Which of the following is equal to `(3/4-1/3)/(3/4+1/3)`?

Options

A. `5/12`
B. `(-5)/12`
C. `(-5)/13`
D. `5/13`

Answer

To solve the expression \((\frac{3}{4} - \frac{1}{3}) / (\frac{3}{4} + \frac{1}{3})\), we will follow these steps:

### Step 1: Find the LCM of the denominators
The denominators in the fractions are 4 and 3. The least common multiple (LCM) of 4 and 3 is 12.

### Step 2: Simplify the numerator \(\frac{3}{4} - \frac{1}{3}\)
To subtract the fractions, we convert them to have a common denominator of 12:

- Convert \(\frac{3}{4}\):
\[ \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \]

- Convert \(\frac{1}{3}\):
\[ \frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12} \]

Now, we can subtract:
\[ \frac{3}{4} - \frac{1}{3} = \frac{9}{12} - \frac{4}{12} = \frac{9 - 4}{12} = \frac{5}{12} \]

### Step 3: Simplify the denominator \(\frac{3}{4} + \frac{1}{3}\)
Again, using the common denominator of 12:

- Convert \(\frac{3}{4}\):
\[ \frac{3}{4} = \frac{9}{12} \]

- Convert \(\frac{1}{3}\):
\[ \frac{1}{3} = \frac{4}{12} \]

Now, we can add:
\[ \frac{3}{4} + \frac{1}{3} = \frac{9}{12} + \frac{4}{12} = \frac{9 + 4}{12} = \frac{13}{12} \]

### Step 4: Combine the results
Now we can substitute the simplified numerator and denominator back into the expression:
\[ \frac{\frac{5}{12}}{\frac{13}{12}} \]

### Step 5: Simplify the division of fractions
To divide by a fraction, we multiply by its reciprocal:
\[ \frac{5}{12} \div \frac{13}{12} = \frac{5}{12} \times \frac{12}{13} = \frac{5 \times 12}{12 \times 13} \]
The \(12\) in the numerator and denominator cancels out:
\[ = \frac{5}{13} \]

### Final Answer
Thus, the final answer is:
\[ \frac{5}{13} \]