With efficiency E included, it becomes MDHE/W constant.
Build the answer as a product of ratios. m₂ = m₁ × (W₂/W₁) × (D₁/D₂) × (H₁/H₂). The direct quantity (work) has the new value on top; the inverse ones (days, hours) have the old value on top. Here: 28 × 15/10 × 5/6 × 4/10 = 14. Getting a ratio upside down changes the answer without making it look wrong, which is why saying the directions aloud is worth the two seconds.
QuantityRelation to the men neededPosition in the ratio
Amount of workdirectnew / old
Number of daysinverseold / new
Hours per dayinverseold / new
Efficiency per workerinverseold / new
Men, if days are wantedinverseold / new
The invariantMDH/W is constant56 man-hours per house
A ratio invertedsilent wrong answersay the direction aloud

05 Cheat sheet

Model 4 on one page

The invariant route, the proportion route, and the classic questions each handles.

CaseRouteWorked
Total labourM × D × H28×5×4 = 560
Per unit of workMDH / W560/10 = 56 per house
Men needed(per-unit × W₂)/(D₂H₂)840/60 = 14
As one proportionM₁D₁H₁/W₁ = M₂D₂H₂/W₂28×15/10×5/6×4/10
Solving for days insteadsame equation, D₂ unknown—
The cats-and-mice formscale both sides equally100 cats, 100 mice, 100 days
A ratio invertedwrong, and looks finecheck each direction
Predict the direction firstDecide whether the answer should rise or fall before computing. Here work rose but hours rose more, so the headcount fell from 28 to 14.
The invariant route cannot be invertedMultiplying to a total and dividing back has an obvious meaning at every step, unlike a four-ratio proportion. Use it until the directions are automatic.
Scaling everything equally changes nothing100 cats eating 100 mice in 100 days means 1 cat eats 1 mouse in 100 days — the workers and the work scaled together, so the time is untouched.

06 Where & why

Where Model 4 shows up

The chain rule appears wherever several quantities move at once, which is well beyond time and work.

SSC CGL · RRB · CPO
Men, days, hours and work

The standard four-quantity question. Answerable in one line via the proportion, and the trap options correspond to inverting one ratio.

TCS NQT · Infosys
The cats-and-mice puzzle

“100 cats eat 100 mice in 100 days.” Looks like a trick and is just scaling workers and work by the same factor, so the time is unchanged.

Partial-work variants
“After 2/3 of the work…”

Convert the remaining work to a fraction first, then it is an ordinary chain-rule question with a smaller W.

Beyond this chapter
Any multi-variable proportion

Pipes filling tanks at different rates for different hours, or machines with different outputs. The direction discipline transfers directly.

This model has no content to learn, only a discipline: name each quantity’s direction before combining anything. Students who do that never get a chain-rule question wrong; students who do not get about half of them wrong.

07 Interview questions

What gets asked

Nine, and the first two are the two routes through every question in the model.

