# Ex.13.8 Q4 Surface Areas and Volumes Solution - NCERT Maths Class 9

## Question

The diameter of the moon is approximately one-fourth of the diameter of the earth. What fraction of the volume of the earth is the volume of the moon?

## Text Solution

**Reasoning:**

Volume of the sphere \(\begin{align} = \frac{4}{3} \end{align}\) \(\pi r^{3}\). So, the fraction of the volume of the earth is the volume of the moon is the ratio of the volume of the earth to the volume of the moon

**What is known?**

Ratio between the diameters of moon and earth.

**What is unknown?**

Fraction of the volume of the earth is the volume of the moon

**Steps:**

Diameter of the Earth \(=2r\)

Radius of the Earth \(\begin{align} = \frac{1}{2}(2r) = r \end{align}\)

Diameter of moon \(\begin{align} = \frac{1}{4}(2r) = \frac{1}{2}r \end{align}\)

Radius of the moon \(\begin{align} = \frac{1}{2}(\frac{r}{2}) = \frac{r}{4} \end{align}\)

Volume of the earth \(\begin{align}({v_1}) = \frac{4}{3} \end{align}\)

Volume of the moon \(\begin{align}({v_2}) = \frac{4}{3}\pi {(\frac{r}{4})^3} \end{align}\)

\[ = \frac{1}{{64}}\left( {\frac{4}{3}\pi {r^3}} \right)\]

\(\begin{align} = \frac{1}{{64}} \end{align}\) Volume of earth

**Answer:**

Hence the volume of the moon is \(\begin{align}\frac{1}{{64}} \end{align}\)th fraction of the volume of the earth.