`2 1/7+4 3/5-3 1/7+5 1/(10)=?`

Options

A. `9 7/(10)`
B. `7 7/(10)`
C. `8 7/(10)`
D. `8 4/(10)`

Answer

To solve the expression \(2 \frac{1}{7} + 4 \frac{3}{5} - 3 \frac{1}{7} + 5 \frac{1}{10}\), we will first convert all mixed fractions into improper fractions, then perform the arithmetic operations step by step.

### Step 1: Convert Mixed Fractions to Improper Fractions
1. **Convert \(2 \frac{1}{7}\)**:
\[ 2 \frac{1}{7} = \frac{2 \times 7 + 1}{7} = \frac{14 + 1}{7} = \frac{15}{7} \]

2. **Convert \(4 \frac{3}{5}\)**:
\[ 4 \frac{3}{5} = \frac{4 \times 5 + 3}{5} = \frac{20 + 3}{5} = \frac{23}{5} \]

3. **Convert \(3 \frac{1}{7}\)**:
\[ 3 \frac{1}{7} = \frac{3 \times 7 + 1}{7} = \frac{21 + 1}{7} = \frac{22}{7} \]

4. **Convert \(5 \frac{1}{10}\)**:
\[ 5 \frac{1}{10} = \frac{5 \times 10 + 1}{10} = \frac{50 + 1}{10} = \frac{51}{10} \]

### Step 2: Substitute Back into the Expression
Now we can rewrite the original expression:
\[ \frac{15}{7} + \frac{23}{5} - \frac{22}{7} + \frac{51}{10} \]

### Step 3: Find a Common Denominator
The denominators are 7, 5, and 10. The least common multiple (LCM) of these numbers is 70.

### Step 4: Convert Each Fraction to Have the Common Denominator
1. **Convert \(\frac{15}{7}\)**:
\[ \frac{15}{7} = \frac{15 \times 10}{7 \times 10} = \frac{150}{70} \]

2. **Convert \(\frac{23}{5}\)**:
\[ \frac{23}{5} = \frac{23 \times 14}{5 \times 14} = \frac{322}{70} \]

3. **Convert \(\frac{22}{7}\)**:
\[ \frac{22}{7} = \frac{22 \times 10}{7 \times 10} = \frac{220}{70} \]

4. **Convert \(\frac{51}{10}\)**:
\[ \frac{51}{10} = \frac{51 \times 7}{10 \times 7} = \frac{357}{70} \]

### Step 5: Combine the Fractions
Now we can combine all the fractions:
\[ \frac{150}{70} + \frac{322}{70} - \frac{220}{70} + \frac{357}{70} \]
Combine the numerators:
\[ = \frac{150 + 322 - 220 + 357}{70} = \frac{609}{70} \]

### Step 6: Convert Back to Mixed Fraction
To convert \(\frac{609}{70}\) back to a mixed fraction:
1. Divide 609 by 70:
- \(609 \div 70 = 8\) remainder \(49\)
- Thus, \(\frac{609}{70} = 8 \frac{49}{70}\)

### Final Answer
The final answer is:
\[ 8 \frac{49}{70} \]
This corresponds to the option \(8 \frac{7}{10}\) since \(\frac{49}{70} = \frac{7}{10}\).