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When we talk about 'volume reduction,' we're referencing the process of decreasing the amount of space that an object or a collection of objects occupies. This concept is crucial in fields such as waste management, manufacturing, and storage optimization. Imagine squishing a sponge in your hand; the sponge's volume decreases but its mass stays the same because you haven't removed any part of the sponge, just compressed it.
In our trash compactor scenario, volume reduction signifies a decrease in space taken up by rubbish without reducing its mass. The device compacts the trash so tightly that its volume is reduced to 35% of its original size. This change impacts how densely the mass is packed together within that smaller space, which brings us directly to the concept of 'compact mass density'.

Compact mass density refers to how closely packed the mass of a substance is within a specific volume. It’s a measure of how much mass there is per unit volume and is usually expressed in kilograms per cubic meter (kg/m³) or grams per cubic centimeter (g/cm³).
In our exercise, after the compaction, the mass remains constant but the volume into which this mass is squeezed has decreased, resulting in a more 'compact' mass density. This denser state has practical advantages like reduced space for disposal and increased efficiency for transport. It's particularly valuable when considering the costs and environmental impact of waste management.

The rule of 'inverse proportionality' comes into play when two related quantities are multiplied together to get a constant value – meaning, if one goes up, the other must come down to keep the product the same. In mathematics, this is often indicated by the equation xy = k, where x and y are the inversely proportional variables, and k is the constant.
In this problem, we apply inverse proportionality to the relationship between volume and density when mass is held constant. When the volume decreases—as it does with the trash compactor—the density must increase correspondingly to maintain the constant mass. This is why, after compacting, the rubbish's density is found by dividing the original density by 0.350. Simply put, when the volume is squished into just a part of the original, the density experiences a boost in inverse proportion to the reduction in volume.