The radius of the Moon is about one-fourth that of the Earth and accel

The radius of the Moon is about one-fourth that of the Earth and acceleration due to gravity on the Moon is about one-sixth that on the Earth. From this, we can conclude that the ratio of the mass of Earth to the mass of the Moon is about

10
100
1,000
10,000
This question was previously asked in
UPSC NDA-1 – 2015
The acceleration due to gravity (g) on a celestial body with mass M and radius R is given by the formula g = GM/R², where G is the gravitational constant. We are given that the radius of the Moon (Rm) is about one-fourth that of the Earth (Re), i.e., Rm ≈ Re/4. We are also given that the acceleration due to gravity on the Moon (gm) is about one-sixth that on the Earth (ge), i.e., gm ≈ ge/6.
From g = GM/R², we can write M = gR²/G.
The ratio of the mass of Earth (Me) to the mass of the Moon (Mm) is:
Me/Mm = (ge * Re² / G) / (gm * Rm² / G)
Me/Mm = (ge/gm) * (Re/Rm)²
Substitute the given ratios: ge/gm ≈ 6 and Re/Rm ≈ 4.
Me/Mm ≈ 6 * (4)² = 6 * 16 = 96.
Among the options, 96 is closest to 100.
The ratio of masses can be calculated using the relationship between acceleration due to gravity, mass, and radius (g = GM/R²).
The actual ratio of Earth’s mass to Moon’s mass is approximately 81.3, so the given approximate values lead to a result (96) that is closest to 100, indicating the question uses approximate figures typical for simplified calculations.