Simplify : `(2(3)/(4))/(1(5)/(6)) +7/8 xx(1/3 + 1/4) + 5/7 + 3/4 ` of `3/7`

Options

A. `(56)/(77)`
B. `113/28`
C. `2/3`
D. `3(2)/(9)`

Answer

To simplify the expression `(2(3)/(4))/(1(5)/(6)) + 7/8 * (1/3 + 1/4) + 5/7 + 3/4 * (3/7)`, we will follow the order of operations (BODMAS/BIDMAS). Let's break it down step by step.

### Step 1: Simplify the first term `(2(3)/(4))/(1(5)/(6))`

1. Convert the mixed numbers to improper fractions:
- \(2 \frac{3}{4} = \frac{2 \times 4 + 3}{4} = \frac{8 + 3}{4} = \frac{11}{4}\)
- \(1 \frac{5}{6} = \frac{1 \times 6 + 5}{6} = \frac{6 + 5}{6} = \frac{11}{6}\)

2. Now, we can rewrite the first term:
\[ \frac{\frac{11}{4}}{\frac{11}{6}} = \frac{11}{4} \times \frac{6}{11} = \frac{6}{4} = \frac{3}{2} \]

### Step 2: Simplify the second term `7/8 * (1/3 + 1/4)`

1. Find the sum \(1/3 + 1/4\):
- The LCM of 3 and 4 is 12.
- Convert each fraction:
\[ 1/3 = \frac{4}{12}, \quad 1/4 = \frac{3}{12} \]
- Now add them:
\[ \frac{4}{12} + \frac{3}{12} = \frac{7}{12} \]

2. Multiply by \(7/8\):
\[ \frac{7}{8} \times \frac{7}{12} = \frac{49}{96} \]

### Step 3: Simplify the third term `5/7`

This term is already in its simplest form:
\[ \frac{5}{7} \]

### Step 4: Simplify the fourth term `3/4 * (3/7)`

1. Multiply:
\[ \frac{3}{4} \times \frac{3}{7} = \frac{9}{28} \]

### Step 5: Combine all the terms

Now we have:
\[ \frac{3}{2} + \frac{49}{96} + \frac{5}{7} + \frac{9}{28} \]

### Step 6: Find a common denominator

The least common multiple (LCM) of the denominators \(2, 96, 7, 28\) is \(672\).

1. Convert each term:
- For \(\frac{3}{2}\):
\[ \frac{3}{2} = \frac{3 \times 336}{2 \times 336} = \frac{1008}{672} \]
- For \(\frac{49}{96}\):
\[ \frac{49}{96} = \frac{49 \times 7}{96 \times 7} = \frac{343}{672} \]
- For \(\frac{5}{7}\):
\[ \frac{5}{7} = \frac{5 \times 96}{7 \times 96} = \frac{480}{672} \]
- For \(\frac{9}{28}\):
\[ \frac{9}{28} = \frac{9 \times 24}{28 \times 24} = \frac{216}{672} \]

### Step 7: Add the numerators

Now we can add all the numerators:
\[ 1008 + 343 + 480 + 216 = 2047 \]

### Step 8: Write the final result

Thus, the combined fraction is:
\[ \frac{2047}{672} \]

Since \(2047\) is greater than \(672\), we can convert it to a mixed number:
\[ 3 \frac{31}{672} \]

### Final Answer:
\[ 3 \frac{31}{672} \]