Simplify : `((1)/(3) + (1)/(4) [(2)/(5) - (1)/(2)])/(1 (2)/(3)"of" (3)/(4) - (3)/(4) "of" (4)/(5))`

Options

A. `(37)/(78)`
B. `(37)/(13)`
C. `(74)/(78)`
D. `(74)/(13)`

Answer

To simplify the expression

\[ \frac{\left(\frac{1}{3} + \frac{1}{4} \left(\frac{2}{5} - \frac{1}{2}\right)\right)}{\left(1 \cdot \frac{2}{3} \text{ of } \frac{3}{4} - \frac{3}{4} \text{ of } \frac{4}{5}\right)} \]

we will follow these steps:

### Step 1: Simplify the numerator

Start with the expression inside the parentheses in the numerator:

\[ \frac{2}{5} - \frac{1}{2} \]

To subtract these fractions, we need a common denominator. The least common multiple of 5 and 2 is 10.

Convert each fraction:

\[ \frac{2}{5} = \frac{4}{10} \quad \text{and} \quad \frac{1}{2} = \frac{5}{10} \]

Now, subtract:

\[ \frac{4}{10} - \frac{5}{10} = \frac{-1}{10} \]

Now substitute back into the numerator:

\[ \frac{1}{3} + \frac{1}{4} \left(\frac{-1}{10}\right) \]

### Step 2: Multiply \(\frac{1}{4}\) by \(\frac{-1}{10}\)

This gives:

\[ \frac{1}{4} \cdot \frac{-1}{10} = \frac{-1}{40} \]

Now, add this to \(\frac{1}{3}\):

\[ \frac{1}{3} + \frac{-1}{40} \]

### Step 3: Find a common denominator for the addition

The least common multiple of 3 and 40 is 120.

Convert each fraction:

\[ \frac{1}{3} = \frac{40}{120} \quad \text{and} \quad \frac{-1}{40} = \frac{-3}{120} \]

Now, add:

\[ \frac{40}{120} - \frac{3}{120} = \frac{37}{120} \]

### Step 4: Simplify the denominator

Now, simplify the denominator:

\[ 1 \cdot \frac{2}{3} \text{ of } \frac{3}{4} - \frac{3}{4} \text{ of } \frac{4}{5} \]

Calculating \(1 \cdot \frac{2}{3} \text{ of } \frac{3}{4}\):

\[ \frac{2}{3} \cdot \frac{3}{4} = \frac{2 \cdot 3}{3 \cdot 4} = \frac{2}{4} = \frac{1}{2} \]

Now calculate \(\frac{3}{4} \text{ of } \frac{4}{5}\):

\[ \frac{3}{4} \cdot \frac{4}{5} = \frac{3 \cdot 4}{4 \cdot 5} = \frac{3}{5} \]

Now subtract:

\[ \frac{1}{2} - \frac{3}{5} \]

### Step 5: Find a common denominator for the subtraction

The least common multiple of 2 and 5 is 10.

Convert each fraction:

\[ \frac{1}{2} = \frac{5}{10} \quad \text{and} \quad \frac{3}{5} = \frac{6}{10} \]

Now, subtract:

\[ \frac{5}{10} - \frac{6}{10} = \frac{-1}{10} \]

### Step 6: Combine the results

Now we have:

\[ \frac{\frac{37}{120}}{\frac{-1}{10}} = \frac{37}{120} \cdot \frac{-10}{1} = \frac{-370}{120} \]

### Step 7: Simplify the fraction

To simplify \(\frac{-370}{120}\), we can divide both the numerator and the denominator by 10:

\[ \frac{-37}{12} \]

### Final Answer

Thus, the simplified expression is:

\[ \frac{-37}{12} \]

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