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

Options

A. `(1)/(63)`
B. `(23)/(40)`
C. `(23)/(55)`
D. `(23)/(63)`

Answer

To simplify the expression

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

we will follow these steps:

### Step 1: Simplify the numerator

First, we need to simplify the expression inside the brackets in the numerator:

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

To do this, we find a common denominator, which is 15:

\[ \frac{2}{5} = \frac{6}{15}, \quad \frac{1}{3} = \frac{5}{15}. \]

Now, subtract:

\[ \frac{6}{15} - \frac{5}{15} = \frac{1}{15}. \]

Now substitute this back into the numerator:

\[ \frac{1}{3} + \frac{3}{4} \left( \frac{1}{15} \right). \]

Next, we calculate \(\frac{3}{4} \times \frac{1}{15}\):

\[ \frac{3}{4} \times \frac{1}{15} = \frac{3}{60} = \frac{1}{20}. \]

Now, we have:

\[ \frac{1}{3} + \frac{1}{20}. \]

To add these fractions, we find a common denominator, which is 60:

\[ \frac{1}{3} = \frac{20}{60}, \quad \frac{1}{20} = \frac{3}{60}. \]

Now, add them:

\[ \frac{20}{60} + \frac{3}{60} = \frac{23}{60}. \]

### Step 2: Simplify the denominator

Now we simplify the denominator:

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

First, convert \(1 \frac{2}{3}\) to an improper fraction:

\[ 1 \frac{2}{3} = \frac{5}{3}. \]

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

\[ \frac{5}{3} \times \frac{3}{4} = \frac{15}{12} = \frac{5}{4}. \]

Next, calculate \(\frac{1}{4} \text{ of } \frac{4}{5}\):

\[ \frac{1}{4} \times \frac{4}{5} = \frac{4}{20} = \frac{1}{5}. \]

Now substitute these back into the denominator:

\[ \frac{5}{4} - \frac{1}{5}. \]

To subtract these fractions, find a common denominator, which is 20:

\[ \frac{5}{4} = \frac{25}{20}, \quad \frac{1}{5} = \frac{4}{20}. \]

Now, subtract them:

\[ \frac{25}{20} - \frac{4}{20} = \frac{21}{20}. \]

### Step 3: Combine the results

Now we can substitute the simplified numerator and denominator back into the original expression:

\[ \frac{\frac{23}{60}}{\frac{21}{20}}. \]

To divide by a fraction, we multiply by its reciprocal:

\[ \frac{23}{60} \times \frac{20}{21} = \frac{23 \times 20}{60 \times 21}. \]

Now simplify:

\[ \frac{460}{1260}. \]

To simplify \(\frac{460}{1260}\), we can divide both the numerator and the denominator by 20:

\[ \frac{23}{63}. \]

### Final Answer

Thus, the simplified expression is

\[ \frac{23}{63}. \]