Aptitude · Problems on Ages · Model 2
Fractions and percentages are ratios in a costume
Model 2 questions look harder than Model 1 questions and are not. The examiner has taken a plain ratio and printed it as a fraction, a decimal or a percentage. Convert it back in one line and the rest of the question is arithmetic you already know.
Take both masks off and watch a Model 1 question appear →01 The idea
150% of ten thousand is a ratio of 3 : 2
Neha's starting salary is ₹10,000 a month. HR tells Vijay his salary will be exactly 150% of Neha's, so his is ₹15,000. Now ask what the ratio of their salaries is: 15,000 to 10,000, which is 3 : 2. HR could have said "your salaries are in the ratio 3 : 2" and meant precisely the same thing.
Age questions do the same thing on purpose. "Vijay's age is three-fifths of his father's age" is 3 : 5. "Vijay's age is 60% of his father's" is 3 : 5. "Vijay is 40% younger than his father" is 3 : 5 as well, and so is "his father is 66⅔% older than Vijay". Four sentences, one fact.
This lesson uses one pair throughout: Vijay is 15 and his father is 25. Every mask above is a true statement about 15 and 25, and you should check each one as you read it. ₹15,000 to ₹10,000 was 3 : 2; fifteen years to twenty-five years is 3 : 5. The arithmetic of unmasking never depends on what the quantity is.
There are exactly two places to be careful. "Less than" and "more than" work off 100, not off the number quoted: 40% less is 60% of, and 50% more is 150% of. And the direction matters — "A is 3/5 of B" gives A : B = 3 : 5, while the same fact read from B's side gives B : A = 5 : 3. Write down which name is which before you reduce anything.
02 Worked example
One mask off, then a plain time shift
Vijay's present age is three-fifths of his father's present age. Five years from now the ratio of their ages will be 2 : 3. Find Vijay's present age. The fraction is the entire difficulty, and it dies in line one.
Reread step 1 and notice how little it cost. One line converted "three-fifths" into 3 : 5, and from step 2 onward the question had no fractions in it at all. If instead you carry 3F/5 through the cross-multiplication you get the same answer after three messier lines, which is three more chances to drop a 5.
03 The method
The mask table, checked against 15 and 25
Every row below is a true statement about Vijay at 15 and his father at 25. Verify each one as you read it — a conversion table you have checked against real numbers is one you will trust in an exam.
| The mask in the question | Unmasked ratio | Check on Vijay 15, father 25 |
|---|---|---|
| Vijay is 3/5 of his father | 3 : 5 | (3/5) × 25 = 15 ✓ |
| Vijay's age is 60% of his father's | 3 : 5 | 60% of 25 = 15 ✓ |
| Vijay is 0.6 times his father | 3 : 5 | 0.6 × 25 = 15 ✓ |
| Vijay is 40% less than his father | 3 : 5 | 100 − 40 = 60% of 25 = 15 ✓ |
| The father is 5/3 of Vijay | 5 : 3 | (5/3) × 15 = 25 ✓ |
| The father's age is 166⅔% of Vijay's | 5 : 3 | 166⅔% of 15 = 25 ✓ |
| The father is 66⅔% more than Vijay | 5 : 3 | 15 + (2/3) × 15 = 25 ✓ |
| "40% less" read as 40% of | never legal | 0.4 × 25 = 10, not 15 |
05 Cheat sheet
Model 2 on one page
Six conversions and two traps. If you can do the conversions in your head, Model 2 costs you the same time as Model 1.
| Mask | Rip it to | On Vijay 15, father 25 |
|---|---|---|
| a/b of | a : b | 3/5 of → 3 : 5 |
| a decimal | write as a fraction, reduce | 0.6 → 6/10 → 3 : 5 |
| P% of | P : 100 reduced | 60% → 60 : 100 → 3 : 5 |
| P% more than | (100 + P) : 100 | 66⅔% more → 5 : 3 |
| P% less than | (100 − P) : 100 | 40% less → 3 : 5 |
| A/m = B/n | nA = mB, so A : B = m : n | A/3 = B/5 → 3 : 5 |
| P% less read as P% of | never legal | 40% of 25 = 10 ≠ 15 |
06 Where & why
Where the mask shows up
Model 2 is a difficulty setting rather than a separate topic. Papers use it when they want a Model 1 question to look worth two marks.
The default style: "50% more than" now and "25% more than" later. Unmasking gives 3 : 2 and 5 : 4, both with a 1-unit gap, so the unit-shift shortcut finishes it.
"Two-thirds of A's age equals 75% of B's age" is one cross-multiplication away from 9 : 8. Papers like it because it looks like two facts and is only one.
"X is 33⅓% less than Y, and Y is 25% more than Z" gives X : Y = 2 : 3 and Y : Z = 5 : 4, which merge to 10 : 15 : 12. Three people, still one ratio.
"Marked 40% above cost and sold at 25% discount" is the identical move in a different chapter. Percentage-to-ratio is one of the highest-yield habits in the whole syllabus.
07 Interview questions
The masks interviewers reach for
Ten in escalating order — the conversions, then the two directions people get backwards, then whether the ratio route is really faster.
Convert: "A's age is three-fifths of B's age."
Convert: "A's age is 60% of B's age."
What is the difference between "120% of" and "20% more than"?
Convert: "A is 40% less than B."
Convert: "One-third of Vijay's age equals half of Neha's age."
Why does 40% of A = 50% of B give A : B = 5 : 4 rather than 4 : 5?
Convert: "Ravi's age is 1.5 times Kavi's age."
Compare unmasking first with carrying the fraction into the algebra.
When is a Model 2 question actually harder than a Model 1 question?
Is any of this useful outside an exam?
08 Practice problems
Six masked problems
Unmask every sentence before you solve. Two of these mask both statements, and the hard one masks three people at once.