Aptitude · Problems on Ages · Foundations
The gap is frozen. The ratio never stops moving.
Every question in this chapter is built on two facts that pull in opposite directions: the difference between two people's ages is the same number forever, and their ratio changes every single year. This lesson makes both facts concrete, and fixes the one reading error that costs more marks here than any algebra.
Watch one gap hold while the ratio slides →01 The idea
One frozen number and one moving number
Vijay is 10 and Radha is 15. Radha is five years older, and she will be five years older on every birthday either of them ever has. Wind forward eighty-five years and they are 95 and 100 — still five apart. Time runs at one speed for everybody, so a difference in ages is a permanent fact about a pair of people.
Their ratio behaves nothing like that. Five years ago they were 5 and 10, a ratio of 1 : 2. Today they are 10 and 15, a ratio of 2 : 3. In five years they will be 15 and 20, which is 3 : 4. Same two people, same five-year gap, three different ratios — and it keeps going: 4 : 5, then 7 : 8, then 19 : 20.
There is a reason the ratio always moves in that direction. The gap of five years is a third of Vijay's age when he is 15, but only a nineteenth of it when he is 95. A fixed difference is a shrinking share of a growing age, so the ratio is always crawling towards 1 : 1 and can never turn back. Once you see that, a question saying "the ratio was 3 : 1 and later it is 2 : 1" stops looking like a contradiction and starts looking like the only thing that could have happened.
That leaves one reading error to kill. "Five years ago their ages were in the ratio 1 : 2" is a statement about 5 and 10, not about 10 and 15. Students routinely write the ratio against the present ages and then add five to the answer, which is a different problem with a different answer. Pin every fact to its moment on the timeline before you write an equation.
02 Worked example
Two ratios, two moments, one pair of ages
The present ages of Vijay and Radha are in the ratio 2 : 3. After 5 years the ratio of their ages will be 3 : 4. Find their present ages. Two ratios of the same two people at two different moments is the standard shape of an ages question, and this is the shortest honest route through it.
Look at what the answer does to the gap. Today 15 − 10 = 5; in five years 20 − 15 = 5. The gap never moved, and yet the ratio went from 2 : 3 to 3 : 4. That is the signature of a correct answer in this chapter — if your two moments give two different gaps, you have made an arithmetic error, not discovered a strange family.
03 The method
The bracket method, and the shortcut that skips it
The bracket method above always works and you should be able to run it cold. But when the two ratios have the same difference in units, there is a two-line route that gets the same answer without an equation.
| Moment | Vijay | Radha | Difference | Ratio |
|---|---|---|---|---|
| 5 years ago | 5 | 10 | 5 | 1 : 2 |
| Today | 10 | 15 | 5 | 2 : 3 |
| In 5 years | 15 | 20 | 5 | 3 : 4 |
| In 10 years | 20 | 25 | 5 | 4 : 5 |
| In 25 years | 35 | 40 | 5 | 7 : 8 |
| In 85 years | 95 | 100 | 5 | 19 : 20 |
05 Cheat sheet
The time-travel rules on one page
Six rows and three habits. If you can apply the first four rows without thinking, the six models that follow are all variations on arithmetic you already have.
| The words | What you write | On Vijay 10 and Radha 15 |
|---|---|---|
| t years ago | x − t | 10 − 5 = 5 |
| t years hence / later / after | x + t | 10 + 5 = 15 |
| Ratio a : b | ax and bx | 2x and 3x, x = 5 |
| Ratio a : b, t years ago | (A − t) : (B − t) = a : b | 5 : 10 = 1 : 2 |
| Difference of their ages | A − B, at any moment | 5 years, always |
| Ratio of their ages | changes every year | 2 : 3 today, 3 : 4 in 5 years |
| "3x + 5" collapsed to "8x" | never legal | (2x + 5) stays as it is |
06 Where & why
Where these two rules earn their marks
Ages is a small chapter with a reliable one or two questions in almost every Indian placement and government paper, and it is scored on speed rather than depth.
The house style: a present ratio and a ratio some years later, four options, ninety seconds. The unit-shift shortcut answers most of them without writing an equation.
Almost always anchored in the past, because writing the past ratio against present ages is the error the setter is fishing for. Reading discipline is the whole question.
One ratio at one moment is one equation in two unknowns, so it is never sufficient alone. Two ratios at two different moments almost always are — unless they are the same ratio, in which case they still are not.
A total that grows by a fixed amount while an average moves is the identical idea in a different chapter. Getting it here saves you learning it three more times.
07 Interview questions
What an interviewer asks about ages
Ten in escalating order — the two rules, then the reading traps, then the sufficiency question that separates people who understand the drift from people who have memorised a method.
What is the one fact that never changes in an ages problem?
Then why does the ratio of their ages keep changing?
A question says the ratio was 3 : 1 ten years ago and will be 2 : 1 ten years from now. Is that consistent?
Five years ago their ages were in the ratio 1 : 2. Can you write Vijay = x and Radha = 2x?
Why do you insist on brackets when you shift an age in time?
Is one ratio at one moment ever enough to find two ages?
What if the two ratios I am given are the same ratio at two different moments?
What is the fastest route when both ratios have the same difference in units?
A student gets a present age of 42.5 years. Is that a mistake?
When would you actually use any of this outside an exam?
08 Practice problems
Six that test the reading, not the algebra
Every one of these is solvable with the bracket method and a gap check. Two of them have answers that are not whole numbers, on purpose — do not talk yourself out of a correct answer.