Simplify : `((2)/(3) div (3)/(4)"of" (5)/(6))/((2)/(3) div (3)/(4) xx (5)/(6)) + (2 + 2 xx 2)/(2 div 2 xx 2)div ((1)/(2) div (1)/(2) "of" (1)/(2))/((1)/(2) + (1)/(2) "of" (1)/(2))`

Options

A. 2
B. 2.5
C. 4
D. 4.5

Answer

To simplify the expression given in the question, we will follow the order of operations (BODMAS/BIDMAS) which stands for Brackets, Orders (i.e., powers and square roots, etc.), Division and Multiplication (from left to right), Addition and Subtraction (from left to right).

Let's break down the expression step by step:

### Step 1: Simplify the first part of the expression
The expression is:
\[ \frac{\left(\frac{2}{3} \div \frac{3}{4} \text{ of } \frac{5}{6}\right)}{\left(\frac{2}{3} \div \frac{3}{4} \times \frac{5}{6}\right)} \]

#### Sub-step 1.1: Calculate \(\frac{2}{3} \div \frac{3}{4}\)
To divide fractions, we multiply by the reciprocal:
\[ \frac{2}{3} \div \frac{3}{4} = \frac{2}{3} \times \frac{4}{3} = \frac{8}{9} \]

#### Sub-step 1.2: Calculate \(\frac{8}{9} \text{ of } \frac{5}{6}\)
The term "of" means to multiply:
\[ \frac{8}{9} \text{ of } \frac{5}{6} = \frac{8}{9} \times \frac{5}{6} = \frac{40}{54} = \frac{20}{27} \text{ (after simplification)} \]

#### Sub-step 1.3: Calculate \(\frac{2}{3} \times \frac{3}{4} \times \frac{5}{6}\)
\[ \frac{2}{3} \times \frac{3}{4} \times \frac{5}{6} = \frac{2 \times 3 \times 5}{3 \times 4 \times 6} = \frac{30}{72} = \frac{5}{12} \text{ (after simplification)} \]

#### Sub-step 1.4: Combine the results
Now we have:
\[ \frac{\frac{20}{27}}{\frac{5}{12}} = \frac{20}{27} \times \frac{12}{5} = \frac{240}{135} = \frac{16}{9} \text{ (after simplification)} \]

### Step 2: Simplify the second part of the expression
The second part is:
\[ \frac{(2 + 2 \times 2)}{\left(\frac{2}{2} \times 2\right)} \div \left(\frac{1}{2} \div \frac{1}{2} \text{ of } \frac{1}{2}\right) \div \left(\frac{1}{2} + \frac{1}{2} \text{ of } \frac{1}{2}\right) \]

#### Sub-step 2.1: Calculate \(2 + 2 \times 2\)
Following BODMAS:
\[ 2 + 2 \times 2 = 2 + 4 = 6 \]

#### Sub-step 2.2: Calculate \(\frac{2}{2} \times 2\)
\[ \frac{2}{2} \times 2 = 1 \times 2 = 2 \]

#### Sub-step 2.3: Combine the results
Now we have:
\[ \frac{6}{2} = 3 \]

#### Sub-step 2.4: Calculate \(\frac{1}{2} \div \frac{1}{2} \text{ of } \frac{1}{2}\)
First, calculate \(\frac{1}{2} \div \frac{1}{2} = 1\) and then:
\[ 1 \text{ of } \frac{1}{2} = \frac{1}{2} \]

#### Sub-step 2.5: Calculate \(\frac{1}{2} + \frac{1}{2} \text{ of } \frac{1}{2}\)
\[ \frac{1}{2} \text{ of } \frac{1}{2} = \frac{1}{4} \]
So:
\[ \frac{1}{2} + \frac{1}{4} = \frac{2}{4} + \frac{1}{4} = \frac{3}{4} \]

#### Sub-step 2.6: Combine the results
Now we have:
\[ 3 \div \frac{1}{2} = 3 \times 2 = 6 \]
And:
\[ 6 \div \frac{3}{4} = 6 \times \frac{4}{3} = 8 \]

### Step 3: Combine both parts
Now we combine the results from Step 1 and Step 2:
\[ \frac{16}{9} + 8 \]
Convert 8 to a fraction:
\[ 8 = \frac{72}{9} \]
So:
\[ \frac{16}{9} + \frac{72}{9} = \frac{88}{9} \]

### Final Result
The simplified expression is:
\[ \frac{88}{9} \]