The product of sum of `2/3` and `3/2` and difference of `2/3 and 3/2` is

Options

A. `65/36`
B. `36/65`
C. `56/65`
D. `53/66`

Answer

To solve the problem, we will follow these steps:

### Step 1: Find the sum of \( \frac{2}{3} \) and \( \frac{3}{2} \)

To find the sum, we need a common denominator. The least common multiple (LCM) of the denominators 3 and 2 is 6.

- Convert \( \frac{2}{3} \) to have a denominator of 6:
\[ \frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6} \]

- Convert \( \frac{3}{2} \) to have a denominator of 6:
\[ \frac{3}{2} = \frac{3 \times 3}{2 \times 3} = \frac{9}{6} \]

Now, we can add the two fractions:
\[ \frac{4}{6} + \frac{9}{6} = \frac{4 + 9}{6} = \frac{13}{6} \]

### Step 2: Find the difference of \( \frac{2}{3} \) and \( \frac{3}{2} \)

Again, we will use the common denominator of 6.

- Convert \( \frac{2}{3} \) to have a denominator of 6:
\[ \frac{2}{3} = \frac{4}{6} \]

- Convert \( \frac{3}{2} \) to have a denominator of 6:
\[ \frac{3}{2} = \frac{9}{6} \]

Now, we can subtract the two fractions:
\[ \frac{3}{2} - \frac{2}{3} = \frac{9}{6} - \frac{4}{6} = \frac{9 - 4}{6} = \frac{5}{6} \]

### Step 3: Find the product of the sum and the difference

Now we will multiply the results from Step 1 and Step 2:
\[ \text{Product} = \left(\frac{13}{6}\right) \times \left(\frac{5}{6}\right) = \frac{13 \times 5}{6 \times 6} = \frac{65}{36} \]

### Final Answer

The product of the sum of \( \frac{2}{3} \) and \( \frac{3}{2} \) and the difference of \( \frac{2}{3} \) and \( \frac{3}{2} \) is:
\[ \frac{65}{36} \]

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