Two earthquakes A and B happen to be of magnitude 5 and 6 respectively on…

Concept — On the Richter scale, magnitude is a logarithmic measure of the energy an earthquake releases: log10 E = 4.8 + 1.5M with E in joules (equivalently log10 E = 11.8 + 1.5M with E in ergs). For a ratio of two energies only the coefficient 1.5 matters: raising the magnitude by one whole unit multiplies the released energy by 101.5, and that factor is the same anywhere on the scale.

Application

1. Write the relation for each event: log10 EA = 4.8 + 1.5 × 5 and log10 EB = 4.8 + 1.5 × 6.

2. Subtract the first from the second, so the constant 4.8 cancels: log10 EB − log10 EA = 1.5 × (6 − 5) = 1.5.

3. Combine the two logarithms into one: log10 (EB/EA) = 1.5.

4. Undo the logarithm: EB/EA = 101.5.

5. Evaluate the power: 101.5 = 10 × √10 = 10 × 3.162 = 31.62, which to the nearest offered value is about 32.

Cross-check — 101.5 = √1000 ≈ 31.6, so the ratio sits just under 32. Applying the same step twice checks out as well: a two-unit gap in magnitude gives 103 = 1000, and 31.6 × 31.6 ≈ 1000.

Contrast — The familiar “ten times bigger per magnitude unit” describes the recorded ground-motion amplitude, not the energy. Amplitude scales as 10ΔM while energy scales as 101.5ΔM. This item asks for energy, so the 101.5 factor per unit of magnitude is the one to use.

Result — EB/EA = 101.5 ≈ 31.6, so the energy released by the magnitude-6 event is approximately 32 times that of the magnitude-5 event.