Aptitude · Problems on Ages · Model 4

A birth or a wedding is a number in disguise

Model 4 buries the age gap inside a life event. "The mother was 26 when her son was born" is not a story about 26 years ago — it is a statement that she will be 26 years older than him forever. Extract that one number and the question collapses.

Turn a life event into one number and solve →

A parent's age at a child's birth is the permanent gap between them. Read it off, write it down, and never look at the event again.

01 The idea

The number that survives the story

Vijay is 30 today and his daughter Neha is born this morning. At this instant she is 0 and he is 30, so they are 30 years apart. When she turns 10 he is 40. When she is 50 he is 80. The gap never moves, because a gap never moves — and the number 30 was handed to you by the birth itself.

That is the entire content of Model 4. An examiner writes "Vijay was 30 when Neha was born" instead of "Vijay is 30 years older than Neha" because the first sentence sounds like it needs a timeline and the second obviously does not. Both say the same thing.

This lesson uses one pair throughout: a mother of 39 and her son of 13, a gap of 26. The question that produces them reads "the mother was 26 years old when her son was born, and today his age is one-third of hers". Step one turns the milestone into 26. Step two turns "one-third" into a ratio of 3 : 1 whose gap is 2 units. Two units are 26 years, so a unit is 13, and the ages are 39 and 13.

The same trick hides behind weddings. "At the time of their marriage ten years ago, the husband was four years older than his wife" tells you he is four years older today, and the ten years are decoration. It hides behind siblings too: "the sister was born when the brother was four" fixes a four-year gap between them. Any sentence that pins two people to one moment is a gap in disguise.

Every life event in an ages question exists to hand you one permanent difference. Convert the event into that number in your first line, then solve an ordinary ratio or sum problem.

MilestoneA sentence that fixes two people's ages at one specific moment — a birth, a wedding, a sibling's arrival. Its only use is the difference it implies, which is permanent.

Hidden gapThe difference the milestone supplies. For a birth it is the parent's age at that birth, because the child was 0. For a wedding or any other event it is whatever difference the sentence states, and the date is irrelevant.

AnchoringChoosing one person's age as the unknown and expressing everyone else through the gaps. With several milestones, anchor on whoever appears in the most of them — usually the youngest child.

02 Worked example

One milestone, one ratio, four lines

A mother was 26 years old when her son was born. Today his age is exactly one-third of hers. Find both present ages. The event contributes exactly one number, and then this is a ratio question with a known difference.

Convert the milestone into a gapAt the son's birth he was 0 and she was 26, so she is 26 years older than him. That is true today and at every other moment.Mother − Son = 26 years

Convert the second fact into a ratio"His age is one-third of hers" means Mother : Son = 3 : 1. Write the bigger age first so the ratio's difference is positive.Mother : Son = 3 : 1

Match the gap in units to the gap in yearsIn 3 : 1 the difference is 2 units. Those 2 units are the same 26 years, because the two facts describe the same present moment.3 − 1 = 2 units = 26 years → 1 unit = 13 years

Price both agesMultiply each share by the unit. Neither answer is 26 — that number was only ever the difference.Mother = 3 × 13 = 39 Son = 1 × 13 = 13

Rebuild the milestone as a checkWind both ages back thirteen years to the son's birth. The mother must land exactly on 26.39 − 13 = 26 ✓ 39 / 13 = 3 ✓

Watch where the 26 went. It never became an age; it became a difference, and it was spent in step 3 against the ratio's 2 units. A student who writes "mother = 26 + something" and tries to track thirteen years of history is solving a harder problem than the one that was set.

03 The method

Every milestone phrasing, one constant

Each row below is a different way of stating the same 26-year gap, and every one is a true sentence about a mother of 39 and a son of 13. The dates in them are noise.

Birth milestone: parent's age at the child's birth = the permanent gap, because the child was 0. Any other milestone: the difference stated at that moment is the difference at every moment. Then with a ratio a : b, 1 unit = gap / (a − b); with a sum S, older = (S + gap)/2 and younger = (S − gap)/2.

Two habits. Ignore the date of the event entirely — "ten years ago at their marriage" contributes nothing but the gap, and adding ten to it is the classic error. And when a question carries several milestones, anchor everything on the youngest person and convert each event into a gap from that anchor, one sentence at a time.

The milestone sentence	Hidden constant	Check on mother 39, son 13
The mother was 26 when her son was born	gap = 26	39 − 13 = 26 ✓
The son was born 26 years after his mother	gap = 26	same statement, reworded
At their meeting 10 years ago she was 26 years older	gap = 26	the 10 years contribute nothing
When the son was 5, the mother was 31	gap = 26	31 − 5 = 26 ✓
Subtracting 26 from her age gives his age	gap = 26	39 − 26 = 13 ✓
... and today his age is one-third of hers	2 units = 26	1 unit = 13 → 39 and 13
"She was 26 then, so the gap today is 26 + 10"	never legal	the gap is 26, not 36

05 Cheat sheet

Model 4 on one page

Four conversions and three traps. Every conversion produces one number, and after that you are in Model 1 territory.

