`(1+2i)/(1+3i)` is equal to

Options

A. `7/10 -i/10`
B. `7/10 +i/10`
C. `7-i`
D. `7/2 + i/2`

Answer

To solve the expression \((1 + 2i)/(1 + 3i)\), we will eliminate the imaginary unit \(i\) from the denominator by multiplying the numerator and the denominator by the complex conjugate of the denominator.

### Step-by-step Solution:

1. **Identify the complex conjugate of the denominator:**
The denominator is \(1 + 3i\). The complex conjugate of this expression is \(1 - 3i\).

2. **Multiply both the numerator and denominator by the complex conjugate:**
\[ \frac{1 + 2i}{1 + 3i} \cdot \frac{1 - 3i}{1 - 3i} = \frac{(1 + 2i)(1 - 3i)}{(1 + 3i)(1 - 3i)} \]

3. **Calculate the denominator:**
The denominator can be simplified using the formula for the difference of squares:
\[ (1 + 3i)(1 - 3i) = 1^2 - (3i)^2 = 1 - 9(-1) = 1 + 9 = 10 \]

4. **Calculate the numerator:**
Expand the numerator:
\[ (1 + 2i)(1 - 3i) = 1 \cdot 1 + 1 \cdot (-3i) + 2i \cdot 1 + 2i \cdot (-3i) \]
Simplifying this gives:
\[ = 1 - 3i + 2i - 6i^2 \]
Since \(i^2 = -1\), we can substitute:
\[ = 1 - 3i + 2i + 6 = 7 - i \]

5. **Combine the results:**
Now we can write the expression as:
\[ \frac{7 - i}{10} \]

6. **Separate the real and imaginary parts:**
This can be expressed as:
\[ \frac{7}{10} - \frac{1}{10}i \]

### Final Answer:
Thus, \((1 + 2i)/(1 + 3i) = \frac{7}{10} - \frac{1}{10}i\).

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