`(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 \(\frac{1 + 2i}{1 + 3i}\), we will follow these steps:

### Step 1: Multiply by the conjugate
We will multiply the numerator and the denominator by the conjugate of the denominator, which is \(1 - 3i\).

\[ \frac{1 + 2i}{1 + 3i} \cdot \frac{1 - 3i}{1 - 3i} \]

### Step 2: Expand the numerator
Now we will expand the numerator:

\[ (1 + 2i)(1 - 3i) = 1 \cdot 1 + 1 \cdot (-3i) + 2i \cdot 1 + 2i \cdot (-3i) \]
\[ = 1 - 3i + 2i - 6i^2 \]

Since \(i^2 = -1\), we can replace \(-6i^2\) with \(6\):

\[ = 1 - 3i + 2i + 6 = 7 - i \]

### Step 3: Expand the denominator
Now we will expand the denominator:

\[ (1 + 3i)(1 - 3i) = 1^2 - (3i)^2 = 1 - 9i^2 \]

Again, replacing \(-9i^2\) with \(9\):

\[ = 1 + 9 = 10 \]

### Step 4: Combine the results
Now we can combine the results from the numerator and the denominator:

\[ \frac{7 - i}{10} \]

### Step 5: Write the final result
This can be expressed as:

\[ \frac{7}{10} - \frac{1}{10}i \]

Thus, the final answer is:

\[ \frac{7}{10} - \frac{1}{10}i \]