Find the product of `(3)/(7)` and reciprocal of `(2)/(7) + (1)/(14).`

Options

A. `6/5`
B. `(15)/(98)`
C. `(-6)/(5)`
D. `(5)/(14)`

Answer

To solve the problem, we need to find the product of \(\frac{3}{7}\) and the reciprocal of \(\frac{2}{7} + \frac{1}{14}\). Let's break it down step by step.

### Step 1: Find the sum of \(\frac{2}{7}\) and \(\frac{1}{14}\)

To add these two fractions, we need to have a common denominator. The least common multiple (LCM) of 7 and 14 is 14.

- Convert \(\frac{2}{7}\) to a fraction with a denominator of 14:
\[ \frac{2}{7} = \frac{2 \times 2}{7 \times 2} = \frac{4}{14} \]

Now we can add:
\[ \frac{4}{14} + \frac{1}{14} = \frac{4 + 1}{14} = \frac{5}{14} \]

### Step 2: Find the reciprocal of \(\frac{5}{14}\)

The reciprocal of a fraction is obtained by flipping the numerator and the denominator. Therefore:
\[ \text{Reciprocal of } \frac{5}{14} = \frac{14}{5} \]

### Step 3: Find the product of \(\frac{3}{7}\) and \(\frac{14}{5}\)

Now we multiply \(\frac{3}{7}\) by \(\frac{14}{5}\):
\[ \frac{3}{7} \times \frac{14}{5} = \frac{3 \times 14}{7 \times 5} = \frac{42}{35} \]

### Step 4: Simplify \(\frac{42}{35}\)

To simplify \(\frac{42}{35}\), we can divide both the numerator and the denominator by their greatest common divisor (GCD), which is 7:
\[ \frac{42 \div 7}{35 \div 7} = \frac{6}{5} \]

### Step 5: Convert \(\frac{6}{5}\) to a mixed number

\(\frac{6}{5}\) can also be expressed as a mixed number:
\[ 6 \div 5 = 1 \quad \text{(remainder 1)} \]
So, \(\frac{6}{5} = 1 \frac{1}{5}\).

### Final Answer

The product of \(\frac{3}{7}\) and the reciprocal of \(\frac{2}{7} + \frac{1}{14}\) is:
\[ \frac{6}{5} \text{ or } 1 \frac{1}{5} \]

---