The value of `(3)/(4) xx 2 (2)/(3) -: (5)/(9) " of " 1(1)/(5) + (2)/(23) xx 3 (5)/(6) -: (2)/(7) " of " 2(1)/(3)` is :

Options

A. `3(1)/(2)`
B. `1(2)/(3)`
C. `4(5)/(6)`
D. `1(5)/(6)`

Answer

To solve the given expression step by step, we will follow the order of operations (BODMAS/BIDMAS: Brackets, Orders, Division/Multiplication, Addition/Subtraction).

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we convert all mixed numbers to improper fractions:
- \(2 \frac{2}{3} = \frac{8}{3}\)
- \(1 \frac{1}{5} = \frac{6}{5}\)
- \(3 \frac{5}{6} = \frac{23}{6}\)
- \(2 \frac{1}{3} = \frac{7}{3}\)

Now, rewrite the expression:
\[ \frac{3}{4} \times \frac{8}{3} \div \frac{5}{9} \text{ of } \frac{6}{5} + \frac{2}{23} \times \frac{23}{6} \div \frac{2}{7} \text{ of } \frac{7}{3} \]

### Step 2: Replace "of" with Multiplication
The term "of" indicates multiplication. Thus, we rewrite the expression:
\[ \frac{3}{4} \times \frac{8}{3} \div \frac{5}{9} \times \frac{6}{5} + \frac{2}{23} \times \frac{23}{6} \div \frac{2}{7} \times \frac{7}{3} \]

### Step 3: Simplify Each Part
1. For the first part:
\[ \frac{3}{4} \times \frac{8}{3} \div \frac{5}{9} \times \frac{6}{5} \]
- First, calculate \(\frac{3}{4} \times \frac{8}{3}\):
\[ = \frac{3 \times 8}{4 \times 3} = \frac{8}{4} = 2 \]
- Now, calculate \(2 \div \frac{5}{9} = 2 \times \frac{9}{5} = \frac{18}{5}\).
- Then, multiply by \(\frac{6}{5}\):
\[ = \frac{18}{5} \times \frac{6}{5} = \frac{108}{25} \]

2. For the second part:
\[ \frac{2}{23} \times \frac{23}{6} \div \frac{2}{7} \times \frac{7}{3} \]
- First, calculate \(\frac{2}{23} \times \frac{23}{6} = \frac{2 \times 23}{23 \times 6} = \frac{2}{6} = \frac{1}{3}\).
- Now, calculate \(\frac{1}{3} \div \frac{2}{7} = \frac{1}{3} \times \frac{7}{2} = \frac{7}{6}\).
- Then, multiply by \(\frac{7}{3}\):
\[ = \frac{7}{6} \times \frac{7}{3} = \frac{49}{18} \]

### Step 4: Combine the Results
Now we combine the two parts:
\[ \frac{108}{25} + \frac{49}{18} \]
To add these fractions, we need a common denominator. The least common multiple of 25 and 18 is 450.

- Convert \(\frac{108}{25}\) to have a denominator of 450:
\[ \frac{108 \times 18}{25 \times 18} = \frac{1944}{450} \]
- Convert \(\frac{49}{18}\) to have a denominator of 450:
\[ \frac{49 \times 25}{18 \times 25} = \frac{1225}{450} \]

Now add the two fractions:
\[ \frac{1944 + 1225}{450} = \frac{3169}{450} \]

### Step 5: Final Result
The final result is:
\[ \frac{3169}{450} \]