If `1 le x le 3` and `2 le y le 4`, what is the maximum value of `((x)/(y))` ?

Options

A. A)`(2)/(3)`
B. B)4
C. C)`(3)/(2)`
D. D)2

Answer

To find the maximum value of \(\frac{x}{y}\) given the constraints \(1 \leq x \leq 3\) and \(2 \leq y \leq 4\), we can follow these steps:

### Step 1: Identify the ranges of \(x\) and \(y\)
- The variable \(x\) can take any value from 1 to 3, inclusive.
- The variable \(y\) can take any value from 2 to 4, inclusive.

### Step 2: Set up the expression to maximize
We want to maximize the expression \(\frac{x}{y}\).

### Step 3: Analyze the expression
To maximize \(\frac{x}{y}\), we should maximize \(x\) and minimize \(y\). This is because a larger numerator and a smaller denominator will yield a larger fraction.

### Step 4: Determine the maximum value of \(x\)
The maximum value of \(x\) in the given range is:
\[ x = 3 \]

### Step 5: Determine the minimum value of \(y\)
The minimum value of \(y\) in the given range is:
\[ y = 2 \]

### Step 6: Calculate the maximum value of \(\frac{x}{y}\)
Now, substituting the maximum value of \(x\) and the minimum value of \(y\) into the expression:
\[ \frac{x}{y} = \frac{3}{2} \]

### Step 7: Conclusion
Thus, the maximum value of \(\frac{x}{y}\) is:
\[ \frac{3}{2} \]

### Final Answer
The maximum value of \(\frac{x}{y}\) is \(\frac{3}{2}\).

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