Aptitude · Problems on Ages · Model 3

One age goes back, the other goes forward

Q: Translate: "A's age 4 years ago was equal to B's age 3 years hence."

(A − 4) = (B + 3) , so A − B = 7 . This is the 1 : 1 split, and it is generous: it hands you a permanent age gap dressed up as time travel. Write both brackets and it falls out in one line.

Model 3 breaks the habit every earlier lesson built. The two ages in the ratio are not in the same year, so the age gap between the two people is no longer the constant you can spend. There is still a constant — it is just a different one.

Split the two time zones and watch each unwind →

In a split problem the two bracketed ages are not a photograph of one moment. Write each bracket separately, and unwind each one by its own shift in its own direction.

01 The idea

A selfie where the two faces are from different years

Imagine Vijay and Radha taking one photograph. Vijay uses a filter that shows him as he was five years ago. Radha uses one that shows her as she will be ten years from now. Looking at that single image, can you subtract the two faces to get the age gap between them? No — they are not in the same year.

That is exactly what a Model 3 question hands you. "The ratio of Neha's age five years ago to Vijay's age five years hence is 1 : 3." The 1 belongs to a moment in the past and the 3 to a moment in the future. Every reflex from Models 1 and 2 — scale the gaps, match the units, spend the constant difference — assumes both numbers come from one moment, and here they do not.

So write each age in its own bracket and touch nothing else. Neha five years ago is (N − 5). Vijay five years hence is (V + 5). The ratio goes between those two brackets exactly as printed, and the second given fact — a gap, a sum or a present ratio — is what makes it solvable.

There is still an invariant, and it is worth finding because it turns the algebra into two lines. The difference between the two brackets is a fixed number: (V + 5) − (N − 5) is (V − N) + 10. If the question tells you Vijay is 10 years older than Neha, that difference is 20 — and a ratio of 1 : 3 with a difference of 20 gives brackets of 10 and 30 immediately.

The gap between two people is constant, but in a split problem the two numbers in the ratio are not their ages at one moment. What is constant is the gap between the two brackets: the age gap plus every year the shifts pulled them apart.

Split shiftOne age moved backwards and the other forwards, or two ages moved by different amounts. The defining feature of Model 3, and the reason the ordinary gap trick does not apply.

The bracketOne person's age at one specific moment, written with its shift attached and left alone: (N − 5), (V + 5). The ratio is a statement about brackets, not about present ages.

Bracket difference(B + q) − (A − p) = (B − A) + p + q. Still a constant, and still computable from a gap or a sum — which is what makes the two-line route possible.

02 Worked example

Two brackets, one link, four lines

The ratio of Neha's age 5 years ago to Vijay's age 5 years hence is 1 : 3. Vijay is 10 years older than Neha. Find Vijay's present age. Notice what you are not allowed to do: the 1 and the 3 are three years apart in ratio but ten years apart in time, so they cannot be compared as a snapshot.

Write each age in its own bracketNeha goes five years back; Vijay goes five years forward. Two shifts, two directions, and no attempt yet to combine them.(N − 5) / (V + 5) = 1 / 3

Use the link to get down to one unknown"Vijay is 10 years older than Neha" is a present-tense fact, so it holds at every moment. Substitute it into the bracket that mentions Vijay.V = N + 10 → (N − 5) / (N + 10 + 5) = 1 / 3

Simplify the second bracket, then cross-multiplyCollapse N + 10 + 5 into N + 15 before cross-multiplying, so there is only one chance to make an arithmetic slip instead of two.(N − 5) / (N + 15) = 1 / 3 → 3(N − 5) = N + 15 → 3N − 15 = N + 15

Solve, then unwind each shift separatelyN is a present age already. Vijay comes from the link, not from the bracket — the bracket held his age five years from now.2N = 30 → N = 15, so V = 15 + 10 = 25

Rebuild the split ratio to checkTake Neha back five and Vijay forward five, exactly as the question said, and confirm the ratio comes out at 1 : 3.(15 − 5) : (25 + 5) = 10 : 30 = 1 : 3 ✓

Two numbers to keep apart. Neha and Vijay are 10 years apart in age. The two brackets, 10 and 30, are 20 apart — the 10-year age gap plus the 5 years one went back plus the 5 the other went forward. A student who uses 10 where 20 belongs gets a clean-looking wrong answer, and that single confusion is what Model 3 tests.

03 The method

The bracket-difference route in four lines

The equation above always works. But when the link is a gap or a sum, the bracket difference gives you the same answer without ever writing an equation, and it keeps the constant-gap idea alive in a form that survives the split.

Let X = (A − p) and Y = (B + q), with X : Y = m : n. Then Y − X = (B − A) + p + q and X + Y = (A + B) − p + q. A gap link prices the first of those, a sum link the second — and then 1 unit = constant / (units it spans).

The two traps. First, X and Y are not present ages: unwind each by its own shift, A = X + p and B = Y − q, in opposite directions. Second, when the link is itself a ratio there is no constant to spend, so fall back to substitution — the shortcut is not universal and pretending otherwise is worse than not knowing it.

Line	What you do	On the worked example
1	Name the two brackets	X = Neha 5 years ago, Y = Vijay 5 years hence
2	Write the split ratio against them	X : Y = 1 : 3
3	Price the bracket difference from the link	Y − X = 10 + 5 + 5 = 20 years
4	Spend it on the units	3 − 1 = 2 units = 20, so 1 unit = 10
5	Read the brackets	X = 10 and Y = 30
6	Unwind each shift in its own direction	Neha = 10 + 5 = 15, Vijay = 30 − 5 = 25
7	Never subtract X from Y as an age gap	20 is the bracket gap; 10 is the age gap

05 Cheat sheet

Model 3 on one page

The first three rows are the method. The last three are the Model 1 habits that have to be switched off, and they are where the marks go.