What the question says	The constant it gives	On mother 39, son 13
Parent was P at the child's birth	`gap = P`	26 → 39 − 13
Sibling B born when sibling A was k	`gap = k`	a sibling milestone works the same
At an event, one was d years older	`gap = d`	the date is irrelevant
Gap plus a ratio a : b	`1 unit = gap / (a − b)`	26 / 2 = 13
Gap plus a sum S	`(S ± gap) / 2`	(52 + 26)/2 = 39
Sum quoted t years ago	add 2t for two people	40 six years ago → 52 today
Adding the event's date to the gap	never legal	gap is 26, not 26 + 10

A newborn is 0, which is why a birth gives the gapThe child contributes nothing to the difference at that instant, so the parent's age is the whole of it. That is the only reason the trick works, and it is worth being able to say out loud.

The date of the event never enters the arithmetic"Ten years ago at their wedding" and "at their wedding" give identical answers. If the date appears in your working, you have imported a constraint the question did not set.

With several milestones, anchor on the youngestSet the youngest child's age as the unknown and express every parent and sibling as that plus a gap. It keeps every sign positive and stops you re-deriving the same interval twice.

06 Where & why

Where milestones show up

Model 4 questions are reading tests wearing arithmetic. The sums are the easiest in the chapter and the wrong answers come from the sentences.

Bank PO · IBPS · RRB

"was 26 when the son was born"

The standard phrasing, usually paired with a present ratio. One line of translation and one division finishes it, so it is set as a fifteen-second question.

TCS Digital · Amazon

Two milestones, two parents

"The father was 34 at the daughter's birth, the mother 36 at the son's, the son is 5 years older than the daughter." Two gaps from one anchor and the parents' difference falls out.

Data sufficiency sets

A milestone alone is never enough

"The mother was 26 when her son was born" fixes a difference and no age, so it needs a second statement. That pairing is one of the most common sufficiency questions in the chapter.

Long multi-generation chains

Grandfather, father, child

Each generation contributes one milestone and therefore one gap. Chain them from the youngest upwards and the question becomes addition, however long the paragraph is.

The only skill here is reading a sentence about a wedding or a birth and writing a single subtraction. Once the gap is on paper you are solving a question you already know how to solve.

07 Interview questions

The milestone questions interviewers ask

Ten in escalating order — the conversion, why it works, the multi-milestone case, and the sufficiency question that catches people.

"A father was 30 when his son was born." What does that give you?

That the father is 30 years older than the son, permanently. At that instant the son was 0, so the whole difference was the father's age. The sentence is a difference wearing a story, and the date it happened is not needed.

Why is the parent's age at the birth exactly the gap and not something else?

Because a newborn is 0 years old, so the child contributes nothing to the difference at that moment. Difference is invariant in time, so whatever it was at the birth it still is. That is the one sentence to be able to say if an interviewer pushes on it.

"At their marriage ten years ago the husband was 4 years older than his wife." What is the gap today?

Four years. The ten is decoration — a difference stated at any moment is the difference at every moment. Adding the ten to get 14 is the standard trap and it is a difference-versus-shift confusion, not an arithmetic slip.

A mother was 26 at her son's birth and his age is now one-third of hers. Solve it.

The ratio 3 : 1 has a gap of 2 units, and that gap is 26 years, so a unit is 13. She is 39 and he is 13. Check by winding back thirteen years: she was 26 and he was 0.

The gap is 26 and their ages total 52. Find both.

Half the total plus half the gap gives the older: (52 + 26)/2 = 39. Half the total minus half the gap gives the younger: (52 − 26)/2 = 13. A sum and a difference together is the easiest pair of facts in all of quant.

A question gives a sum from six years ago. What do you do first?

Add 12, not 6, because there are two people and each has gained six years. So a total of 40 six years ago is a total of 52 today. Forgetting to multiply by the head count is the same error that dominates Model 5.

"A father was 34 at his daughter's birth; the mother was 36 at her son's birth; the son is 5 years older than the daughter." How much older is the mother than the father?

Anchor on the daughter. The father is 34 years older than her. The son is 5 years older than her, and the mother is 36 years older than the son, so she is 41 years older than the daughter. The mother is therefore 41 − 34 = 7 years older than the father.

Is a milestone ever sufficient on its own?

No. It gives one equation — a difference — in two unknowns, so it fixes how far apart two people are and nothing about how old either is. It needs a ratio, a sum or one actual age alongside it, which is exactly why it appears so often in data sufficiency sets.

Compare Model 4 with Model 1.

Hard

My father was 45 years old when my brother was born. My mother was 34 when I was born. My sister is 5 years older than me, and my brother was 7 years old when my sister was born. What was my father's age when my sister was born?

Follow-up

Two of the four facts are never needed, and finding out which two is the actual question. A long milestone paragraph is a filtering exercise: identify what the question asks for, find the shortest chain of milestones that reaches it, and refuse to compute anything else.

Show the hint

The question asks for the father at the sister's birth. Which single sentence links the father to the brother, and which links the brother to the sister?