Three years ago, the age of a person was six times the age of his younger brother. Eight years later, the age of the person will be three years less than twice the age of his younger brother. Find the present ages of the person and his younger brother.
- (a)4 years and 12 years
- (b)12 years and 4 years
- (c)5 years and 15 years
- (d)15 years and 5 years
Answer
Why
Correct — D. Let P be the person's present age and B the brother's present age.
Three years ago: P − 3 = 6(B − 3)
so P = 6B − 18 + 3 = 6B − 15
Eight years later: P + 8 = 2(B + 8) − 3
Substitute P: 6B − 15 + 8 = 2B + 16 − 3
6B − 7 = 2B + 13, so 4B = 20 and B = 5
Then P = 6 × 5 − 15 = 15
Check both conditions: three years ago 12 and 2, and 12 = 6 × 2. Eight years later 23 and 13, and 2 × 13 − 3 = 23.
The present ages are 15 years and 5 years → option (d).
Why the others are wrong
- (a)4 years and 12 years — 4 and 12 makes the person the younger of the two, against the wording. It also fails the first condition: three years ago the ages were 1 and 9, and 1 is not six times 9.
- (b)12 years and 4 years — 12 and 4 keeps the right order but not the arithmetic. Three years ago the ages were 9 and 1, and 9 is not six times 1, so the first sentence of the question already rejects it.
- (c)5 years and 15 years — 5 and 15 is option (d) with the two ages swapped, making the person the younger brother. Three years ago that reads 2 and 12, and 2 is not six times 12.
Concept
Age problems are linear equations in disguise. Give each person one letter for the present age, then shift both ages by the same amount at every time reference.
Three years ago means subtract 3 from both before applying the multiple — six times the brother's age then, not now. That is the step most wrong solutions lose.
Three years less than twice becomes 2(B + 8) − 3, a subtraction from the multiple and not from the age. Two conditions give two equations, and substitution finishes them.
Every option is a pair of numbers, so back-substitution beats algebra under time pressure. Testing only the first condition, that the person was six times the brother three years ago, eliminates the other three options on its own.
Key facts
- The present ages are 15 years for the person and 5 years for his younger brother.
- Three years ago they were 12 and 2, and 12 is six times 2.
- Eight years from now they will be 23 and 13, and 23 is three less than twice 13.
- The two conditions reduce to P = 6B − 15 and P + 8 = 2B + 13, which meet at B = 5.
Study next
Common traps
- Applying the multiple to the present ages instead of to the ages three years ago
- Reading three years less than twice as twice the difference of the two ages
- Answering with the pair reversed, since two options carry the same two numbers
SSC gives two time references and one multiple, and both references shift both people equally. Comparable setups are asked 9 Sep 2024, 12:30, Quant Q.6 for Prachi and her daughter, and 23 Sep 2024, 09:00, Quant Q.1 for Sunita and Tanya. This shift also runs an ages-in-arithmetic-progression item at Quant Q.23.
Related PYQs
No directly related past PYQ was found.