If `2w = (3)/(2) x = y = (5)/(3) z` for four positive integers w,x,y and z which of the following expression can represent an integer ?

Options

A. `(wx)/(yz)`
B. `(x)/(w)`
C. `(4x)/(y)`
D. `(x^(2))/(y)`

Answer

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

\[ 2w = \frac{3}{2}x = y = \frac{5}{3}z \]

Let's denote this common value as \( k \). Therefore, we can express \( w \), \( x \), \( y \), and \( z \) in terms of \( k \):

1. From \( 2w = k \):
\[ w = \frac{k}{2} \]

2. From \( \frac{3}{2}x = k \):
\[ x = \frac{2}{3}k \]

3. From \( y = k \):
\[ y = k \]

4. From \( \frac{5}{3}z = k \):
\[ z = \frac{3}{5}k \]

Now we have all variables expressed in terms of \( k \):
- \( w = \frac{k}{2} \)
- \( x = \frac{2}{3}k \)
- \( y = k \)
- \( z = \frac{3}{5}k \)

Next, we will evaluate each of the given expressions to determine which one can represent an integer.

### Option A: \( \frac{wx}{yz} \)
\[ wx = \left(\frac{k}{2}\right) \left(\frac{2}{3}k\right) = \frac{k^2}{3} \]
\[ yz = k \left(\frac{3}{5}k\right) = \frac{3k^2}{5} \]
Now substituting into the expression:
\[ \frac{wx}{yz} = \frac{\frac{k^2}{3}}{\frac{3k^2}{5}} = \frac{k^2}{3} \cdot \frac{5}{3k^2} = \frac{5}{9} \]
This is not an integer.

### Option B: \( \frac{x}{w} \)
\[ \frac{x}{w} = \frac{\frac{2}{3}k}{\frac{k}{2}} = \frac{2}{3}k \cdot \frac{2}{k} = \frac{4}{3} \]
This is also not an integer.

### Option C: \( \frac{4x}{y} \)
\[ \frac{4x}{y} = \frac{4 \cdot \frac{2}{3}k}{k} = \frac{8}{3} \]
This is not an integer.

### Option D: \( \frac{x^2}{y} \)
\[ x^2 = \left(\frac{2}{3}k\right)^2 = \frac{4}{9}k^2 \]
\[ \frac{x^2}{y} = \frac{\frac{4}{9}k^2}{k} = \frac{4}{9}k \]
For this expression to be an integer, \( k \) must be a multiple of 9 (e.g., \( k = 9, 18, 27, \ldots \)).

### Conclusion
The only expression that can represent an integer is:
\[ \frac{x^2}{y} \text{ when } k \text{ is a multiple of } 9. \]