Problems on Ages — The Equation Decoder

Problems on Ages · 25 min

Aptitude · Problems on Ages · Foundations

Ages questions are lost in translation, not in algebra

Once an ages question is a pair of equations it is school arithmetic. Turning the English into those equations is the actual test. This lesson is a phrasebook for every phrasing examiners use, including the one word that quietly changes a multiplier.

Change one word and watch the equation change →
“N times as old as” means ×N. “N times more than” means ×(N+1). They are different sentences and they have different answers.

01 The idea

One pair of ages, eight ways of saying it

A father is 40 and his son is 10. Now read these eight sentences: the father is 4 times as old as the son; the father is 3 times more than the son; the father is 30 years older; the father's age is 400% of the son's; the father's age is 300% more than the son's; one-fourth of the father's age equals the son's; in 20 years the father will be twice as old; five years ago the father was 7 times as old.

Every one of those eight sentences is true of 40 and 10. An examiner picks whichever phrasing hides the arithmetic best and builds the question around it. So your job at the start of an ages question is not to solve anything — it is to get each sentence onto paper as an equation without losing anything in the move.

Most of the phrasebook is mechanical. "Older than", "senior to", "exceeds by", "surpasses by" and "greater than by" all mean add. "Younger", "junior to", "falls short of" and "less than" all mean subtract. "Times", "twice", "thrice", a fraction and a percentage all mean multiply. Learn the synonyms once and they stop being obstacles.

Two of them are not mechanical and they are where the marks go. The first is the word more: "3 times more than" is the son's age plus three more of it, which is four times as old, not three. The second is a time shift under a multiplier: twice the age five years hence is 2(x + 5) = 2x + 10, and writing 2x + 5 changes the answer. Both are handled by the same discipline — decode the multiplier, then bracket, then expand.

Translate one sentence at a time. Decode the multiplier first, wrap the age in brackets second, expand third. The equation you write in that order is the equation the examiner meant.
Direct formThe sentence written with one age alone on the left: F = S + 30, F = 4S. Fastest to substitute into another equation, which is what you nearly always do next.
Difference formThe same sentence written as a gap: F − S = 30. Worth writing whenever a difference appears, because a difference is the one quantity that survives every time shift untouched.
Effective multiplierThe number an age is actually multiplied by after the phrasing is decoded. "4 times as old" gives 4; "4 times more", "300% more" and "400% of" all give 4 as well, by three different routes.

02 Worked example

Two sentences, two equations, one substitution

A father is 4 times as old as his son. In 20 years he will be twice as old as his son. Find their present ages. Nothing here is hard once both sentences are equations — so the worked solution spends its effort on the translation and almost none on the algebra.

1
Name the letters and translate sentence one"4 times as old as" is a multiplication, and the multiplier attaches to the smaller age. F + 4 = S and F = S + 4 are both wrong readings of this sentence.F = 4S
2
Decode sentence two's multiplier"Twice as old as" is the plain reading, so the multiplier is 2. Had it said "twice more than", the multiplier would have been 3 and the answer would change.multiplier = 2
3
Bracket both ages 20 years forwardThe same 20 years is added to each of them, and the multiplier goes outside the bracket so that it multiplies the 20 as well.(F + 20) = 2(S + 20)
4
Substitute sentence one and solveReplace F with 4S and one unknown is left. Note the right-hand side becomes 2S + 40, not 2S + 20.4S + 20 = 2S + 40 → 2S = 20 → S = 10, so F = 4(10) = 40
5
Read both ages back into the EnglishCheck against the sentences as written, not against your equations. In 20 years they are 60 and 30, and 60 is twice 30.40 = 4 × 10 ✓    (40 + 20) = 2(10 + 20) → 60 = 60 ✓

Now change one word. If sentence two had read "in 20 years he will be twice more than his son", the multiplier would be 3, the equation would be 4S + 20 = 3S + 60, and the son would be 40 with a 160-year-old father — visibly absurd. That absurdity is useful: an impossible answer is usually a mistranslated sentence rather than bad arithmetic.

03 The method

The phrasebook, and the one row that catches people

Check every row against the same pair of numbers — father 40, son 10. A phrasebook you have verified against one concrete pair is a phrasebook you will trust under time pressure.

