If distinct numbers `x, y,z and p` are chosen from the numbers -2, 2, 1/2, -1/3, what is the largest possible value of the expression `(x^2y)/(z - p)` ?

Options

A. `26//4`
B. `34//5`
C. `38//5`
D. `48//5`

Answer

To find the largest possible value of the expression \(\frac{x^2y}{z - p}\) using distinct numbers \(x, y, z,\) and \(p\) chosen from the set \(-2, 2, \frac{1}{2}, -\frac{1}{3}\), we will follow these steps:

### Step 1: Analyze the Expression
The expression consists of a numerator \(x^2y\) and a denominator \(z - p\). To maximize the overall expression, we need to maximize the numerator and minimize the denominator.

### Step 2: Choose Values for \(z\) and \(p\)
To minimize the denominator \(z - p\), we should choose \(z\) as the smallest number and \(p\) as the largest number from the remaining options.

From the available numbers:
- The smallest number is \(-\frac{1}{3}\).
- The largest number is \(\frac{1}{2}\).

Thus, we can set:
- \(z = \frac{1}{2}\)
- \(p = -\frac{1}{3}\)

Now, we calculate \(z - p\):
\[ z - p = \frac{1}{2} - \left(-\frac{1}{3}\right) = \frac{1}{2} + \frac{1}{3} \]
To add these fractions, we find a common denominator (which is 6):
\[ \frac{1}{2} = \frac{3}{6}, \quad \frac{1}{3} = \frac{2}{6} \]
So,
\[ z - p = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \]

### Step 3: Choose Values for \(x\) and \(y\)
Now we have two numbers left to choose from: \(-2\) and \(2\). We want to maximize \(x^2y\).

Since \(x^2\) will always be positive, we should choose:
- \(y = 2\) (the largest value)
- \(x = -2\) (the remaining value)

Now we calculate \(x^2y\):
\[ x^2y = (-2)^2 \cdot 2 = 4 \cdot 2 = 8 \]

### Step 4: Calculate the Final Expression
Now we can substitute our values back into the expression:
\[ \frac{x^2y}{z - p} = \frac{8}{\frac{5}{6}} = 8 \cdot \frac{6}{5} = \frac{48}{5} \]

### Conclusion
Thus, the largest possible value of the expression \(\frac{x^2y}{z - p}\) is \(\frac{48}{5}\).