Model 7: Men, Women and Children

Time and Work · 25 min

Aptitude · Time and Work · Model 7

One little word decides the whole question: or

“3 men or 6 women” gives you a conversion rate between two kinds of worker. “3 men and 6 women” gives you a combined rate and no conversion at all. Reading which one you have is most of the model.

Set the equivalence and price any mixed group →
OR means the two groups are interchangeable — convert one into the other. AND means they worked together — you have one combined rate.

01 The idea

Two currencies, one exchange rate

If 3 men or 6 women can do a piece of work in 7 days, then 3 men and 6 women are equally strong — either group, on its own, finishes the same job in the same time. Divide both sides by 3 and one man is worth two women.

That exchange rate is the whole model. Once you have it, pick one worker type as your unit and convert everything into it. Take a woman as 1 unit, so a man is 2. The job is 6 women for 7 days, which is 42 woman-days.

Now any group can be priced. Five men and four women is 5 × 2 + 4 = 14 woman-units per day, and 42 woman-days at 14 a day is 3 days. Two kinds of worker became one, and the question became ordinary counting.

The word matters absolutely. “3 men and 6 women complete a work in 7 days” is a different sentence: it tells you the combined rate of that mixed group and nothing about how a man compares with a woman. You cannot get a conversion from it, and a question phrased that way needs a second equation before it can be solved.

Convert one worker type into the other, choose the smaller as your unit, and count the job in those units. Two kinds of worker become one.
The OR form“a men or b women take the same time”. Gives the equivalence a men = b women, hence 1 man = b/a women.
The AND form“a men and b women together take T days”. Gives one combined rate and no conversion. Two such statements are needed to separate the two rates.
Woman-daysThe job measured in units of one woman working one day. Any equivalent unit works; choosing the weaker worker keeps the numbers whole.

02 Worked example

3 men or 6 women in 7 days → 5 men and 4 women

The source question. If 3 men or 6 women can do a piece of work in 7 days, in how many days can 5 men and 4 women do the same work?

1
Read the OR as an equivalenceEither group alone does the same job in the same 7 days, so the two groups are equally strong.3 men = 6 women
2
Price one manDivide both sides by three.1 man = 2 women
3
Size the job in woman-daysSix women take 7 days, so the job is that product. Using women as the unit avoids fractions.W = 6 × 7 = 42 woman-days
4
Price the new groupEach of the five men counts as two women, plus the four actual women.5 × 2 + 4 = 14 woman-units per day
5
DivideForty-two woman-days of work at fourteen a day.42 / 14 = 3 days

Check it a second way to be sure the conversion was right: 5 men and 4 women is equivalent to 14 women, and if 6 women take 7 days then 14 women take 6/14 × 7 = 3 days. Agreement between the unit-counting route and the inverse-proportion route means your exchange rate is correct — and the exchange rate is the only thing in this model that can be wrong.

03 The method

OR against AND, and three worker types

The distinction in the title, then what to do when the question gives you the harder form.

OR form: a men = b women, so 1 man = b/a women. Job size = b × T woman-days. AND form: a·m + b·w = 1/T, one equation in two unknowns.
Two AND statements give you both rates. If 3 men and 4 women take 10 days, and 5 men and 2 women take 8 days, you have two simultaneous equations in the men’s and women’s rates — solve them and proceed as usual. A single AND statement is not enough, and a question giving only one is either asking about that exact group or has another clue you have missed.
The question saysWhat you getNext step
3 men OR 6 women, 7 days1 man = 2 womenConvert and count
4 men OR 6 women, 10 days1 man = 1.5 womenConvert and count
3 men AND 4 women, 10 daysone combined rateNeed a second statement
Two AND statementstwo equationsSolve simultaneously
2 men = 3 women = 4 boysa three-way ratePick the weakest as the unit
Treating AND as ORwronginvents a conversion
Double the workdouble the unit-daysThen divide as usual

05 Cheat sheet

Model 7 on one page

The conversion, the two phrasings, and the extensions.

CaseRouteOn 3 men or 6 women, 7 days
Exchange ratea men = b women1 man = 2 women
Job sizeb × T woman-days42 woman-days
Price a groupΣ (count × value)5M + 4W = 14 units
Time for a groupW / group strength42/14 = 3 days
Three worker typeschain the equivalences2M = 3W = 4 boys
Double the workdouble the unit-days84 woman-days
Reading AND as ORwronginvents an exchange rate
OR gives a conversion, AND does not“3 men or 6 women” means the groups are interchangeable. “3 men and 6 women” gives one combined rate and needs a second statement to separate the two.
Use the weaker worker as the unitIt keeps the exchange rate at or above 1 and the arithmetic in whole numbers. Either choice gives the same answer.
Verify by inverse proportionConvert the group into a single worker type and use M₁D₁ = M₂D₂. Agreement with the unit count confirms the exchange rate.

06 Where & why

Where Model 7 shows up

The last of the seven models, and the one that most rewards careful reading over arithmetic.

SSC CGL · RRB · CPO
“x men or y women”

The standard OR form. Two lines once the exchange rate is out, and the whole difficulty is spotting that “or” is doing the work.

Bank PO Mains
Two AND statements

Simultaneous equations in the two rates. Harder, and the giveaway is being handed two mixed groups with two times.

Three worker types
Men, women and boys

“2 men = 3 women = 4 boys.” Chain the equivalences and take the weakest as the unit; nothing else changes.

