What well Centre in place of the question mark (?) in the following questions?
`1 (1)/(5) - 1 (1)/(10) + 1(1)/(20)=?`

Options

A. `1 (1)/(54)`
B. `1(3)/(20)`
C. `1(1)/(20)`
D. `1(5)/(54)`

Answer

To solve the expression \( 1 \frac{1}{5} - 1 \frac{1}{10} + 1 \frac{1}{20} = ? \), we will first convert the mixed numbers into improper fractions and then perform the operations step by step.

### Step 1: Convert mixed numbers to improper fractions
1. For \( 1 \frac{1}{5} \):
\[ 1 \frac{1}{5} = \frac{5 \times 1 + 1}{5} = \frac{6}{5} \]

2. For \( 1 \frac{1}{10} \):
\[ 1 \frac{1}{10} = \frac{10 \times 1 + 1}{10} = \frac{11}{10} \]

3. For \( 1 \frac{1}{20} \):
\[ 1 \frac{1}{20} = \frac{20 \times 1 + 1}{20} = \frac{21}{20} \]

### Step 2: Substitute the improper fractions into the expression
Now we can rewrite the expression:
\[ \frac{6}{5} - \frac{11}{10} + \frac{21}{20} \]

### Step 3: Find the least common multiple (LCM) of the denominators
The denominators are 5, 10, and 20. The LCM of these numbers is 20.

### Step 4: Convert each fraction to have the common denominator of 20
1. Convert \( \frac{6}{5} \):
\[ \frac{6}{5} = \frac{6 \times 4}{5 \times 4} = \frac{24}{20} \]

2. Convert \( \frac{11}{10} \):
\[ \frac{11}{10} = \frac{11 \times 2}{10 \times 2} = \frac{22}{20} \]

3. \( \frac{21}{20} \) is already in terms of the common denominator.

### Step 5: Substitute back into the expression
Now we can rewrite the expression:
\[ \frac{24}{20} - \frac{22}{20} + \frac{21}{20} \]

### Step 6: Combine the fractions
Combine the numerators over the common denominator:
\[ \frac{24 - 22 + 21}{20} = \frac{24 - 22 + 21}{20} = \frac{2 + 21}{20} = \frac{23}{20} \]

### Step 7: Convert the improper fraction to a mixed number
Convert \( \frac{23}{20} \) to a mixed number:
\[ \frac{23}{20} = 1 \frac{3}{20} \]

### Final Answer
Thus, the value of the question mark (?) is:
\[ 1 \frac{3}{20} \]

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