What is the value of `[overset(4)/5divoverset(5)/8-overset(7)/5+overset(15)/2" of "(overset(2)/3divoverset(4)/5)]`?

Options

A. `underset(100)/613`
B. `underset(100)/13`
C. `underset(29)/15`
D. `underset(17)/80`

Answer

To solve the expression \(\left[\frac{4}{5} \div \frac{5}{8} - \frac{7}{5} + \frac{15}{2} \text{ of } \left(\frac{2}{3} \div \frac{4}{5}\right)\right]\), we will follow the order of operations (BODMAS/BIDMAS: Brackets, Orders, Division and Multiplication, Addition and Subtraction).

### Step 1: Solve the expression inside the brackets
We start by solving the term inside the brackets:
\[ \frac{2}{3} \div \frac{4}{5} \]
To divide by a fraction, we multiply by its reciprocal:
\[ \frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6} \]

### Step 2: Substitute back into the main expression
Now we substitute \(\frac{5}{6}\) back into the main expression:
\[ \left[\frac{4}{5} \div \frac{5}{8} - \frac{7}{5} + \frac{15}{2} \times \frac{5}{6}\right] \]

### Step 3: Calculate \(\frac{15}{2} \times \frac{5}{6}\)
Now we compute:
\[ \frac{15}{2} \times \frac{5}{6} = \frac{15 \times 5}{2 \times 6} = \frac{75}{12} = \frac{25}{4} \]

### Step 4: Now, rewrite the expression
The expression now looks like this:
\[ \left[\frac{4}{5} \div \frac{5}{8} - \frac{7}{5} + \frac{25}{4}\right] \]

### Step 5: Solve \(\frac{4}{5} \div \frac{5}{8}\)
Next, we solve:
\[ \frac{4}{5} \div \frac{5}{8} = \frac{4}{5} \times \frac{8}{5} = \frac{32}{25} \]

### Step 6: Substitute back into the expression
Now we rewrite the expression:
\[ \left[\frac{32}{25} - \frac{7}{5} + \frac{25}{4}\right] \]

### Step 7: Convert \(\frac{7}{5}\) to a common denominator
To combine these fractions, we convert \(\frac{7}{5}\) to have a denominator of 25:
\[ \frac{7}{5} = \frac{35}{25} \]

### Step 8: Now, rewrite the expression
The expression now becomes:
\[ \left[\frac{32}{25} - \frac{35}{25} + \frac{25}{4}\right] \]
This simplifies to:
\[ \left[-\frac{3}{25} + \frac{25}{4}\right] \]

### Step 9: Convert \(-\frac{3}{25}\) to have a denominator of 100
To combine these fractions, we convert \(-\frac{3}{25}\):
\[ -\frac{3}{25} = -\frac{12}{100} \]
And convert \(\frac{25}{4}\) to have a denominator of 100:
\[ \frac{25}{4} = \frac{625}{100} \]

### Step 10: Combine the fractions
Now we can combine:
\[ -\frac{12}{100} + \frac{625}{100} = \frac{613}{100} \]

### Final Answer
Thus, the final value of the expression is:
\[ \frac{613}{100} \]