What should come in the place of question mark (?) in the following questions ?
`1(1)/(4) +1 (5)/(9) xx 1(5)/(8) + 6(1)/(2)`= ?

Options

A. 17
B. 27
C. 42
D. None of these

Answer

To solve the equation \( 1\frac{1}{4} + 1\frac{5}{9} \times 1\frac{5}{8} + 6\frac{1}{2} \), we will first convert all mixed numbers to improper fractions and then perform the operations step by step.

### Step 1: Convert mixed numbers to improper fractions
1. \( 1\frac{1}{4} = \frac{4 \times 1 + 1}{4} = \frac{5}{4} \)
2. \( 1\frac{5}{9} = \frac{9 \times 1 + 5}{9} = \frac{14}{9} \)
3. \( 1\frac{5}{8} = \frac{8 \times 1 + 5}{8} = \frac{13}{8} \)
4. \( 6\frac{1}{2} = \frac{2 \times 6 + 1}{2} = \frac{13}{2} \)

Now the equation becomes:
\[ \frac{5}{4} + \left(\frac{14}{9} \times \frac{13}{8}\right) + \frac{13}{2} \]

### Step 2: Calculate \( \frac{14}{9} \times \frac{13}{8} \)
To multiply fractions, multiply the numerators and the denominators:
\[ \frac{14 \times 13}{9 \times 8} = \frac{182}{72} \]

### Step 3: Simplify \( \frac{182}{72} \)
Find the GCD of 182 and 72 to simplify:
- The GCD is 2.
\[ \frac{182 \div 2}{72 \div 2} = \frac{91}{36} \]

### Step 4: Now substitute back into the equation
The equation now looks like:
\[ \frac{5}{4} + \frac{91}{36} + \frac{13}{2} \]

### Step 5: Convert all fractions to have a common denominator
The least common multiple (LCM) of 4, 36, and 2 is 36. Convert each fraction:
1. \( \frac{5}{4} = \frac{5 \times 9}{4 \times 9} = \frac{45}{36} \)
2. \( \frac{91}{36} \) remains the same.
3. \( \frac{13}{2} = \frac{13 \times 18}{2 \times 18} = \frac{234}{36} \)

### Step 6: Add the fractions
Now add:
\[ \frac{45}{36} + \frac{91}{36} + \frac{234}{36} = \frac{45 + 91 + 234}{36} = \frac{370}{36} \]

### Step 7: Simplify \( \frac{370}{36} \)
Find the GCD of 370 and 36:
- The GCD is 2.
\[ \frac{370 \div 2}{36 \div 2} = \frac{185}{18} \]

### Step 8: Convert to mixed number if necessary
To convert \( \frac{185}{18} \) to a mixed number:
- \( 185 \div 18 = 10 \) remainder \( 5 \)
So, \( \frac{185}{18} = 10\frac{5}{18} \).

### Final Answer
Thus, the answer is \( 10\frac{5}{18} \).