To calculate the density ratio of a newly discovered giant planet compared to Earth, the planet's average radius is 13 times that of Earth, and its mass is 963 times greater. The volume of the new planet can be determined using the formula for the volume of a sphere, leading to a density calculation. The user attempts to compute the density using the mass and volume but encounters errors in their calculations. The final density of the new planet is compared to Earth's density, which is approximately 5515 kg/m³. Accurate calculations are essential for determining the correct density ratio.