If x,y and z are three numbers such that x+y=13,y+z=15 and z+x=16,then the value of `(xy+xz)/(xyz)` is:

Options

A. `(5)/(36)`
B. `(18)/(5)`
C. `(5)/(18)`
D. `(36)/(5)`

Answer

To solve the problem, we need to find the value of \((xy + xz) / (xyz)\) given the equations:

1. \(x + y = 13\)
2. \(y + z = 15\)
3. \(z + x = 16\)

### Step 1: Add the three equations
We start by adding all three equations together:

\[ (x + y) + (y + z) + (z + x) = 13 + 15 + 16 \]

This simplifies to:

\[ 2x + 2y + 2z = 44 \]

### Step 2: Simplify the equation
Now, we can divide the entire equation by 2:

\[ x + y + z = 22 \]

### Step 3: Find the individual values of x, y, and z
Now we can use the value of \(x + y + z\) to find the individual variables.

From the first equation \(x + y = 13\), we can express \(z\):

\[ z = 22 - (x + y) = 22 - 13 = 9 \]

Now we have \(z = 9\).

Next, we use the second equation \(y + z = 15\) to find \(y\):

\[ y = 15 - z = 15 - 9 = 6 \]

Now we have \(y = 6\).

Finally, we can find \(x\) using the first equation again:

\[ x = 13 - y = 13 - 6 = 7 \]

Now we have all the values: \(x = 7\), \(y = 6\), \(z = 9\).

### Step 4: Substitute the values into the expression
Now we substitute \(x\), \(y\), and \(z\) into the expression \((xy + xz) / (xyz)\):

First, calculate \(xy\) and \(xz\):

\[ xy = 7 \cdot 6 = 42 \]
\[ xz = 7 \cdot 9 = 63 \]

Now substitute these into the expression:

\[ xy + xz = 42 + 63 = 105 \]

Next, calculate \(xyz\):

\[ xyz = 7 \cdot 6 \cdot 9 = 378 \]

### Step 5: Calculate the final value
Now we can find the value of the expression:

\[ \frac{xy + xz}{xyz} = \frac{105}{378} \]

### Step 6: Simplify the fraction
To simplify \(\frac{105}{378}\), we can divide both the numerator and the denominator by their greatest common divisor (GCD), which is 21:

\[ \frac{105 \div 21}{378 \div 21} = \frac{5}{18} \]

Thus, the final value of \((xy + xz) / (xyz)\) is:

\[ \frac{5}{18} \]

### Final Answer
The value of \((xy + xz) / (xyz)\) is \(\frac{5}{18}\).