CAPF AC 2022 Q3: General Mental Ability – Quantitative Aptitude
A solid spherical ball made of iron is melted and two new balls are made whose diameters are in the ratio of 1 : 2. The ratio of the volume of the smaller new ball to the original ball is
- (a) 1 : 3
- (b) 1 : 5
- (c) 2 : 9
- (d) 1 : 9
Answer: (d) 1 : 9
Explanation
One ball melted into two
Given: a solid iron ball is melted into two new balls whose diameters are in the ratio 1 : 2; find the ratio of the volume of the smaller new ball to the original ball.
How to think: volume of a sphere goes with the cube of the diameter.
The original ball equals the two new balls together.
| Step | Working |
|---|---|
| Diameters | 1 : 2 |
| Volumes of the new balls | 1³ : 2³ = 1 : 8 |
| Volume of the original ball | 1 + 8 = 9 parts |
| Smaller ball : original | 1 : 9 |
Answer: (d) 1 : 9.
Why each option is right or wrong
- (a) Incorrect. 1 : 3 uses the diameters directly (1 + 2) instead of their cubes.
- (b) Incorrect. 1 : 5 uses the squares (1 + 4), which is the rule for area.
- (c) Incorrect. 2 : 9 mixes up the two balls; the smaller one is 1 part.
- (d) CORRECT. Volumes are 1 and 8 parts, so the original is 9 parts and the ratio is 1 : 9.
Key point
Lengths in ratio a : b → areas a² : b² → volumes a³ : b³. Melting conserves volume, so the original equals the sum of the parts.
Remember
Diameters 1 : 2 → volumes 1 : 8 → total 9.
Common mistakes
- Comparing the smaller ball with the bigger ball (1 : 8) instead of the original.
- Using the square of the diameters, which applies to surface area.
- Forgetting that the original ball is the two new balls added together.
Exam tip
Educated guess: the word volume means cube. Cubes of 1 and 2 are 1 and 8, and 1 + 8 = 9. Only (c) and (d) have a 9; the smaller ball is 1 part.
Sources
- NCERT Class IX Mathematics — Surface Areas and Volumes (volume of a sphere)
- NCERT Class X Mathematics — Surface Areas and Volumes (conversion of solids)