What will come in place of question mark (?) in the following questions ?
`2(1)/(4) - 13(4)/(5) + 22(1)/(4) = ?`

Options

A. `10(8)/(9)`
B. `10(7)/(10)`
C. `10(6)/(10)`
D. None of these

Answer

To solve the equation \(2 \frac{1}{4} - 13 \frac{4}{5} + 22 \frac{1}{4} = ?\), we will first convert the mixed numbers into improper fractions, perform the operations, and then simplify the result.

### Step 1: Convert mixed numbers to improper fractions
1. Convert \(2 \frac{1}{4}\):
\[ 2 \frac{1}{4} = \frac{2 \times 4 + 1}{4} = \frac{8 + 1}{4} = \frac{9}{4} \]

2. Convert \(13 \frac{4}{5}\):
\[ 13 \frac{4}{5} = \frac{13 \times 5 + 4}{5} = \frac{65 + 4}{5} = \frac{69}{5} \]

3. Convert \(22 \frac{1}{4}\):
\[ 22 \frac{1}{4} = \frac{22 \times 4 + 1}{4} = \frac{88 + 1}{4} = \frac{89}{4} \]

### Step 2: Rewrite the equation with improper fractions
Now we can rewrite the original equation:
\[ \frac{9}{4} - \frac{69}{5} + \frac{89}{4} \]

### Step 3: Combine the fractions
1. Combine \(\frac{9}{4}\) and \(\frac{89}{4}\):
\[ \frac{9 + 89}{4} = \frac{98}{4} \]

2. Now we have:
\[ \frac{98}{4} - \frac{69}{5} \]

### Step 4: Find a common denominator
The least common multiple (LCM) of 4 and 5 is 20. We will convert both fractions to have a denominator of 20:
1. Convert \(\frac{98}{4}\):
\[ \frac{98}{4} = \frac{98 \times 5}{4 \times 5} = \frac{490}{20} \]

2. Convert \(\frac{69}{5}\):
\[ \frac{69}{5} = \frac{69 \times 4}{5 \times 4} = \frac{276}{20} \]

### Step 5: Perform the subtraction
Now we can subtract:
\[ \frac{490}{20} - \frac{276}{20} = \frac{490 - 276}{20} = \frac{214}{20} \]

### Step 6: Simplify the fraction
To simplify \(\frac{214}{20}\):
1. Find the greatest common divisor (GCD) of 214 and 20, which is 2.
2. Divide both the numerator and the denominator by 2:
\[ \frac{214 \div 2}{20 \div 2} = \frac{107}{10} \]

### Final Answer
Thus, the answer to the equation is:
\[ \frac{107}{10} \text{ or } 10 \frac{7}{10} \]