28 men build 10 houses in 5 days at 4 hours a day. How many for 15 houses in 6 days at 10 hours?
Fourteen. The first job is 28 × 5 × 4 = 560 man-hours for 10 houses, so 56 per house. Fifteen houses need 840 man-hours, and each worker supplies 6 × 10 = 60 hours, giving 840/60 = 14 men.
Give the proportion form.
M₁D₁H₁/W₁ = M₂D₂H₂/W₂, or as a product of ratios: m₂ = 28 × (15/10) × (5/6) × (4/10) = 14. Work has the new value on top because it is direct; days and hours have the old value on top because they are inverse.
List the four directions.
More work needs more men. More days needs fewer men. More hours a day needs fewer men. More efficiency per worker needs fewer men. The first is direct and the other three are inverse, and that is the whole content of the model.
Why did the answer fall from 28 men to 14 despite more houses?
Because the hours per day went from 4 to 10, which more than offsets the extra work. Work rose by a factor of 1.5, days by 1.2 and hours by 2.5 — and since days and hours are inverse, the net effect is 1.5 ÷ 1.2 ÷ 2.5 = 0.5, halving the headcount.
100 cats eat 100 mice in 100 days. How long for 1 cat to eat 1 mouse?
One hundred days. The workers and the work were both divided by 100, and those two changes cancel — work is direct with the time and workers are inverse with it, so scaling both equally leaves the time untouched. It reads as a trick and is really a check on whether you track directions.
Which route should you use under exam pressure?
The invariant one until the directions are automatic: total the man-hours, reduce to a per-unit figure, scale, divide. Every step there has an obvious physical meaning, so it cannot be inverted by accident. The proportion is faster but silently wrong if a ratio goes upside down.
How do you handle a question where two-thirds of the work is already done?
Convert the remainder to a fraction of the job first — one-third here — and use that as W₂. After that it is an ordinary chain-rule question. Trying to work with the completed portion rather than the remaining one is the usual slip.
If 20 girls complete a work in 34 days, how long for 17 girls to do double the work?
Eighty days. Fewer workers pushes the time up by 20/17, and double the work pushes it up by 2, so 34 × (20/17) × 2 = 80 days. Both factors move the same way here, so the answer must be well above 34 — a useful check before computing.
Does the chain rule ever fail?
It assumes every worker contributes an identical, independent rate no matter how many others are present. In reality crowding, coordination costs and indivisible tasks break that, which is why the relation confidently predicts that 120 workers finish a 10-worker, 12-day job in one day. Use it freely on exam questions and cautiously on real ones.

08 Practice problems

Six chains

Before calculating each one, write down whether the answer should rise or fall and why. Then check your arithmetic agrees.

Only the work changes

Easy
If 15 men can build a wall in 12 days, how many men are needed to build three such walls in the same 12 days?
Follow-up
One quantity moves and it is the direct one, so the answer should scale straight up. Use it to confirm you have the direct direction right before the harder ones.
Show the hint
Work is directly proportional to the men needed when the time is fixed.

Only the days change

Easy
If 24 men can complete a job in 15 days, how many men are needed to complete it in 10 days?
Follow-up
The inverse direction, on its own. Predict whether you need more or fewer men before computing, and check the person-days total is unchanged.
Show the hint
Men times days is constant for a fixed job.

Two quantities move

Medium
If 20 girls can complete a piece of work in 34 days, in how many days can 17 girls complete double the work?
Follow-up
Both changes push the time in the same direction, so the answer must be well above 34 days. If you get something smaller, one of the two ratios is inverted.
Show the hint
Scale for the change in workers and for the change in work separately, then combine.

Hours enter

Medium
If 12 men working 8 hours a day can finish a job in 15 days, how many days will 18 men working 10 hours a day take?
Follow-up
Three quantities and two of them inverse. Compute the total man-hours first rather than setting up a proportion — it is harder to invert something with a physical meaning.
Show the hint
Total the man-hours in the first job, then see how many the new crew supplies per day.

The scaling puzzle

Medium
If 8 machines can produce 8 widgets in 8 minutes, how long do 5 machines take to produce 5 widgets? Explain the result in one sentence.
Follow-up
This is the cats-and-mice shape and the answer is not 5 minutes. The one-sentence explanation is the real question — it is about which two changes cancel.
Show the hint
Machines are inverse with the time and widgets are direct, so scaling both by the same factor does what to the time?

Four quantities and a partial job

Hard
A contractor undertakes to build a road with 40 men working 8 hours a day, expecting to finish in 30 days. After 10 days he finds that only one-quarter of the road is complete. (a) Find how many man-hours the remaining work requires. (b) If he keeps 40 men but extends the working day, how many hours a day must they work to finish on time? (c) If instead he keeps 8-hour days, how many extra men must he hire? (d) Show that (b) and (c) supply exactly the same amount of extra labour, and say what that tells you about how the chain rule treats men and hours.
Follow-up
Part (d) is the payoff. Both fixes supply the same additional man-hours, which is why the chain rule treats extra men and longer days as interchangeable — and noticing that they are interchangeable in the arithmetic but obviously not in reality is the honest reading of the model.
Show the hint
Work out the man-hours actually spent in the first 10 days and what that bought, then price the remaining three-quarters.