Aptitude · Problems on Ages · Model 5

A newborn adds nothing to the total and one to the count

This is the model students get wrong most often, and always for the same reason: they work in averages. An average is a ratio of two things that change independently. Track the total and the head count separately, and divide once, at the end.

Watch the total and the head count move separately →

Never add to an average. Total = average × head count in the first line, track the total, and divide only when you have the answer.

01 The idea

Why a birthday can lower the family average

Three friends share a flat and each earns ₹30,000 a month, so the average income in the room is ₹30,000. A student brother moves in earning nothing. Nobody's salary changed, but the average is now ₹90,000 over four people, which is ₹22,500. Adding a zero pulled the average down.

A newborn does exactly that to a family's ages. The baby adds 0 to the total age of the household and 1 to the number of people in it. Those are two different numbers landing on two different counters, and an average — which is one divided by the other — cannot represent that with a single addition. This is the whole reason Model 5 is hard.

Here is the scenario this lesson uses. A couple's average age at their marriage was 28, so their total was 56. Two years later they are 60 between them. A child is born: the total is still 60 but there are now three of them, so the average drops from 30 to 20 in an instant. Five years on, three people have each gained five years, so the total is 75 and the average is 25.

Read that last number carefully. Seven years after the wedding, every member of that family is seven years older than they were, and the family average has gone down from 28 to 25. Nothing is wrong. The average is not a person's age — it is a total divided by a head count, and the head count changed.

A total is additive; a head count is a separate counter; an average is neither. Convert to a total in line one, do every event on the total and the count separately, and divide once at the end.

Total ageThe sum of every member's age, average × head count. The only quantity in this model that behaves additively, which is why every step works on it and not on the average.

Head countHow many people the average is taken over. It changes on a birth, a marriage, a death or a departure — and it is what makes the average move without anybody having a birthday.

The per-year gainEach year the total rises by the current head count, not by one. Two people gain 2 years a year between them; three people gain 3. Using the old count after a birth is the second most common error in the model.

02 Worked example

Seven years pass and the average falls by three

The average age of a couple was 28 years at the time of their marriage. Two years later a child was born. Find the total age and the average age of the family when the child is 5 years old. Follow the total and the head count in separate columns and resist every temptation to add anything to 28.

Turn the average into a totalTwo people averaging 28 is 56 years between them. This is the only line in which the number 28 appears.total = 2 × 28 = 56 years members = 2

Age the couple to the child's birthTwo years pass with two members, so the total gains 2 × 2 = 4 years. Note it is four, not two — each of them had two birthdays.total = 56 + (2 × 2) = 60 years members = 2

Add the child: nothing to the total, one to the countAt the instant of birth the child is 0 years old, so the total is untouched. The head count goes from 2 to 3, and the average therefore drops from 30 to 20 without anybody ageing.total = 60 + 0 = 60 members = 3 average = 60 / 3 = 20

Age the whole family five more years, at three membersNow three people are ageing, so the total gains 3 × 5 = 15 years. Using the old count of two here would give 70 and a wrong answer.total = 60 + (3 × 5) = 75 years members = 3

Divide, onceSeventy-five years across three people. The total went up by 19 and the average went down by 3, which is the whole lesson in two numbers.average = 75 / 3 = 25 years (started at 28)

Compare it with the same couple having no child. Seven years after the wedding their total would be 56 + (2 × 7) = 70 and their average 35 — exactly 28 plus the seven years, because the head count never moved. The birth cost the average ten years. When membership is constant an average tracks the years; the moment it is not, the average stops being a thing you can add to.

03 The method

Every event, priced on the total and the count

Six events and one shortcut. The middle column is what you actually write down; the right-hand column checks each one against our family at the moment of the birth, when the total was 60 over 2 members.

Total = average × head count. Then, per event: a year passing adds the current head count to the total; a person of age x joining adds x to the total and 1 to the count; a person of age x leaving subtracts both; a replacement changes the total by (new − old) and the count by nothing.

The one-line version, when you only need the shift. Somebody of age x joining a group of N people averaging m moves the average by (x − m) / (N + 1). Our newborn: N = 2, m = 30, x = 0, so the shift is −30/3 = −10 and the average goes 30 → 20. For a straight replacement it is (new − old) / N, with no change of count.

Event	Total and head count	On our family (total 60, 2 members)
One year passes	total `+ N`, count unchanged	60 → 62, still 2
A baby is born	total `+ 0`, count `+ 1`	60 stays 60, but 2 → 3
A person aged x joins	total `+ x`, count `+ 1`	a bride of 24: 60 → 84, 2 → 3
A person aged x leaves or dies	total `− x`, count `− 1`	a 16-year-old: 60 → 44, 2 → 1
Old age r replaced by new age a	total `+ (a − r)`, count unchanged	55 out, 35 in: 60 → 40, still 2
A misrecorded age is corrected	total `+ (true − recorded)`, count unchanged	24 recorded as 14: total falls 10
"The average rises by the years that pass"	only if the count never changes	28 became 25 over seven years

05 Cheat sheet

Model 5 on one page

The first row is the one to internalise. The last two rows are the errors that account for nearly every wrong answer in this model.

