`(x)/(z)=(1)/(3)`
If in the equation above x and z are integers which are possible values of `(x^(2))/(z)`?
I.`(1)/(9)`
II.`(1)/(3)`
III. 3

Options

A. II only
B. III only
C. I and III only
D. II and III only

Answer

To solve the problem step by step, we start with the given equation:

\[ \frac{x}{z} = \frac{1}{3} \]

From this equation, we can express \( z \) in terms of \( x \):

\[ z = 3x \]

Now, we need to evaluate the possible values of \( \frac{x^2}{z} \) for the given statements.

### Step 1: Evaluate Statement I: \( \frac{x^2}{z} = \frac{1}{9} \)

Cross-multiplying gives us:

\[ 9x^2 = z \]

Substituting \( z = 3x \) into the equation:

\[ 9x^2 = 3x \]

Now, we can simplify this equation:

\[ 9x^2 - 3x = 0 \]

Factoring out \( 3x \):

\[ 3x(3x - 1) = 0 \]

This gives us two solutions:

1. \( 3x = 0 \) → \( x = 0 \)
2. \( 3x - 1 = 0 \) → \( x = \frac{1}{3} \) (not an integer)

Since \( x \) must be an integer, the only solution is \( x = 0 \). If \( x = 0 \), then \( z = 3(0) = 0 \). However, \( \frac{x^2}{z} \) becomes undefined (as division by zero is not possible). Thus, Statement I is **not true**.

### Step 2: Evaluate Statement II: \( \frac{x^2}{z} = \frac{1}{3} \)

Cross-multiplying gives us:

\[ 3x^2 = z \]

Substituting \( z = 3x \):

\[ 3x^2 = 3x \]

Simplifying this gives:

\[ 3x^2 - 3x = 0 \]

Factoring out \( 3x \):

\[ 3x(x - 1) = 0 \]

This gives us two solutions:

1. \( 3x = 0 \) → \( x = 0 \)
2. \( x - 1 = 0 \) → \( x = 1 \)

Both solutions are integers. If \( x = 1 \), then:

\[ z = 3(1) = 3 \]

Thus, both \( x \) and \( z \) are integers, making Statement II **true**.

### Step 3: Evaluate Statement III: \( \frac{x^2}{z} = 3 \)

Cross-multiplying gives us:

\[ x^2 = 3z \]

Substituting \( z = 3x \):

\[ x^2 = 3(3x) \]

This simplifies to:

\[ x^2 = 9x \]

Rearranging gives:

\[ x^2 - 9x = 0 \]

Factoring out \( x \):

\[ x(x - 9) = 0 \]

This gives us two solutions:

1. \( x = 0 \)
2. \( x - 9 = 0 \) → \( x = 9 \)

Both solutions are integers. If \( x = 9 \), then:

\[ z = 3(9) = 27 \]

Thus, both \( x \) and \( z \) are integers, making Statement III **true**.

### Conclusion

- Statement I: **Not true**
- Statement II: **True**
- Statement III: **True**

The correct answer is that Statements II and III are possible values of \( \frac{x^2}{z} \).