CAPF AC 2020 Q114: General Mental Ability – Quantitative Aptitude
- (a) ₹280 profit
- (b) ₹280 loss
- (c) ₹720 profit
- (d) ₹720 loss
Answer: (d) ₹720 loss
Explanation
Cost varies as the cube of weight
Given: the cost of gold varies directly as the cube of its weight.
A piece weighing 20 decigram costs ₹1,000.
It is broken into two pieces with weights in the ratio 2 : 3.
How to think: breaking a piece always loses value when cost depends on the cube.
So the answer must be a loss.
Then find the new total cost.
| Step | Working |
|---|---|
| Weights of the pieces | 2 : 3 of 20 gives 8 and 12 decigram |
| Rule | Cost = k × weight³ |
| Find k | 1,000 = k × 20³ = k × 8,000, so k = 1/8 |
| Cost of the 8 piece | 8³ ÷ 8 = 512 ÷ 8 = ₹64 |
| Cost of the 12 piece | 12³ ÷ 8 = 1,728 ÷ 8 = ₹216 |
| New total | 64 + 216 = ₹280 |
| Loss | 1,000 − 280 = ₹720 |
The trap: ₹280 is the new value, not the loss.
Answer: (d) ₹720 loss.
Why each option is right or wrong
- (a) Incorrect. Breaking the piece cannot give a profit, and ₹280 is the new value.
- (b) Incorrect. ₹280 is what the two pieces are worth, not the loss incurred.
- (c) Incorrect. The amount is right, but it is a loss, not a profit.
- (d) CORRECT. The pieces are worth ₹64 + ₹216 = ₹280, a loss of ₹720.
Key point
Cost = k × weight³. Whole: 20³ gives ₹1,000. Pieces 8 and 12: ₹64 + ₹216 = ₹280. Loss = ₹720. Breaking always loses value when cost rises with a power of weight.
Remember
Value as cube of weight: broken pieces are worth far less than the whole.
Common mistakes
- Marking ₹280 as the loss; it is the new value.
- Splitting the cost in the ratio 2 : 3 instead of cubing the weights.
- Calling the change a profit when the pieces are worth less.
Exam tip
Educated guess: breaking always loses here, so both profit options go. Shortcut: (2³ + 3³) ÷ 5³ = 35 ÷ 125 = 28%, so 72% is lost.
Sources
- NCERT Class VIII Mathematics — Direct and Inverse Proportions
- NCERT Class VIII Mathematics — Cubes and Cube Roots