2`3/7`-2`1/4`-1`1/14`+1`1/28`=?

Options

A. `1/7`
B. 1`1/7`
C. 1`1/14`
D. `2/7`

Answer

To solve the expression \( 2\frac{3}{7} - 2\frac{1}{4} - 1\frac{1}{14} + 1\frac{1}{28} \), we will first convert the mixed fractions into improper fractions and then perform the arithmetic operations step by step.

### Step 1: Convert Mixed Fractions to Improper Fractions

1. **Convert \( 2\frac{3}{7} \)**:
\[ 2\frac{3}{7} = \frac{(2 \times 7) + 3}{7} = \frac{14 + 3}{7} = \frac{17}{7} \]

2. **Convert \( 2\frac{1}{4} \)**:
\[ 2\frac{1}{4} = \frac{(2 \times 4) + 1}{4} = \frac{8 + 1}{4} = \frac{9}{4} \]

3. **Convert \( 1\frac{1}{14} \)**:
\[ 1\frac{1}{14} = \frac{(1 \times 14) + 1}{14} = \frac{14 + 1}{14} = \frac{15}{14} \]

4. **Convert \( 1\frac{1}{28} \)**:
\[ 1\frac{1}{28} = \frac{(1 \times 28) + 1}{28} = \frac{28 + 1}{28} = \frac{29}{28} \]

Now, we rewrite the expression with the improper fractions:
\[ \frac{17}{7} - \frac{9}{4} - \frac{15}{14} + \frac{29}{28} \]

### Step 2: Find a Common Denominator

The denominators are \( 7, 4, 14, \) and \( 28 \). The least common multiple (LCM) of these numbers is \( 28 \).

### Step 3: Rewrite Each Fraction with the Common Denominator

1. **Convert \( \frac{17}{7} \)**:
\[ \frac{17}{7} = \frac{17 \times 4}{7 \times 4} = \frac{68}{28} \]

2. **Convert \( \frac{9}{4} \)**:
\[ \frac{9}{4} = \frac{9 \times 7}{4 \times 7} = \frac{63}{28} \]

3. **Convert \( \frac{15}{14} \)**:
\[ \frac{15}{14} = \frac{15 \times 2}{14 \times 2} = \frac{30}{28} \]

4. **Convert \( \frac{29}{28} \)**:
\[ \frac{29}{28} = \frac{29}{28} \quad \text{(already has the common denominator)} \]

Now, the expression becomes:
\[ \frac{68}{28} - \frac{63}{28} - \frac{30}{28} + \frac{29}{28} \]

### Step 4: Combine the Fractions

Combine the numerators:
\[ \frac{68 - 63 - 30 + 29}{28} = \frac{68 - 63 = 5}{28} \quad \text{then} \quad 5 - 30 = -25 \quad \text{and} \quad -25 + 29 = 4 \]

So, we have:
\[ \frac{4}{28} \]

### Step 5: Simplify the Fraction

Now, simplify \( \frac{4}{28} \):
\[ \frac{4 \div 4}{28 \div 4} = \frac{1}{7} \]

### Final Answer
Thus, the final answer is:
\[ \frac{1}{7} \]