Situation	What to write	On our family
An average is given	`total = average × count`	2 × 28 = 56
t years pass	`total + (count × t)`	56 + (2 × 2) = 60
A newborn arrives	`total + 0`, `count + 1`	60 over 3 → average 20
Someone aged x joins	`total + x`, `count + 1`	a bride of 24 → total 84
Someone aged x leaves	`total − x`, `count − 1`	a 16-year-old → 4 × 25 − 16 = 84 over 3
Find a joiner's age from an average shift	`x = (new count × new avg) − old total`	a family of 4 at 30 rising to 32: 5 × 32 − 4 × 30 = 40
Ageing the family at the old head count	wrong by (t × change in count)	70 instead of 75

A baby contributes 0 and 1Zero to the total, one to the head count. Two different counters, two different numbers, and no single addition to the average that can represent both.

A past total shifts by count times years"Their total was 40 six years ago" with two people means 52 today, not 46. Every average or total quoted at another moment needs the head count applied to the gap.

Divide once, at the endIf an average appears anywhere in the middle of your working, you are carrying a ratio through additions. Convert in, work in totals, convert out.

06 Where & why

Where the shifting average shows up

This is the model that separates the top of an aptitude section from the middle, and it is set as a thinking question rather than a speed one.

Bank PO · IBPS · RRB

"the average drops by 1 year 4 months"

A fractional shift in the average, chosen so that working in averages is unmanageable and working in totals is exact. 6 × (30 − 4/3) is 172 with no rounding.

TCS Digital · Amazon

A birth plus a marriage plus elapsed years

Several events on one family, each changing the head count. The only tractable route is a two-column running tally of total and count.

SSC · CAT

Replacement and misrecorded-age questions

"A teacher aged 45 replaces a student aged 15 and the average rises by 1" gives the head count directly, because a replacement leaves the count alone and 30 / 1 = 30 people.

Averages, mixtures, weighted marks

Any average over a changing population

Batch averages, alligation, a class mean after a re-mark — identical reasoning. The habit of converting to a total is worth far more than the age story it is taught with.

One sentence carries this model: convert the average to a total in your first line and do not write another average until the answer. Every wrong answer here comes from ignoring that.

07 Interview questions

The average questions that catch people

Ten in escalating order — the rule, the birth, the elapsed years, the replacement, and one honest look at a standard textbook answer that does not hold up.

Why is working in totals better than working in averages here?

Because a total is additive and an average is not. Every event in this model changes the head count as well as the sum, and an average is the sum divided by that count — so there is no single number you can add to the average to represent a birth. Convert once at the start, once at the end, and never in between.

What exactly does a newborn do to a family's average?

It adds 0 to the total age and 1 to the head count. So a family of 2 with a total of 60 has an average of 30, and the instant the baby arrives it is 60 over 3, which is 20. Nobody had a birthday and the average fell by ten years.

A family of 4 averages 30, a relative joins, and the average goes up by 2. How old is the relative?

The old total is 4 × 30 = 120 and the new total is 5 × 32 = 160, so the relative is 40. Note that the new average must be multiplied by the new count of five, which is where this question is usually lost.

Five years pass in a family of three. What happens to the total?

It rises by 15, because each of the three people gains five years. The per-year gain is the head count, not one. If the head count changed part-way through, you split the interval and use the count that applied in each part.

The total was 40 six years ago for two people. What is it now?

52. Two people over six years is 12 years of ageing, not 6. Any total or average quoted at another moment has to be walked across with the head count applied — this is the same arithmetic as the previous answer and it accounts for a large share of errors.

A 45-year-old teacher replaces a 15-year-old student and the average rises by exactly 1. How many people are there?

Thirty. The total rose by 30 and the count did not change, so the rise in the average is 30 / N = 1. A replacement is the friendliest event in the model precisely because the head count stays put.

A 60-year-old dies and a baby is born the same day in a family of 6 averaging 25. New average?

Fifteen. The total goes from 150 to 150 − 60 + 0 = 90, and the head count is unchanged at 6 because one person left and one arrived. 90 / 6 = 15. Two events, one net count change of zero.

Is there a shortcut for a single joiner?

Yes: the average shifts by (joiner's age − old average) / (new count). For our newborn joining two people averaging 30, that is (0 − 30) / 3 = −10. It is quick and it is worth knowing, but it only handles one event with no elapsed time, so the total method remains the default.

A standard question says: "the average age of a husband and wife was 25; after a baby was born the average became 18; how old is the baby?" The book says 4. Is it right?

Hard

The average age of a husband and wife was 27 years when their twins were born. When the twins are 4 years old the husband dies, and the average age of the remaining family — the mother and the two twins — is 14 years. How old was the husband when he died?

Follow-up

The head count runs 2, then 4, then 3, and the twins contribute 0 at their birth but 8 to the total four years later. It also closes the loop with Model 4: the four years since the twins' birth is exactly the interval the couple's total has to be carried across, so the milestone and the average are doing the same job in different clothes.

Show the hint

Total for the couple at the twins' birth is 54; carry it forward four years for two people, then add the twins before subtracting the husband.