For every question, four probable answers bear in letters (A), (B), (C) and (D) are given. Only one out of these is correct. You have to choose the correct answer.
Simplified form of expression
`12-{5/3 + (3 7/9 - 2 1/3 + 7 1/9)}` is

Options

A. `9/26`
B. `7/26`
C. `26/9`
D. `16/9`

Answer

To simplify the expression \( 12 - \left\{ \frac{5}{3} + \left( 3 \frac{7}{9} - 2 \frac{1}{3} + 7 \frac{1}{9} \right) \right\} \), we will follow these steps:

### Step 1: Simplify the expression inside the parentheses
We need to simplify \( 3 \frac{7}{9} - 2 \frac{1}{3} + 7 \frac{1}{9} \).

1. Convert mixed numbers to improper fractions:
- \( 3 \frac{7}{9} = \frac{3 \times 9 + 7}{9} = \frac{27 + 7}{9} = \frac{34}{9} \)
- \( 2 \frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{6 + 1}{3} = \frac{7}{3} \)
- \( 7 \frac{1}{9} = \frac{7 \times 9 + 1}{9} = \frac{63 + 1}{9} = \frac{64}{9} \)

2. Now substitute these values into the expression:
\[ \frac{34}{9} - \frac{7}{3} + \frac{64}{9} \]

3. To perform the subtraction and addition, we need a common denominator. The LCM of 9 and 3 is 9. Convert \( \frac{7}{3} \) to have a denominator of 9:
\[ \frac{7}{3} = \frac{7 \times 3}{3 \times 3} = \frac{21}{9} \]

4. Now substitute back:
\[ \frac{34}{9} - \frac{21}{9} + \frac{64}{9} = \frac{34 - 21 + 64}{9} = \frac{77}{9} \]

### Step 2: Substitute back into the original expression
Now we substitute \( \frac{77}{9} \) back into the expression:
\[ 12 - \left\{ \frac{5}{3} + \frac{77}{9} \right\} \]

### Step 3: Simplify the curly bracket expression
1. Convert \( \frac{5}{3} \) to have a denominator of 9:
\[ \frac{5}{3} = \frac{5 \times 3}{3 \times 3} = \frac{15}{9} \]

2. Now substitute back:
\[ \frac{15}{9} + \frac{77}{9} = \frac{15 + 77}{9} = \frac{92}{9} \]

### Step 4: Final calculation
Now we have:
\[ 12 - \frac{92}{9} \]

1. Convert 12 to a fraction with a denominator of 9:
\[ 12 = \frac{12 \times 9}{1 \times 9} = \frac{108}{9} \]

2. Now perform the subtraction:
\[ \frac{108}{9} - \frac{92}{9} = \frac{108 - 92}{9} = \frac{16}{9} \]

### Final Answer
The simplified form of the expression is \( \frac{16}{9} \).

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