Aptitude · Time and Work · Relations

More hands, fewer days — and the map of what comes next

Before the models, two proportionalities: persons against days, and persons against work. Get the direction of each right and the chain rule later becomes bookkeeping rather than guesswork. Then a map of the seven models, so you know what you are looking at.

Change the workforce and watch person-days stay put →

For a fixed job, persons × days is constant. For a fixed time, persons are proportional to work. Those two sentences generate the whole chapter.

01 The idea

Two proportionalities, opposite directions

Five people finish a job in 8 days. Ten people, working at the same speed, finish it in 4. Double the workers, halve the time — persons and days are inversely proportional when the job is fixed, so their product stays constant at 40 person-days.

That product is worth naming, because it is what you actually carry between the two halves of a question. Forty person-days is the size of the job in labour terms, and it is unchanged by how you staff it.

The second relation runs the other way. If the time is fixed instead, then doing more work needs more people: persons are directly proportional to the work. Twice the wall in the same fortnight needs twice the painters.

Almost every mistake in the chain rule comes from getting one of these directions wrong. So before combining anything, ask of each quantity: if this goes up, does the thing I want go up or down? More men, fewer days. More work, more days. More hours a day, fewer days. More efficiency, fewer days.

Ask of every quantity: if it increases, does my answer increase or decrease? Get the four directions right and the chain rule cannot go wrong.

Person-daysPersons multiplied by days — the labour size of a job. Constant for a fixed job, which is what lets you move between different staffing levels.

Inverse proportionPersons against days, and hours-per-day against days. One goes up, the other goes down, and their product is fixed.

Direct proportionPersons against work, and days against work. More of one needs more of the other. In the chain rule these sit the opposite way up from the inverse pairs.

02 Worked example

Five people, 8 days — then ten people

The relation in its simplest form, then read backwards. If 5 persons can complete a work in 8 days, how many days will 10 persons take? And how many persons are needed to finish it in 5 days?

Find the labour size of the jobFive people for eight days is forty person-days of labour, and that total does not change however you staff it.5 × 8 = 40 person-days

Restaff it with ten peopleThe same forty person-days, now supplied by ten people.40 / 10 = 4 days

Check the directionTwice the people, half the time. Persons and days moved in opposite directions, as they must.5 → 10 persons, 8 → 4 days ✓

Now fix the time at 5 days insteadForty person-days spread over five days.40 / 5 = 8 persons

State it as the standard relationAll three answers are the same equation rearranged.M₁D₁ = M₂D₂ ⇒ 5×8 = 10×4 = 8×5 = 40

Person-days is the useful intermediate. Rather than setting up a proportion each time, compute the job’s labour size once and then divide it by whatever you are given. It generalises directly: when hours per day enter, the invariant becomes person-day-hours, and when the job changes size you scale it. That is the whole of the chain rule, arrived at without a formula.

03 The method

The four directions, and the seven models ahead

The table is the reference you need for the chain rule. Below it is the map of the module, so that each model arrives as a named thing rather than a surprise.

M₁D₁/W₁ = M₂D₂/W₂ for the same efficiency, and with hours per day, M₁D₁H₁/W₁ = M₂D₂H₂/W₂. Every chain-rule question is this one equation with the unknown in a different slot.

The seven models, and what marks each one out. 1 Basic — everyone works throughout. 2 Alternate days — a repeating cycle; find the work per cycle. 3 Joining and leaving — split into stages. 4 Chain rule — men, days, hours and work all moving. 5 Wages — pay follows work actually done. 6 Efficiency — rates given as ratios or percentages. 7 Mixed workers — men, women and children with a conversion rate.

If this increases	The days needed	So in the chain rule it is
Number of persons	`decrease`	Inverse
Hours worked per day	`decrease`	Inverse
Efficiency of each worker	`decrease`	Inverse
Amount of work	`increase`	Direct
Persons, with time fixed	`— work increases`	Direct with work
Getting a direction backwards	`the usual error`	check each one aloud

05 Cheat sheet

Relations and roadmap on one page

The relations you need for the chain rule, then the model list so you can name what you are looking at.

