If `M - ((3)/(7)) div ((6)/(5)) xx ((2)/(3)) + ((1)/(5) xx ((3)/(2)) and N = ((2)/(5))xx ((5)/(6)) div ((1)/(3)) + ((3)/(5)) xx ((2)/(3)) div ((3)/(5))`, then what is the value of `(M)/(N)`

Options

A. `(207)/(560)`
B. `(339)/(1120)`
C. `(113)/(350)`
D. `(69)/(175)`

Answer

To solve the problem, we need to calculate the values of \( M \) and \( N \) step by step, and then find the value of \( \frac{M}{N} \).

### Step 1: Calculate \( M \)

Given:
\[ M = \left( \frac{3}{7} \right) \div \left( \frac{6}{5} \right) \times \left( \frac{2}{3} \right) + \left( \frac{1}{5} \times \frac{3}{2} \right) \]

1. **Calculate \( \left( \frac{3}{7} \right) \div \left( \frac{6}{5} \right) \)**:
\[ \frac{3}{7} \div \frac{6}{5} = \frac{3}{7} \times \frac{5}{6} = \frac{3 \times 5}{7 \times 6} = \frac{15}{42} = \frac{5}{14} \]

2. **Now calculate \( \left( \frac{5}{14} \times \frac{2}{3} \right) \)**:
\[ \frac{5}{14} \times \frac{2}{3} = \frac{5 \times 2}{14 \times 3} = \frac{10}{42} = \frac{5}{21} \]

3. **Calculate \( \left( \frac{1}{5} \times \frac{3}{2} \right) \)**:
\[ \frac{1}{5} \times \frac{3}{2} = \frac{1 \times 3}{5 \times 2} = \frac{3}{10} \]

4. **Now add the two results**:
\[ M = \frac{5}{21} + \frac{3}{10} \]
To add these fractions, we need a common denominator. The least common multiple of 21 and 10 is 210.

Convert \( \frac{5}{21} \) and \( \frac{3}{10} \) to have a denominator of 210:
\[ \frac{5}{21} = \frac{5 \times 10}{21 \times 10} = \frac{50}{210} \]
\[ \frac{3}{10} = \frac{3 \times 21}{10 \times 21} = \frac{63}{210} \]

Now add:
\[ M = \frac{50}{210} + \frac{63}{210} = \frac{113}{210} \]

### Step 2: Calculate \( N \)

Given:
\[ N = \left( \frac{2}{5} \times \frac{5}{6} \right) \div \left( \frac{1}{3} \right) + \left( \frac{3}{5} \times \frac{2}{3} \right) \div \left( \frac{3}{5} \right) \]

1. **Calculate \( \left( \frac{2}{5} \times \frac{5}{6} \right) \)**:
\[ \frac{2}{5} \times \frac{5}{6} = \frac{2 \times 5}{5 \times 6} = \frac{10}{30} = \frac{1}{3} \]

2. **Now divide by \( \frac{1}{3} \)**:
\[ \frac{1}{3} \div \frac{1}{3} = 1 \]

3. **Calculate \( \left( \frac{3}{5} \times \frac{2}{3} \right) \)**:
\[ \frac{3}{5} \times \frac{2}{3} = \frac{3 \times 2}{5 \times 3} = \frac{6}{15} = \frac{2}{5} \]

4. **Now divide by \( \frac{3}{5} \)**:
\[ \frac{2}{5} \div \frac{3}{5} = \frac{2}{5} \times \frac{5}{3} = \frac{2 \times 5}{5 \times 3} = \frac{2}{3} \]

5. **Now add the two results**:
\[ N = 1 + \frac{2}{3} \]
Convert 1 to have a denominator of 3:
\[ 1 = \frac{3}{3} \]
Now add:
\[ N = \frac{3}{3} + \frac{2}{3} = \frac{5}{3} \]

### Step 3: Calculate \( \frac{M}{N} \)

Now we have:
\[ M = \frac{113}{210} \]
\[ N = \frac{5}{3} \]

Calculate \( \frac{M}{N} \):
\[ \frac{M}{N} = \frac{\frac{113}{210}}{\frac{5}{3}} = \frac{113}{210} \times \frac{3}{5} = \frac{113 \times 3}{210 \times 5} = \frac{339}{1050} \]

### Final Answer
The value of \( \frac{M}{N} \) is \( \frac{339}{1050} \).