Question

A big metal sphere is melted to make spheres of half of the radius of big sphere. How many such small spheres can be made?

• A. 64
• B. 8
• C. 24
• D. 6

Answer 1

Answer:

B. 8

The correct answer is Option B (8).

The Ratio Method

Let’s define the dimensions:

• Big Sphere Radius ($R$): Let's call it $R$.

• Small Sphere Radius ($r$): The problem says it's half of $R$, so $r = \frac{R}{2}$.

To find the number of spheres, you divide the Total Volume by the Volume of one small sphere:

$\text{Number of Spheres} = \frac{\text{Volume of Big Sphere}}{\text{Volume of Small Sphere}}$

Substitute the volume formula $\frac{4}{3}\pi (\text{radius})^3$:

$\text{Number} = \frac{\frac{4}{3}\pi(R)^3}{\frac{4}{3}\pi(\frac{R}{2})^3}$

Simplifying the Math

1. Notice that the constant parts ($\frac{4}{3}\pi$) are in both the top and bottom, so they cancel out immediately.

2. Now you are left with:
   $\text{Number} = \frac{R^3}{(\frac{R}{2})^3}$

3. Apply the cube to the bottom fraction:

4. $\text{Number} = \frac{R^3}{\frac{R^3}{8}}$

5. When you divide by a fraction, you flip it and multiply:
   $\text{Number} = R^3 \times \frac{8}{R^3}$

6. The $R^3$ cancels out, leaving you with 8.