x, y and z are three quantities. x varies inversely with y when z is constant.…

When one quantity varies inversely with a second while a third stays fixed, and that second quantity in turn varies inversely with the third while the first stays fixed, the three quantities together obey one joint invariant: the product of all three, x·y·z, stays the same across every state of the system — because each pairwise inverse relationship only fixes the product of two quantities while the third is frozen, and chaining both statements shows the full triple product itself must be constant.

1. x varies inversely with y when z is constant, so x·y = constant whenever z is fixed.

2. y varies inversely with z when x is constant, so y·z = constant whenever x is fixed.

3. Combining the two partial invariants shows the full product x·y·z is constant across both given states.

4. In the first state, x = 30, y = 8, z = 7, so x·y·z = 30 × 8 × 7 = 1680.

5. In the second state, y = 16 and z = 21, so y·z = 16 × 21 = 336.

6. Since x·y·z must still equal 1680, the new x satisfies x × 336 = 1680, giving x = 1680 ÷ 336 = 5.

Check: 5 × 16 × 21 = 1680, exactly matching the first state's product 30 × 8 × 7 = 1680, confirming the invariant holds.

So x = 5.