Question:
Let A and B be two solid spheres such that the surface area of B is 300% higher than the surface area of A. The volume of A is found to be k% lower than the volume of B. The value of k must be
Let A and B be two solid spheres such that the surface area of B is 300% higher than the surface area of A. The volume of A is found to be k% lower than the volume of B. The value of k must be
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"300% higher" means the new value is the original plus 300% of it. Find how the radii compare first, then cube the ratio for volume.
Updated On: Nov 7, 2026
- 85.5
- 92.5
- 90.5
- 87.5
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The Correct Option is D
Solution and Explanation
Step 1: Understand the wording.
"300% higher" means 300% more than the original, so the new value is 100% + 300% = 400% of the original. This is the part students often get wrong, because they read it as 300% of the original.
So the surface area of B is 4 times the surface area of A.
Step 2: Key formulas.
For a sphere of radius \(r\):
1. Surface area \(= 4\pi r^{2}\).
2. Volume \(= \dfrac{4}{3}\pi r^{3}\).
Step 3: Find the ratio of the radii.
Let the radii of A and B be \(r_A\) and \(r_B\). Then \[ \frac{4\pi r_B^{2}}{4\pi r_A^{2}} = 4 \implies \frac{r_B^{2}}{r_A^{2}} = 4 \implies \frac{r_B}{r_A} = 2 \]
So B has twice the radius of A.
Step 4: Find the ratio of the volumes.
Volume depends on the cube of the radius, so \[ \frac{V_B}{V_A} = \left(\frac{r_B}{r_A}\right)^{3} = 2^{3} = 8 \]
Take \(V_A = 1\) unit. Then \(V_B = 8\) units.
Step 5: Find how much lower A is than B.
The volume of A is lower than the volume of B, so the base for the percentage is the volume of B. \[ k = \frac{V_B - V_A}{V_B} \times 100 = \frac{8 - 1}{8} \times 100 = \frac{7}{8} \times 100 = 87.5 \]
So A is 87.5% lower than B.
Step 6: Why the other options fail.
Options 1, 2 and 3 (85.5, 92.5 and 90.5) do not match the value \(\frac{7}{8} \times 100 = 87.5\) that the calculation gives. If a student wrongly treats "300% higher" as 3 times the area, the radius ratio becomes \(\sqrt{3}\) and the answer comes out near 80.8, which is also not in the list. Only 87.5 agrees with the correct reading.
Final Answer:
The volume of A is 87.5% lower than the volume of B, so \(k = 87.5\), which is option 4. \[ \boxed{87.5\ \text{(Option 4)}} \]
"300% higher" means 300% more than the original, so the new value is 100% + 300% = 400% of the original. This is the part students often get wrong, because they read it as 300% of the original.
So the surface area of B is 4 times the surface area of A.
Step 2: Key formulas.
For a sphere of radius \(r\):
1. Surface area \(= 4\pi r^{2}\).
2. Volume \(= \dfrac{4}{3}\pi r^{3}\).
Step 3: Find the ratio of the radii.
Let the radii of A and B be \(r_A\) and \(r_B\). Then \[ \frac{4\pi r_B^{2}}{4\pi r_A^{2}} = 4 \implies \frac{r_B^{2}}{r_A^{2}} = 4 \implies \frac{r_B}{r_A} = 2 \]
So B has twice the radius of A.
Step 4: Find the ratio of the volumes.
Volume depends on the cube of the radius, so \[ \frac{V_B}{V_A} = \left(\frac{r_B}{r_A}\right)^{3} = 2^{3} = 8 \]
Take \(V_A = 1\) unit. Then \(V_B = 8\) units.
Step 5: Find how much lower A is than B.
The volume of A is lower than the volume of B, so the base for the percentage is the volume of B. \[ k = \frac{V_B - V_A}{V_B} \times 100 = \frac{8 - 1}{8} \times 100 = \frac{7}{8} \times 100 = 87.5 \]
So A is 87.5% lower than B.
Step 6: Why the other options fail.
Options 1, 2 and 3 (85.5, 92.5 and 90.5) do not match the value \(\frac{7}{8} \times 100 = 87.5\) that the calculation gives. If a student wrongly treats "300% higher" as 3 times the area, the radius ratio becomes \(\sqrt{3}\) and the answer comes out near 80.8, which is also not in the list. Only 87.5 agrees with the correct reading.
Final Answer:
The volume of A is 87.5% lower than the volume of B, so \(k = 87.5\), which is option 4. \[ \boxed{87.5\ \text{(Option 4)}} \]
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