CAPF AC 2025 Q56: General Mental Ability – Quantitative Aptitude
For any three natural numbers a, b and c, if (bc + 1)/(abc + a + c) = 11/43, then the value of abc is
- (a) 30
- (b) 45
- (c) 60
- (d) 75
Answer: (a) 30
Explanation
Finding abc
Given: a, b, c are natural numbers, and (bc + 1) / (abc + a + c) = 11/43.
How to think: turn the fraction upside down and split off a whole number.
| Step | Working | Result |
|---|---|---|
| Flip the fraction | (abc + a + c) / (bc + 1) = 43/11 | Same fraction, upside down |
| Split the numerator | a(bc + 1) + c, so the left side is a + c/(bc + 1) | Whole part plus fraction |
| Whole and fractional parts | 43/11 = 3 + 10/11 | a = 3, c/(bc + 1) = 10/11 |
| Solve the fraction | c = 10, bc + 1 = 11, so b = 1 | b = 1, c = 10 |
| Product | 3 × 1 × 10 | abc = 30 |
Check: (1 × 10 + 1) / (3 × 10 + 3 + 10) = 11/43. ✓
Bank key corrected: the stored key was (c) 60; the right answer is (a) 30.
Answer: (a) 30.
Why each option is right or wrong
- (a) CORRECT. a = 3, b = 1, c = 10 gives abc = 30.
- (b) Incorrect. 45 does not come from any natural numbers that satisfy the equation.
- (c) Incorrect. 60 fails the check: the fraction would not equal 11/43.
- (d) Incorrect. 75 fails the check.
Key point
Method: reciprocal of (bc+1)/(abc+a+c) is a + c/(bc+1). With 43/11 = 3 + 10/11 we get a = 3, c = 10, b = 1. Then abc = 30.
Remember
Flip, split: a + c/(bc+1) = 3 + 10/11, so abc = 30.
Common mistakes
- Cross-multiplying and trying values at random, which wastes time.
- Forgetting that the fractional part must be a proper fraction less than 1.
- Assuming a is the larger digit without splitting 43/11.
Exam tip
Educated guess: 43/11 is a little less than 4, so a is 3. Try c = 10: it makes the fraction part 10/11 immediately, and the product is 3 × 1 × 10 = 30.
Sources
- NCERT Class VII Mathematics, Chapter 2 (Fractions and Decimals: improper fractions and mixed numbers)
- NCERT Class VIII Mathematics, Chapter 1 (Rational Numbers)