If ` ( c - d) = (( c + d))/( 5) = ( (cd))/(3)and c, d ne 0` then what is the value of cd ?

Options

A. `(1)/(2)`
B. `(3)/(2)`
C. `(5)/(2)`
D. `(5)/(2)`

Answer

To solve the equation \( c - d = \frac{c + d}{5} = \frac{cd}{3} \) where \( c \) and \( d \neq 0 \), we can follow these steps:

### Step 1: Set up the equations
From the given information, we can write:
1. \( c - d = \frac{c + d}{5} \)
2. \( c - d = \frac{cd}{3} \)

### Step 2: Solve the first equation
Starting with the first equation:
\[ c - d = \frac{c + d}{5} \]
Multiply both sides by 5 to eliminate the fraction:
\[ 5(c - d) = c + d \]
This simplifies to:
\[ 5c - 5d = c + d \]
Rearranging gives:
\[ 5c - c = d + 5d \]
\[ 4c = 6d \]
Dividing both sides by 2:
\[ 2c = 3d \quad \text{or} \quad \frac{c}{d} = \frac{3}{2} \]

### Step 3: Express \( c \) in terms of \( d \)
From \( \frac{c}{d} = \frac{3}{2} \), we can express \( c \) as:
\[ c = \frac{3}{2}d \]

### Step 4: Substitute \( c \) into the second equation
Now substitute \( c = \frac{3}{2}d \) into the second equation:
\[ c - d = \frac{cd}{3} \]
Substituting \( c \):
\[ \frac{3}{2}d - d = \frac{\left(\frac{3}{2}d\right)d}{3} \]
This simplifies to:
\[ \frac{3}{2}d - \frac{2}{2}d = \frac{3d^2}{6} \]
\[ \frac{1}{2}d = \frac{d^2}{2} \]

### Step 5: Solve for \( d \)
Multiply both sides by 2 to eliminate the fraction:
\[ d = d^2 \]
Rearranging gives:
\[ d^2 - d = 0 \]
Factoring out \( d \):
\[ d(d - 1) = 0 \]
Since \( d \neq 0 \), we have:
\[ d = 1 \]

### Step 6: Find \( c \)
Now substitute \( d = 1 \) back into the equation for \( c \):
\[ c = \frac{3}{2}(1) = \frac{3}{2} \]

### Step 7: Calculate \( cd \)
Now we can find \( cd \):
\[ cd = \left(\frac{3}{2}\right)(1) = \frac{3}{2} \]

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

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