Situation	What to do	On Neha 15, Vijay 25
Ratio across two different moments	one bracket per person, unwound separately	(15 − 5) : (25 + 5) = 1 : 3
Link is a gap	`Y − X = gap + p + q`	20 = 10 + 5 + 5
Link is a sum	`X + Y = sum − p + q`	40 = 40 − 5 + 5
Link is itself a ratio	no constant — substitute	3 : 5 present → B = (5/3)A
Scaling the two ratios to a common gap	does not apply here	the brackets are not one snapshot
Reading X and Y as present ages	wrong by p and q	10 and 30 are not 15 and 25
Present gap vs bracket gap	never the same number	10 versus 20

Write the equation exactly as printedThe order of the words fixes which age goes back and which goes forward. There is no symmetry to exploit here, so copying the sentence literally into brackets is the whole of step one.

The bracket gap is the constant, not the age gapIt equals the age gap plus both shifts. Using the age gap where the bracket gap belongs produces an answer that is wrong by exactly p + q units of the ratio.

Unwind in opposite directionsA = X + p and B = Y − q. Adding the same number to both, out of Model 1 habit, is the most common single error in this model.

06 Where & why

Where the split shows up

Model 3 is the discriminating question in an ages set. Papers use it when they want to separate candidates who understood Model 1 from candidates who memorised it.

TCS Digital · Amazon

"the ratio of A's age 3 years ago to B's age 7 years hence"

Printed exactly like a Model 1 question and answered exactly unlike one. Candidates who reach for the gap-matching shortcut lose the mark in under ten seconds.

Bank PO · IBPS

Cross-time equalities

"A's age 4 years ago equalled B's age 3 years hence" is the 1 : 1 case, and it collapses to A − B = 7 in one line — a free constant if you write both brackets.

CAT · XAT

Split ratio plus a present ratio

The version with no constant to spend, so it needs a genuine substitution. Two ratios and two time zones is about as hard as this chapter gets.

Any question with two different shifts

"A after 5 years" against "B after 12 years"

Both forward but by different amounts is the same model. The test is whether the two ages share a moment, not whether the signs match.

One question decides which model you are in: are the two numbers in the ratio the ages of two people at the same moment? If yes, Model 1. If no, write brackets and forget every shortcut except the bracket difference.

07 Interview questions

What gets asked about split shifts

Ten in escalating order — how to spot the model, why the usual shortcut dies, what replaces it, and the honest limit of the replacement.

How do you recognise a Model 3 question?

The two ages in the ratio belong to different moments — one is shifted back and the other forward, or both are shifted by different amounts. If the two numbers in the ratio are not a snapshot of one instant, the gap-matching shortcut from Model 1 does not apply.

Why can't you match the gaps in units the way Model 1 does?

Because that method rests on the two ratios describing one fixed age difference. Here the two bracketed ages are not the two people's ages at any single moment, so there is no common age gap for the ratios to share. The constant has to be rebuilt from the shifts.

So what is constant in a split problem?

The difference between the two brackets. (B + q) − (A − p) = (B − A) + p + q, which is the age gap plus both shifts. In the worked example that is 10 + 5 + 5 = 20, and a ratio of 1 : 3 with a difference of 20 gives brackets of 10 and 30 straight away.

Neha is 15 and Vijay is 25. Their gap is 10, but the brackets differ by 20. Which is right?

Both, and they are answers to different questions. The age gap between the two people is 10 and always will be. The two bracketed ages differ by 20 because one was pulled five years back and the other five years forward. Confusing the two is the single defining error of this model.

Translate: "A's age 4 years ago was equal to B's age 3 years hence."

(A − 4) = (B + 3), so A − B = 7. This is the 1 : 1 split, and it is generous: it hands you a permanent age gap dressed up as time travel. Write both brackets and it falls out in one line.

When does the bracket-difference shortcut fail?

When the linking fact is itself a ratio rather than a gap or a sum. A present ratio gives no absolute number to spend on the units, so you substitute B = (d/c)A into the split equation and solve normally. Knowing the shortcut's limit matters as much as knowing the shortcut.

A split question comes out with an age of 84. Have you gone wrong?

Probably not. Split problems genuinely produce large answers, because the bracket difference can be many times the age gap and a ratio one unit apart then prices each unit very high. Check it by rebuilding the ratio from your answer — that is decisive, and eyeballing the size is not.

Compare Model 1 and Model 3 in one sentence each.

Model 1: both people move by the same amount, so the age gap is shared by both ratios and you match the gaps in units. Model 3: they move by different amounts, so what is shared is the bracket difference, and each bracket has to be unwound by its own shift in its own direction.

Are both ages having to move for it to be Model 3?

Hard

(a) The ratio of A's age 4 years ago to B's age 4 years hence is 3 : 5, and B is 6 years older than A. Find both present ages. (b) Now suppose the second reference had read "4 years ago" instead of "4 years hence". Solve again, and explain why version (b) can be done by matching gaps in units while version (a) cannot.

Follow-up

The two versions differ by one word and by eight years of bracket difference. Part (b) is a Model 1 question in disguise — both ages sit at the same moment, so the 6-year age gap is directly spendable on the ratio's 2 units. Part (a) is not, and seeing exactly where the extra 8 years comes from is the point of the whole lesson.

Show the hint

For (a) the bracket difference is 6 + 4 + 4; for (b) both brackets sit four years ago, so it is just 6.