Problems on Ages: Ratios, Sums and Differences
Problems on Ages: Ratios, Sums and Differences
Age problems rest on one fact: everyone ages at the same rate. The difference between two people's ages never changes, and n years later every age rises by n. NTPC sets ratios now and some years ago or later, sums with ratios, and differences or products of ages.
1. Rules Box
Present ages in ratio a : b: take ax and bx
n years later: ax + n and bx + n; n years ago: ax − n and bx − n
The age difference is constant at all times
Sum of k people's ages rises by k × n after n years
| Given | Equation |
|---|---|
| Ratio now a : b, after n years c : d | (ax + n)/(bx + n) = c/d |
| Ratio p years ago a : b, q years hence c : d | (ax + p + q)/(bx + p + q) = c/d, then add p for present ages |
| Sum S and ratio a : b | Ages = S × a/(a + b) and S × b/(a + b) |
| Difference D and ratio a : b | (b − a)x = D |
✗ Ratio 4 : 5 now becomes 4 + 6 : 5 + 6 after 6 years | ✓ Add 6 years to the ages 4x and 5x, not to the ratio terms
✗ Sum of two ages is 50 now, so 5 years ago it was 45 | ✓ Each person is 5 years younger: the sum was 40
हिंदी नोट: दो व्यक्तियों की आयु का अंतर हमेशा समान रहता है। n वर्ष बाद हर व्यक्ति की आयु n बढ़ती है, इसलिए दो लोगों के योग में 2n जुड़ता है। अनुपात a : b हो तो आयु ax और bx मानिए।
Exam Pointer: Verified NTPC pattern: the ratio of two ages some years ago and some years hence (April 2016 CBT-1). Sum-with-ratio and difference-or-product questions had no verified NTPC shift in our check and are tagged syllabus-based.
Pariksha Pattern: Every Way NTPC Asks This Topic
Pattern 1: Ratio now and after or before some years
[PYQ: NTPC CBT-1 2-Apr-2016 Shift-2]
EXAM LEVEL
Q. The present ages of A and B are in the ratio 4 : 5. After 6 years the ratio will be 5 : 6. Find their present ages.
(4x + 6)/(5x + 6) = 5/6 gives 24x + 36 = 25x + 30, so x = 6. Ages: 24 and 30 years.
Answer: 24 and 30 years
EXAMATLAS LEVEL
Q. Five years ago the ages of P and Q were in the ratio 5 : 7. Three years from now the ratio will be 3 : 4. Find their present ages.
Five years ago, ages 5x and 7x; eight years after that, (5x + 8)/(7x + 8) = 3/4 gives 20x + 32 = 21x + 24, so x = 8. Five years ago they were 40 and 56, so now 45 and 61 years. Check: 48 : 64 = 3 : 4.
Answer: 45 and 61 years
Pattern 2: Sum of ages with a ratio
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. The sum of the ages of two brothers is 60 years and their ages are in the ratio 7 : 5. Find their ages.
60 × 7/12 = 35 and 60 × 5/12 = 25.
Answer: 35 and 25 years
EXAMATLAS LEVEL
Q. The sum of the present ages of A and B is 50 years. Five years ago their ages were in the ratio 7 : 3. Find the ratio of their ages 5 years from now.
Five years ago the sum was 50 − 10 = 40, split 28 and 12. Now they are 33 and 17; five years on, 38 and 22, a ratio of 19 : 11.
Answer: 19 : 11
Pattern 3: Difference or product of ages
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. A is 8 years older than B, and their ages are in the ratio 7 : 5. Find their ages.
7x − 5x = 8, so x = 4: 28 and 20 years.
Answer: 28 and 20 years
EXAMATLAS LEVEL
Q. The product of the ages of two children is 96 and the elder is 4 years older. Find the ratio of their ages 4 years ago and 4 years from now.
Ages 12 and 8 (12 × 8 = 96, difference 4). Four years ago: 8 : 4 = 2 : 1. Four years later: 16 : 12 = 4 : 3.
Answer: 2 : 1 and 4 : 3
Pattern 4: Three people linked by ratios
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. The ages of A, B and C are in the ratio 2 : 3 : 5 and their sum is 80 years. Find C's age.
80 × 5/10 = 40 years.
Answer: 40 years
EXAMATLAS LEVEL
Q. A : B = 3 : 4 and B : C = 6 : 7 in age, and C is 20 years older than A. Find the sum of their ages 4 years ago.
Make B common: A : B : C = 9 : 12 : 14. C − A = 5 parts = 20, so one part = 4 and the ages are 36, 48 and 56 years. Their sum now is 140, so 4 years ago it was 140 − 12 = 128 years.
Answer: 128 years
60-Second Revision
- Ratio a : b: ages ax and bx; add or subtract the same years to both.
- Age difference never changes.
- A sum of k ages changes by k × n in n years.
- Combine chained ratios before using a difference or sum.