Aptitude · Problems on Ages · Model 1
Both people jump together, so the gap is your ruler
Model 1 gives you two ratios of the same two people at two moments, with the same number of years added to each of them. Because the jump is symmetrical the age gap is common to both ratios, and that turns a pair of simultaneous equations into one division.
Watch two ratios get scaled to a common gap →01 The idea
A three-year senior is still three years older in the office
Vijay is 18 and in his first year of B.Tech. Radha is 21 and in her fourth. Ten years later they both work at the same company; Vijay is 28 and Radha is 31. Their years of study, their salaries and their job titles all changed. The three-year gap did not, because the ten years were added to both of them.
That is the shape of every Model 1 question. You are handed two ratios of the same pair at two different moments, and the shift between those moments applies to both people equally. "Ten years ago the ratio was 3 : 1" and "ten years from now it will be 2 : 1" are two photographs of one family taken twenty years apart.
The naive route is to set the first ratio as 3x and x, add twenty to each, cross-multiply and solve. That always works and it is worth being able to do cold. But it throws away the fact that makes the model easy: both ratios describe the same gap in years, so once their gaps agree in units, one unit means the same thing in both.
Here is the family this lesson uses. A mother and daughter were in the ratio 3 : 1 ten years ago, and the mother will be twice as old ten years from now. They are 70 and 30 today, a gap of 40 years. Ten years ago they were 60 and 20 — gap 40. Ten years hence they are 80 and 40 — gap 40. Three ratios, one gap.
02 Worked example
The long way, once, so the shortcut makes sense
Ten years ago the ages of a mother and daughter were in the ratio 3 : 1. Ten years from now the mother's age will be twice the daughter's. Find their present ages. This is the bracket method in full. Section 03 does the same question in four lines.
Look at what x turned out to be: 20 years, which is exactly the size of the jump. That is not a coincidence. In 3 : 1 the gap is 2 units and in 2 : 1 it is 1 unit, so rescaling the second to 4 : 2 makes both gaps 2 units — and then the mother's share moves from 3 units to 4 units, one unit, across twenty years. The shortcut in section 03 reads that off without ever writing an equation.
03 The method
The unit-shift shortcut in four lines
This is the method to use in the exam. It works whenever the jump is symmetrical, which in Model 1 is always, and it replaces the whole equation with one division.
| Line | What you do | On 3 : 1 then, 2 : 1 ten years hence |
|---|---|---|
| 1 | Read the gap in units from each ratio | 3 − 1 = 2 units; 2 − 1 = 1 unit |
| 2 | Scale both to the LCM of those gaps | 3 : 1 stays; 2 : 1 becomes 4 : 2 (gap 2) |
| 3 | Read the shift in the first person's share | 3 units → 4 units, shift = 1 unit |
| 4 | Divide the elapsed years by the shift | 20 years / 1 unit = 20 years per unit |
| 5 | Price the ages at the earlier moment | 3 × 20 = 60 and 1 × 20 = 20 |
| 6 | Add the years to reach the present | 60 + 10 = 70 and 20 + 10 = 30 |
| 7 | Check the gap at both moments | 60 − 20 = 40 and 80 − 40 = 40 ✓ |
05 Cheat sheet
Model 1 on one page
The first four rows are the shortcut. The last three are the errors that turn a thirty-second question into a wrong answer.
| Situation | What to do | On mother 70, daughter 30 |
|---|---|---|
| Two ratios, same gap in units | 1 unit = years / shift | 7 : 3 now, 2 : 1 in 10 yr → scale to 4 : 2, shift 1, unit 10 |
| Two ratios, different gaps | scale to the LCM of the gaps first | 3 : 1 (gap 2) with 2 : 1 (gap 1) → 4 : 2 |
| Ratio plus a gap in years | gap in units = gap in years | 3 : 1 ten years ago, gap 40 → 2 units = 40, unit 20 |
| Ratio plus a sum of ages | total units = total years | 7 : 3 today, sum 100 → 10 units = 100, unit 10 |
| "10 years ago" to "10 years hence" | jump is 20, not 10 | 60 and 20 → 80 and 40 |
| Ratio moving away from 1 : 1 over time | impossible | 2 : 1 now cannot be 3 : 1 later |
| Answer found at the earlier moment | not the answer yet | 60 and 20 are past ages; add 10 |
06 Where & why
Where Model 1 shows up
This is the most-set ages model in Indian placement papers, and it is set as a speed question with four widely spaced options.
The house pattern. The unit-shift shortcut answers it in the time it takes to write the two gaps, which is what the section timing assumes you will do.
One ratio and one absolute number — a difference of 40 years, or a total of 100. Both reduce to "how many units is that number worth", which is the same division.
Each ratio alone is one equation in two unknowns and never sufficient. Together they are, unless they are the same ratio at different moments — the case the setter hides in one option.
Solve the two-person Model 1 core first, then the extra people are one subtraction each. The extra names add reading time, not difficulty.
07 Interview questions
What gets asked about symmetrical jumps
Ten in escalating order — the setup, the shortcut, why the shortcut is valid, and the two traps that account for most wrong answers.
What makes a question a Model 1 question?
Solve it: the ratio is 3 : 1 today and will be 2 : 1 in fifteen years.
Why is it legitimate to scale one ratio and not the other?
A question says ten years ago and ten years hence. What is the jump?
Can the ratio 2 : 1 become 3 : 1 later?
You are given one ratio and the difference in years. Is that enough?
What if the two given ratios are the same ratio at different times?
Compare the shortcut with just writing 3x and x and cross-multiplying.
A question adds C and D, defined relative to A and B. Does the method change?
When would you actually use this outside an exam?
08 Practice problems
Six symmetrical jumps
Try each one with the unit-shift shortcut first, then confirm with the equation. Two of them have unequal gaps in units, so the scaling step is not optional.