Simplify : `(3)/(4 + (5)/(6 + (7)/(8)))-(3)/(5) + (1)/(2) "of 1"(1)/(5) + 1(3)/(26)`

Options

A. `(3)/(4)`
B. `(1)/(2)`
C. `(3)/(5)`
D. `(3)/(7)`

Answer

To simplify the expression \( \frac{3}{4 + \frac{5}{6 + \frac{7}{8}}} - \frac{3}{5} + \frac{1}{2} \cdot \frac{1}{5} + \frac{1}{3} \cdot \frac{1}{26} \), we will follow these steps:

### Step 1: Simplify the innermost fraction
First, we simplify \( \frac{7}{8} \):
\[ 6 + \frac{7}{8} = \frac{6 \cdot 8 + 7}{8} = \frac{48 + 7}{8} = \frac{55}{8} \]

### Step 2: Substitute back into the expression
Now substitute this back into the next fraction:
\[ \frac{5}{6 + \frac{7}{8}} = \frac{5}{\frac{55}{8}} = 5 \cdot \frac{8}{55} = \frac{40}{55} = \frac{8}{11} \]

### Step 3: Substitute again
Now substitute \( \frac{8}{11} \) back into the expression:
\[ 4 + \frac{5}{6 + \frac{7}{8}} = 4 + \frac{8}{11} = \frac{4 \cdot 11 + 8}{11} = \frac{44 + 8}{11} = \frac{52}{11} \]

### Step 4: Simplify the first term
Now simplify the first term:
\[ \frac{3}{4 + \frac{5}{6 + \frac{7}{8}}} = \frac{3}{\frac{52}{11}} = 3 \cdot \frac{11}{52} = \frac{33}{52} \]

### Step 5: Simplify the remaining terms
Next, we simplify \( \frac{3}{5} \) and \( \frac{1}{2} \cdot \frac{1}{5} \):
\[ \frac{1}{2} \cdot \frac{1}{5} = \frac{1}{10} \]
And \( \frac{1}{3} \cdot \frac{1}{26} = \frac{1}{78} \).

### Step 6: Combine all terms
Now we have:
\[ \frac{33}{52} - \frac{3}{5} + \frac{1}{10} + \frac{1}{78} \]

### Step 7: Find a common denominator
The least common multiple (LCM) of \( 52, 5, 10, \) and \( 78 \) is \( 780 \).

### Step 8: Convert each fraction
Convert each fraction to have a denominator of \( 780 \):
\[ \frac{33}{52} = \frac{33 \cdot 15}{780} = \frac{495}{780} \]
\[ \frac{3}{5} = \frac{3 \cdot 156}{780} = \frac{468}{780} \]
\[ \frac{1}{10} = \frac{1 \cdot 78}{780} = \frac{78}{780} \]
\[ \frac{1}{78} = \frac{1 \cdot 10}{780} = \frac{10}{780} \]

### Step 9: Combine the fractions
Now combine all fractions:
\[ \frac{495}{780} - \frac{468}{780} + \frac{78}{780} + \frac{10}{780} = \frac{495 - 468 + 78 + 10}{780} = \frac{115}{780} \]

### Step 10: Simplify the final fraction
Now simplify \( \frac{115}{780} \):
\[ \frac{115 \div 5}{780 \div 5} = \frac{23}{156} \]

Thus, the simplified expression is:
\[ \frac{23}{156} \]