Double-work variants
“…half the work” or “…twice the work”

Scale the job in unit-days before dividing. It is the chain rule’s direct relation appearing inside this model.

Circle the word “or” or “and” before you start. It is the only thing that decides whether you have an exchange rate or a combined rate, and everything downstream depends on which.

07 Interview questions

What gets asked

Nine, and the second is the distinction the whole model turns on.

3 men or 6 women do a job in 7 days. How long for 5 men and 4 women?
Three days. The OR means 3 men = 6 women, so 1 man = 2 women. The job is 6 × 7 = 42 woman-days, and 5 men plus 4 women is 5 × 2 + 4 = 14 woman-units a day, giving 42/14 = 3 days.
What is the difference between “3 men or 6 women” and “3 men and 6 women”?
The OR says either group alone does the job in that time, which makes them equally strong and gives you an exchange rate. The AND says that mixed group together takes that time, which gives one combined rate and no way to compare a man with a woman. They are completely different statements.
Which worker type should you use as the unit?
The weaker one, usually. It keeps the exchange rate at or above 1 and the arithmetic in whole numbers. Either choice gives the same final answer, so it is purely about convenience.
How do you handle a question that only gives an AND statement?
You cannot separate the two rates from one AND statement — it is one equation in two unknowns. Either the question is asking only about that exact group, or there is a second statement you have not used. Two AND statements give simultaneous equations you can solve.
4 men or 6 women do a job in 10 days. How long for 6 men and 1 woman?
About 6.15 days. Here 1 man = 1.5 women, so the job is 60 woman-days and the group is 6 × 1.5 + 1 = 10 woman-units a day, giving 6 days. The exchange rate being fractional is fine — it just means men are not a whole number of women.
How do you extend this to three worker types?
Chain the equivalences. From “2 men = 3 women = 4 boys”, take a boy as 3 units, a woman as 4 and a man as 6, so all three are whole. Then count and divide exactly as with two types.
What if the second group has to do double the work?
Double the job size in unit-days before dividing. That is the direct proportion from the chain rule appearing inside this model: more work needs more unit-days, and the group strength is unchanged.
How would you verify a Model 7 answer?
Convert the mixed group into a single worker type and use inverse proportion. If 5 men and 4 women is 14 women, and 6 women take 7 days, then 14 women take 6/14 × 7 = 3 days. Agreement with the unit-counting route confirms the exchange rate, which is the only thing that can be wrong.
Is there anything questionable about the premise of this model?
Yes, and it is worth being aware of. The arithmetic treats worker categories as having fixed relative productivity, which is a modelling convenience rather than a fact about people. Exam questions use the convention, so answer them on their own terms — but the conversion rate is a given in the problem, not a claim about the world.

08 Practice problems

Six conversions

Circle the “or” or the “and” in each one first. Two of these give you the harder form.

The standard OR

Easy
If 4 men or 6 women can do a piece of work in 10 days, in how many days can 6 men and 1 woman do the same work?
Follow-up
The exchange rate comes out fractional here, which is fine and is the point of including it. Check your answer against the inverse-proportion route.
Show the hint
Four men equal six women, so divide both sides by four.

Convert and scale

Easy
If 2 men or 5 boys can complete a job in 20 days, in how many days can 4 men and 5 boys complete it?
Follow-up
The exchange rate is 1 man = 2.5 boys. Take the boy as the unit and everything stays manageable.
Show the hint
Size the job in boy-days first, then price the mixed group.

Three worker types

Medium
Given that 2 men = 3 women = 4 boys in working capacity, and that 4 men can complete a job in 9 days, in how many days can 3 women and 4 boys complete it?
Follow-up
Chain the two equivalences into one common unit before counting anything. Choosing the unit so all three come out whole is the step that makes this easy rather than fiddly.
Show the hint
Take a boy as 3 units; then a woman is 4 and a man is 6.

Double the work

Medium
If 3 men or 6 women can complete a piece of work in 7 days, in how many days can 4 men and 6 women complete twice that work?
Follow-up
Two changes at once: a different group and a bigger job. Scale the job in unit-days before dividing, and predict whether the answer should be above or below 7 days.
Show the hint
The job is now 84 woman-days rather than 42.

The AND form

Medium
2 men and 3 women can complete a job in 3 days, while 1 man and 3 women can complete the same job in 4 days. Find how long 1 man alone and 1 woman alone would each take.
Follow-up
Two AND statements, so two simultaneous equations in the two rates. Note that neither statement alone would have been enough — that is the difference from the OR form.
Show the hint
Let a man’s daily work be m and a woman’s be w, then write both statements as equations in m and w.

When can you not get a conversion?

Hard
(a) From “6 men and 8 women can complete a job in 5 days”, explain why you cannot determine how many women one man is worth. (b) Show that adding “3 men and 4 women can complete it in 10 days” still does not determine it, and explain what is special about that second statement. (c) Give a second statement that would determine it, and solve for the conversion.
Follow-up
Part (b) is the trap: the second group is exactly half the first, so the second equation is a multiple of the first and adds no information. Recognising a dependent equation — rather than grinding at two equations that cannot be solved — is a genuinely useful skill and it is why examiners occasionally set it.
Show the hint
Write both statements as equations in m and w and compare them. What happens when one is a scalar multiple of the other?