Suppose x, y and z are variables taking positive real numbers as their possible values. It is given that y is directly proportional to x² and x is inversely proportional to z. For z = 7/25, the values of x and y are 5 and 50, respectively. If y = 98, what is z equal to?

This problem relies on the algebraic properties of direct and inverse variation.

According to the fundamental rules of direct variation, when y is directly proportional to x², it is mathematically expressed as y = k₁x², where k₁ is a constant of proportionality. Similarly, inverse variation dictates that when x is inversely proportional to z, it is expressed as x = k₂/z, where k₂ is another constant.

First, we determine k₂ using the given conditions x = 5 and z = 7/25. Substituting these into the second equation gives 5 = k₂/(7/25). Solving for k₂ yields k₂ = 5 × 7/25 = 7/5. Therefore, the exact relationship is x = (7/5)/z.

Next, we determine k₁ using the initial conditions x = 5 and y = 50. Substituting these into the first equation gives 50 = k₁(5)², which simplifies to 50 = 25k₁, meaning k₁ = 2. Therefore, the exact relationship is y = 2x².

To find z when y = 98, we must first solve for x: 98 = 2x² ⇒ x² = 49. Since the variables only take positive real numbers, x = 7. Substituting x = 7 into the inverse variation equation: 7 = (7/5)/z ⇒ 7z = 7/5 ⇒ z = ⅕. Thus, Option B is correct.

Why the other options are incorrect:

• Option A (1/7): Occurs if an aspirant erroneously assumes k₂ = 1 and solves 7 = 1/z, yielding 1/7.
• Option C (5/7): Results from incorrectly taking the reciprocal of the constant k₂ (7/5) without solving the final equation algebraically.
• Option D (1): Occurs if an aspirant ignores calculating the respective constants of proportionality altogether.

Takeaway: When solving chain proportionality questions, always calculate the individual constants of proportionality (k₁, k₂) using the initial baseline values before attempting to solve for the final target variable.