Aptitude · Problems on Ages · Model 6

Find the one hard number, then climb the ladder

Model 6 is a paragraph linking three, four or five people. It looks unmanageable and it is the least mathematical model in the chapter: every sentence is one link, exactly one sentence carries a real number, and the rest is substitution in order.

Decode a four-person paragraph one sentence at a time →

Read one sentence, write one equation, stop. Then find the sentence with an actual number in it — that is your anchor — and climb the chain one link at a time.

01 The idea

The gadget shop, and why the paragraph is not the problem

The Apple AirPods cost ₹2,000 more than the Sony earbuds. The Sony earbuds cost three times as much as the boAt Airdopes. The boAt Airdopes cost ₹1,500. Nobody finds that hard. You write Apple = Sony + 2000, then Sony = 3 × boAt, then spot that boAt has a real price attached — so Sony is ₹4,500 and Apple is ₹6,500.

Model 6 age questions are that shopping trip with names instead of brands. "In 10 years P will be 50. P is twice as old as Q. Q is 5 years older than R. R is half the age of S." Four people, three comparisons and one hard number. The hard number is the anchor, and the answer is three substitutions away.

What makes these questions feel hard is reading them as one object. A paragraph with four names in it does not fit in your head, so you try to hold it there and lose track. The fix is mechanical: take one sentence, write the equation it gives you, and refuse to look ahead. Three short lines on paper beat one long sentence in your head.

This lesson uses P 40, Q 20, R 15 and S 30 throughout. Those four also happen to add up to 105, which matters, because sometimes the paragraph gives you no hard number at all — only a total. Then the anchor is the sum, and the first move is to write everybody in terms of one person before using it.

One sentence, one equation. Then find the anchor — the only sentence with a real number, or the total — and substitute your way along the chain. Never solve two equations at once.

LinkA sentence comparing exactly two people: "P is twice as old as Q", "Q is 5 years older than R". It gives one equation in two unknowns and is never solvable by itself, which is why there is nothing to be gained by staring at it.

AnchorThe one fact carrying a real number — "R is 20 years old", "in 10 years P will be 50" — or, when no person has one, the total of all the ages. Every Model 6 question has exactly one anchor and finding it is the whole strategy.

ClimbingSubstituting the anchor into the link that touches it, then that result into the next link, and so on. Each step is one line and uses only the previous line, so a five-person chain is no harder than a two-person one.

02 Worked example

Four people, three substitutions

In 10 years P will be 50. P is twice as old as Q. Q is 5 years older than R. R is half the age of S. Find S's age. Take the sentences strictly in order, and notice that you never write two equations on the same line.

Write each sentence as a link, solving nothingThree comparisons. Each one is one equation in two unknowns, so none of them can be solved yet — and trying to is what makes this feel difficult.P = 2Q · Q = R + 5 · R = S / 2

Find the anchorOne sentence has an actual number in it rather than a comparison. That is where the arithmetic starts, and it is worth scanning for it before writing anything.P + 10 = 50 → P = 40

Climb to QSubstitute P into the link that mentions P. Note the direction: P is the bigger one, so Q is P halved, not P doubled.P = 2Q → 40 = 2Q → Q = 20

Climb to R, then to SEach line uses only the number from the line above and the sentence that mentions it. "R is half the age of S" makes S the bigger one, so S is R doubled.Q = R + 5 → R = 20 − 5 = 15 R = S/2 → S = 2 × 15 = 30

Read all four back into the paragraphP 40, Q 20, R 15, S 30. Check every sentence, because a single reversed comparison propagates down the whole chain without ever looking wrong.40 + 10 = 50 ✓ 40 = 2(20) ✓ 20 = 15 + 5 ✓ 15 = 30/2 ✓

The total of those four ages is 105. That matters, because a common variant gives you no hard number at all and instead says "the four ages add up to 105". Then write Q, R and S in terms of P — here Q = P/2, R = P/2 − 5, S = P − 10 — add them to get 3P − 15 = 105, and P is 40 again. Same chain, different anchor.

03 The method

The chain decoder, checked on all four ages

Every row is a sentence that could appear in this question, with the equation it produces and a check against P 40, Q 20, R 15, S 30. Note rows 2 and 3, and 4 and 5: different words, identical link.

Three moves, in order. (1) One sentence, one equation — write all the links down before solving anything. (2) Find the anchor: the sentence with a real number, or the total. (3) Substitute along the chain, one link per line. If the anchor is a total, first write every age as (multiple of P) + (shift) and add.

When the chain has no anchor at all, look for a loop. "A grandfather is 5 times as old as his grandson; the father is 26 years younger than the grandfather; the grandson is 22 years younger than the father" closes back on itself, so substituting all the way round leaves one equation in one unknown. And if two links multiply rather than add, expect a quadratic and discard the negative root — there is always exactly one usable answer.

Sentence	The link	Check on P 40, Q 20, R 15, S 30
In 10 years P will be 50	`P + 10 = 50`	P = 40 ✓
P is twice as old as Q	`P = 2Q`	40 = 2 × 20 ✓
P is 20 years older than Q	`P = Q + 20`	same link, different words
Q is 5 years older than R	`Q = R + 5`	20 = 15 + 5 ✓
Q is 4/3 times as old as R	`3Q = 4R`	60 = 60 ✓ — same link again
R is half the age of S	`S = 2R`	30 = 2 × 15 ✓
The four ages add up to 105	`P + Q + R + S = 105`	40 + 20 + 15 + 30 = 105 ✓
Reading the whole paragraph before writing	guarantees a lost link	four sentences, four lines, in order

05 Cheat sheet

Model 6 on one page

Five habits and two traps. None of it is mathematics; all of it is bookkeeping, which is exactly why it is worth writing down.