Three moves, in this order. (1) Decode the multiplier: "N times as old" → N, "N times more" → N + 1, "P% of" → P/100, "P% more" → 1 + P/100. (2) Bracket each age into the moment the sentence describes: (x + t) for later, (x − t) for earlier. (3) Only then expand and cross-multiply.
The trap worth memorising: "more" adds one to the multiplier. "3 times more" = 4 times as old, "100% more" = twice, "200% more" = 3 times, "half a time more" = 1.5 times. And a fraction reverses: "A is two-thirds of B" is 3A = 2B, so cross-multiply immediately rather than carrying the fraction through three lines.
The sentenceThe equationCheck on father 40, son 10
F is 4 times as old as SF = 4S40 = 4 × 10 ✓
F is 3 times more than SF = 4S3 + 1 = 4, so 40 = 4 × 10 ✓
F is 30 years older than SF = S + 3040 = 10 + 30 ✓
F’s age is 400% of S’sF = 4S400% of 10 = 40 ✓
F’s age is 300% more than S’sF = 4S1 + 3 = 4, so 40 ✓
One-fourth of F’s age equals S’sF / 4 = S40 / 4 = 10 ✓
In 20 years F will be twice SF + 20 = 2(S + 20)60 = 2 × 30 ✓
5 years ago F was 7 times SF − 5 = 7(S − 5)35 = 7 × 5 ✓
F is 7 times S, written F + 7 = Snever legal47 ≠ 10 — "times" is not "plus"

05 Cheat sheet

The decoder on one page

The synonym rows are free marks. The two tinted rows are where the chapter actually sets its traps, and they are worth reading aloud before an exam.

Words in the questionWhat you writeOn father 40, son 10
older / elder / senior / exceeds / surpasses+F = S + 30
younger / junior / falls short of / less than−S = F − 30
ago / back / beforex − t5 years ago: 35 and 5
hence / later / after / from nowx + tin 20 years: 60 and 30
N times as old as / thrice / twice×NF = 4S
N times more than / P% more than×(N+1)3 times more → F = 4S
A is a/b of BbA = aBS = F/4 → 4S = F
multiplier on top of a shiftN(x + t), never Nx + t2(10 + 20) = 60, not 40
Decode the multiplier before you touch the algebraEverything downstream depends on whether the multiplier is N or N + 1. Fix it in step one and the rest of the solution is safe; get it wrong and every subsequent line is wasted.
Write the difference form whenever a gap appearsA difference is the one quantity a time shift cannot change, so F − S = 30 stays true in every sentence of the question. A direct form has to be re-shifted each time.
An absurd answer is a mistranslationA 160-year-old father or a negative age almost never comes from bad arithmetic. Go back to the sentence and re-read the multiplier word before you re-check the sums.

06 Where & why

Where the translation itself is the question

Papers that want to test reading rather than arithmetic use ages, because the numbers are trivially small and the sentences can be made as slippery as the setter likes.

Bank PO · RRB · IBPS
"times more than" in the stem

The phrase appears deliberately, and both answers — the N reading and the N + 1 reading — are in the options. There is no arithmetic route around it; you either know the rule or you guess.

TCS Digital · Infosys
Two-clause sentences

"P is 5 years junior to twice Q's age" packs a multiplier and a shift into one clause. Splitting it into 2Q, then minus 5, is the entire difficulty.

CAT · verbal-quant hybrids
Cross-time equalities

"A's age 4 years ago equalled B's age 3 years hence" collapses to A − B = 7 in one line if you write both brackets, and looks unsolvable if you do not.

Percentages, ratios, profit and loss
The same "more than" trap

"20% more than" and "20% of" differ by exactly the same off-by-one everywhere in quant. Fixing the habit here fixes it in four other chapters.

You will not remember twenty phrasings on exam day. You will remember two rules — "more" adds one to the multiplier, and brackets come before multipliers — and those two cover almost every sentence a setter can write.

07 Interview questions

The translation questions interviewers ask

Ten in escalating order — the plain phrasings, then the traps, then the honest question about whether any of this is worth memorising.

