The given relation “x varies directly as y and inversely as z” can be written as x = k·y/z for some constant k. Using x = 16 when y = 8 and z = 3, we have 16 = k·8/3, so k = 16 × 3 / 8 = 6. For y = 9 and z = 2, x = 6·9/2 = 54/2 = 27. Hence, the new value of x is 27.
Option A:
Option A, 18, would correspond to x = 6·y/z = 6·9/? with some other z value, but it is not obtained when z = 2. If we plug x = 18 into x = k·y/z with k = 6, we get 18 = 6·9/z, implying z = 3, which contradicts the specified z = 2. So 18 does not satisfy the given condition.
Option B:
Option B, 24, arises if we mistakenly treat the change in y as proportional but ignore the change in z. With k = 6, x would be 24 only when y/z = 4, which is not the case for y = 9 and z = 2 where y/z = 4.5. Thus, 24 is not consistent with the correct variation formula.
Option C:
Option C is correct because it uses the same constant of proportionality k = 6 found from the first condition and correctly substitutes y = 9 and z = 2. The resulting value 27 is obtained by straightforward application of x = k·y/z and maintains both the direct dependence on y and inverse dependence on z.
Option D:
Option D, 30, would require y/z to be 5 when k = 6, since 30 = 6·5. But the actual ratio y/z is 9/2 = 4.5 in the new situation. Therefore, 30 does not follow from the prescribed relationship between x, y and z and is not an acceptable solution.
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