Suppose x, y and z are variables taking positive real numbers as their possible values. It is given that y is directly proportional to $x^2$ and x is inversely proportional to z. For $z={\displaystyle \frac{7}{25}}$, the values of x and y are 5 and 50, respectively. If $y=98$, what is z equal to?

The correct option is B - 1/5.

[as per provisional answerkey]

Solution

Step 1: Establish the relationships between variables.
1. y is directly proportional to $x^2$: $y=k_1\cdot x^2$
2. x is inversely proportional to z: $x=k_2/z$
Combining these, we get: $y=k_1\cdot (k_2/z{)}^2=K/z^2$ (where K is a constant).

Step 2: Find the constant K using the given values.
Given: $z=7/25$, $x=5$, and $y=50$.
Using $y=K/z^2$:
$50=K/(7/25{)}^2$
$50=K/(49/625)$
$K=50\cdot (49/625)$
$K=(2\cdot 25\cdot 49)/(25\cdot 25)$
$K=(2\cdot 49)/25=98/25$.

Step 3: Calculate z for the new value of y.
Given: $y=98$. Find z.
$y=K/z^2$
$98=(98/25)/z^2$
Divide both sides by 98:
$1=(1/25)/z^2$
$z^2=1/25$
Since z is a positive real number:
$z=\sqrt{1/25}=1/5$.

Why the other options are incorrect

• Option (a) - 1/7: This value would result if the relationship was $y\propto 1/z$ instead of $y\propto 1/z^2$, or if a calculation error occurred while squaring the fraction.
• Option (c) - 5/7: This value is the reciprocal of the ratio used in the initial conditions ($7/25\times 5/1$) and does not satisfy the inverse square relationship between y and z.
• Option (d) - 1: This value assumes a direct equality or a specific constant K that does not align with the provided data points ($z=7/25,y=50$).

Key Concept

The principle of combined variation, where the relationship between the first and last variable is derived by substituting the intermediate variable ($y\propto x^2$ and $x\propto 1/z$ implies $y\propto 1/z^2$).