Translate: "A father is 7 times as old as his son."
F = 7S. The multiplier attaches to the smaller age, so the bigger age sits alone on the left. F + 7 = S is the classic wrong reading — "times" is a multiplication and never an addition.
What is the difference between "3 times as old as" and "3 times more than"?
"3 times as old as" is A = 3B. "3 times more than" is B plus three more of B, so A = 4B. The word "more" adds one to the multiplier, and examiners put both answers in the options precisely because most candidates read them as the same sentence.
So what does "200% more than" mean?
Three times as old. 200% more is the original plus two more of it, which is 300% of it. In the same family, "100% more" is twice, "50% more" is 1.5 times and "a quarter more" is 1.25 times. Compare that with "200% of", which is only twice — the word "of" does not add anything.
Translate: "A is two-thirds as old as B."
A = (2/3)B, and immediately cross-multiply it to 3A = 2B. Carrying a fraction through three more lines is how arithmetic slips creep in, and the cross-multiplied form is also the form you substitute into the next equation.
Why is 2(x + 5) not the same as 2x + 5?
Because the multiplier applies to the age at that moment, and the age at that moment already includes the 5. Expanding gives 2x + 10. Concretely: twice a 10-year-old's age in 5 years is twice 15, which is 30 — not 25. Bracket first, expand second, always.
Translate: "A's age 4 years ago was equal to B's age 3 years hence."
(A − 4) = (B + 3), which simplifies to A − B = 7. That is the useful form, because a difference does not change with time, so it can be reused in every other sentence of the question without reshifting.
"5 years ago, A was 12 years older than B." What does that tell you?
That A is 12 years older than B today, and will be forever. Writing it out as (A − 5) − (B − 5) = 12 cancels the shift completely. A difference stated at any moment is a difference at every moment, so a sentence like this is a free constant dressed up as a time-travel clause.
Translate: "P is 5 years junior to twice Q's age."
P = 2Q − 5. Two operations in one clause: build 2Q first, then subtract 5. "Junior to" and "younger than" both mean subtract, and "senior to" and "exceeds" both mean add — the synonym list is short and worth learning once.
You get a father aged 160 and a son aged 40. What do you do?
Go back to the sentence, not to the arithmetic. An impossible family almost always means a multiplier was decoded as N when it was N + 1, or a bracket was expanded wrongly. Re-reading the multiplier word is faster than re-checking three lines of algebra, and it is the more likely fault.
Is memorising twenty phrasings actually worth it?
No, and you should not try. Two rules cover nearly all of them: "more" adds one to the multiplier, and brackets go on before multipliers. The rest is synonyms you already know from English. What is worth practising is the habit of writing one equation per sentence before solving anything — that is what stops a long question from becoming unmanageable.

08 Practice problems

Six pure translation problems

None of these needs anything harder than a linear equation. Every one of them can be got wrong by misreading a single word, so write one equation per sentence before you solve.

As much younger as older

Easy
X is as much younger than Y as he is older than Z. If the sum of the ages of Y and Z is 48, find X's age.
Follow-up
The sentence looks like it needs three unknowns and a lot of work. Written as one equation it collapses, and neither Y nor Z is ever needed individually.
Show the hint
"As much younger than Y as he is older than Z" says Y − X = X − Z.

The word that changes the answer

Easy
A's age is 3 times more than B's age, and the difference between their ages is 24 years. Find both ages.
Follow-up
Read "3 times more" as three times as old and the difference becomes 2 units instead of 3, giving 12 and 36 instead of the correct pair. Both answers will be in the options.
Show the hint
"3 times more" means A = B + 3B, so the difference is 3B, not 2B.

Two sentences, two equations

Medium
Twice A's age added to B's age is 90 years. Thrice B's age added to A's age is 85 years. Find the difference between their ages.
Follow-up
There is no time travel and no ratio here, only two sentences that each have to be read in the right order — and the question asks for a difference, not for either age, so you can stop one step early if you notice.
Show the hint
Write 2A + B = 90 and A + 3B = 85, then substitute one into the other.

One person, two time zones

Medium
Thrice the mother's age 5 years hence is equal to four times her age 5 years ago. Find her present age.
Follow-up
Only one person appears, so there is no gap and no ratio — but there are two brackets and two multipliers, and dropping either bracket gives a clean-looking wrong answer.
Show the hint
Write 3(M + 5) = 4(M − 5) and expand both sides fully before collecting terms.

A sum that grows twice as fast

Medium
A man's age is twice the sum of the ages of his two children. Twenty years later his age will equal the sum of their ages. Find the man's present age.
Follow-up
In twenty years the man gains 20 years but the sum of two children's ages gains 40, because each child gains 20. Treating the sum as one person is the mistake the question is built on.
Show the hint
Let the sum of the children's ages be S. The man is 2S now, and in 20 years compare 2S + 20 with S + 40.

One reading has no answer at all

Hard
A father is 4 times as old as his son today. A second sentence is printed ambiguously: it reads either "in 5 years the father will be 3 times as old as his son" or "in 5 years the father will be 3 times more than his son". (a) Solve under each reading. (b) Explain which reading the examiner must have meant, and why.
Follow-up
One of the two readings produces a flat contradiction rather than a wrong answer — the unknown cancels and you are left with a false numeric statement. That is the strongest possible demonstration that "more" is not decoration, and it also shows why a ratio cannot stay put while time passes.
Show the hint
Under the second reading the multiplier is 4, the same as today's. Ask yourself whether a ratio of 4 : 1 can still be 4 : 1 five years later.