Relation or model	Statement	Example
Persons and days	`M₁D₁ = M₂D₂`	5×8 = 10×4
With hours	`MDH/W constant`	chain rule
Persons and work	`M ∝ W`	double the wall, double the painters
Model 1 Basic	`add the rates`	A 10, B 15 → 6 days
Model 2 Alternate	`work per cycle`	day-wise tracking
Model 3 Join/leave	`split into stages`	remaining-work model
Models 4-7	`chain, wages, efficiency, mixed`	the rest of this module

Person-days is the invariantCompute it once and divide by whatever the question gives you. It beats setting up a fresh proportion each time and it generalises to hours.

Say each direction out loudMore men, fewer days. More work, more days. More hours, fewer days. Getting one of these inverted is the only real difficulty in the chain rule.

Equal workers halve cleanlyTwo identical workers take exactly half the solo time; three take a third. Unequal workers never divide that neatly, which is a useful sanity check.

06 Where & why

Where these relations show up

This lesson is infrastructure. It earns its marks inside the models rather than on its own.

SSC CGL · RRB

Direct persons-and-days questions

“If 5 men take 8 days, how long do 10 take?” Set as a warm-up, answerable from the person-days invariant alone.

Chain rule questions

Men, days, hours and work together

The four directions in the table are the entire content. Model 4 later in this module is nothing but this table applied carefully.

Pipes and cisterns

Tanks instead of walls

Identical relations with an emptying pipe supplying a negative rate. The roadmap here maps almost one-to-one onto that module.

Partnership

Money × time instead of persons × days

The same product-is-what-matters idea. Recognising the shared structure makes both chapters shorter.

You now have the map. Each of the next seven lessons takes one line of the roadmap and makes it routine, and all seven use the LCM habit from the previous lesson.

07 Interview questions

What gets asked

Nine, weighted towards the proportionality directions, because that is where chain-rule marks are actually lost.

If 5 persons finish a job in 8 days, how long do 10 take?

Four days. The job is 5 × 8 = 40 person-days, and ten people supply that in 40/10 = 4 days. Persons and days are inversely proportional for a fixed job, so doubling the workforce halves the time.

State the relation between persons and days.

M₁D₁ = M₂D₂ for the same job and the same efficiency per worker. Their product — the person-days — is the labour size of the job and does not change with staffing.

And between persons and work?

Direct: for a fixed time, more work needs more persons. Twice the wall in the same fortnight needs twice the painters. This sits the opposite way up from persons-against-days, which is why the chain rule has both kinds of term.

List the four chain-rule directions.

More persons means fewer days. More hours per day means fewer days. More efficiency means fewer days. More work means more days. The first three are inverse and the last is direct, and stating them aloud before combining anything is what prevents the standard error.

How many persons are needed to finish that 40 person-day job in 5 days?

Eight. Forty person-days divided by five days. Note the answer need not be a whole number in general — if it comes out fractional, the practical reading is that you round up, since you cannot hire two-thirds of a person.

Do two workers always halve the time?

Only if they are equally efficient. Two people who each take 8 days alone finish in 4. But A taking 10 days and B taking 15 finish in 6, not 5, because B is slower. Equal workers divide cleanly; unequal ones do not.

What is the difference between Model 1 and Model 3 in the roadmap?

In Model 1 everyone works for the whole job, so a single combined rate applies throughout. In Model 3 somebody joins or leaves partway, so the combined rate changes and you must split the job into stages. That split is the defining move of Model 3.

Which model is a question with a repeating pattern of who works each day?

Hard

A contractor observes that 10 workers finish a job in 12 days, and reasons that 120 workers would finish it in 1 day and 240 workers in half a day. (a) Verify that both figures follow from M₁D₁ = M₂D₂. (b) Give two concrete reasons why the second figure is unlikely to hold in practice. (c) State precisely which assumption in the derivation of M₁D₁ = M₂D₂ those reasons violate.

Follow-up

Part (c) is the point: the relation assumes every worker contributes an identical, independent rate regardless of how many others are present. Crowding, coordination overhead and indivisible tasks all break that independence — which is worth knowing so you apply the relation to exam questions confidently and to real ones cautiously.

Show the hint

For (c), look at what has to be true of each worker’s rate for the product to stay constant.