The value of `3(1)/(3) div 2(1)/(2)` of `1(3)/(5) + ((3)/(8) + (1)/(7) xx 1 (3)/(4))` is

Options

A. `(35)/(24)`
B. `(55)/(24)`
C. `(5)/(24)`
D. `(25)/(24)`

Answer

To solve the expression \( \frac{3 \frac{1}{3}}{2 \frac{1}{2}} \text{ of } \left( \frac{1}{3} + \left( \frac{3}{8} + \frac{1}{7} \times 1 \frac{3}{4} \right) \right) \), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we need to convert the mixed numbers into improper fractions.

- \( 3 \frac{1}{3} = \frac{10}{3} \) (because \( 3 \times 3 + 1 = 10 \))
- \( 2 \frac{1}{2} = \frac{5}{2} \) (because \( 2 \times 2 + 1 = 5 \))
- \( 1 \frac{3}{4} = \frac{7}{4} \) (because \( 1 \times 4 + 3 = 7 \))

Now our expression looks like this:
\[ \frac{\frac{10}{3}}{\frac{5}{2}} \text{ of } \left( \frac{1}{3} + \left( \frac{3}{8} + \frac{1}{7} \times \frac{7}{4} \right) \right) \]

### Step 2: Calculate the Division
Next, we calculate the division of the improper fractions:
\[ \frac{10}{3} \div \frac{5}{2} = \frac{10}{3} \times \frac{2}{5} = \frac{20}{15} = \frac{4}{3} \]

### Step 3: Solve the Bracket
Now we need to solve the expression inside the brackets:
\[ \frac{1}{3} + \left( \frac{3}{8} + \frac{1}{7} \times \frac{7}{4} \right) \]
First, calculate \( \frac{1}{7} \times \frac{7}{4} \):
\[ \frac{1}{7} \times \frac{7}{4} = \frac{1}{4} \]
Now substitute this back into the bracket:
\[ \frac{1}{3} + \left( \frac{3}{8} + \frac{1}{4} \right) \]

### Step 4: Find a Common Denominator
To add \( \frac{3}{8} \) and \( \frac{1}{4} \), we need a common denominator. The least common multiple of 8 and 4 is 8.
\[ \frac{1}{4} = \frac{2}{8} \]
Now we can add:
\[ \frac{3}{8} + \frac{2}{8} = \frac{5}{8} \]

### Step 5: Add to \( \frac{1}{3} \)
Now we add \( \frac{1}{3} \) to \( \frac{5}{8} \). The least common multiple of 3 and 8 is 24.
\[ \frac{1}{3} = \frac{8}{24}, \quad \frac{5}{8} = \frac{15}{24} \]
Now add:
\[ \frac{8}{24} + \frac{15}{24} = \frac{23}{24} \]

### Step 6: Multiply by \( \frac{4}{3} \)
Now we multiply the result from the division by the result from the bracket:
\[ \frac{4}{3} \text{ of } \frac{23}{24} = \frac{4}{3} \times \frac{23}{24} = \frac{92}{72} \]

### Step 7: Simplify the Fraction
Now simplify \( \frac{92}{72} \):
\[ \frac{92 \div 4}{72 \div 4} = \frac{23}{18} \]

### Final Answer
The value of the expression is \( \frac{23}{18} \).