Situation	What to do	On P 40, Q 20, R 15, S 30
A paragraph with three or more names	one sentence, one equation, in order	P = 2Q, Q = R + 5, R = S/2
One sentence has a real number	that is the anchor — start there	P + 10 = 50 → P = 40
No person has a real number, but a total is given	write everyone in terms of one person	3P − 15 = 105 → P = 40
The chain closes back on itself	substitute all the way round	grandfather = 5(grandfather − 48)
Two links multiply instead of adding	expect a quadratic	discard the negative root
A group appears on one side	a sum of k people gains kt years	two children gain 40 in 20 years
Solving two links simultaneously	never necessary here	substitute one at a time

Anchor first, alwaysScan for the sentence with a real number before you write anything. Starting at the top of the paragraph and hoping means you build the chain in the wrong direction and have to rearrange every line.

Check the direction of every comparison"P is twice as old as Q" makes Q the smaller one. A single reversed link gives an answer that satisfies every sentence except the one you reversed, so it never looks wrong until you check it.

Length is reading load, not difficultyA five-person chain needs four substitutions and no simultaneous equations. Papers use the length to consume your clock, so the whole defence is writing short lines fast.

06 Where & why

Where the long chain shows up

Model 6 is the standard way to make an easy question expensive. The arithmetic is the lightest in the chapter and the time cost is the highest.

Bank PO · IBPS · RRB

Four-name paragraphs

The house style, usually with the anchor buried in the last sentence so that a candidate reading top-down builds the chain backwards. Scanning for the number first is worth about thirty seconds per question.

TCS Digital · Infosys

Chains with no anchor, only a sum

"P is one-third of Q, Q is 5 years older than R, R is half of S, and the four total 50." One extra step — express everyone in P — and it is the same problem.

SSC · CAT

Closed loops and product links

A loop leaves one equation in one unknown; a product link leaves a quadratic with one positive root. Both look alarming and both collapse in two lines.

Data sufficiency sets

Which two statements complete the chain?

A chain of links is never sufficient without an anchor, and an anchor is never sufficient without the links. That pairing is nearly the whole design of ages sufficiency questions.

There is no technique to learn in Model 6 beyond discipline. Write short lines, take the sentences one at a time, find the anchor before you start, and check the direction of every comparison at the end.

07 Interview questions

The chain questions interviewers ask

Ten in escalating order — the method, the anchorless variant, the loop, the quadratic, and the honest question about whether long questions are worth attempting.

How do you approach a paragraph with four people in it?

One sentence, one equation, in the order printed, solving nothing. Then find the anchor — the only sentence with a real number — and substitute along the chain one link per line. The paragraph is unmanageable in your head and trivial on paper, and that difference is the entire question.

What exactly is the anchor?

The single fact that pins a real number to somebody: "R is 20", "in 10 years P will be 50". Every other sentence is a comparison, which is one equation in two unknowns and unsolvable alone. Exactly one anchor is what makes the system determinate.

What if no sentence gives an actual age?

Then the anchor is the total. Write every age as a multiple of one chosen person plus a shift, add them, and set the result equal to the given sum. For our chain that is 3P − 15 = 105, so P is 40 — one extra line, same answer.

Whom do you choose as the base person when there is no anchor?

Whoever appears in the most links, which is usually one end of the chain. Choosing a middle person means expressing things in both directions and inverting fractions, which is where errors come from. Either end works; the busiest end works best.

"P is twice as old as Q." Which one is older?

P. The multiplier attaches to the smaller age, so Q = P/2. Reversing this is the most damaging error in Model 6, because a reversed link still produces a self-consistent set of numbers — they just do not satisfy the sentence you reversed.

A chain closes back on itself with no anchor anywhere. What then?

Substitute all the way round and the unknown reappears on both sides. "A grandfather is 5 times his grandson; the father is 26 years younger than the grandfather; the grandson is 22 years younger than the father" gives grandson = grandfather − 48, so G = 5(G − 48), and G is 60. A loop is an anchor in disguise.

What happens if a link is a product rather than a sum?

You get a quadratic instead of a linear equation. "A is 5 years older than B, B is 4 years older than C, and the product of A and C is 90" gives (B + 5)(B − 4) = 90, so B² + B − 110 = 0 and B is 10 or −11. Discard the negative root; there is always exactly one usable answer.

A chain has "the sum of his two children's ages" on one side. Anything different?

Yes, and it is the standard trap. A sum of two people gains two years per year, so in 20 years it gains 40 while the parent gains 20. Treating a group total as one person is the same error that dominates Model 5, and it appears here dressed as a chain.

Compare Model 6 with Model 1.

Hard

A is older than B by 5 years. B is older than C by 4 years. The product of the ages of A and C is 90. Find B's age.

Follow-up

The two links are additive but the anchor is multiplicative, so substituting produces a quadratic rather than a linear equation. It has two roots and only one of them is an age, which is worth noticing: the algebra cannot tell you which root to keep, and the physical meaning can. That is the same judgement you use whenever a guarded solver refuses an input.

Show the hint

Write A and C in terms of B, multiply out, and factorise B² + B − 110.