Simplify the following :
`1""13/14-2""5/6+1""6/7`

Answer

To simplify the expression \(1 \frac{13}{14} - 2 \frac{5}{6} + 1 \frac{6}{7}\), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
1. Convert \(1 \frac{13}{14}\) to an improper fraction:
\[ 1 \frac{13}{14} = \frac{1 \times 14 + 13}{14} = \frac{14 + 13}{14} = \frac{27}{14} \]

2. Convert \(2 \frac{5}{6}\) to an improper fraction:
\[ 2 \frac{5}{6} = \frac{2 \times 6 + 5}{6} = \frac{12 + 5}{6} = \frac{17}{6} \]

3. Convert \(1 \frac{6}{7}\) to an improper fraction:
\[ 1 \frac{6}{7} = \frac{1 \times 7 + 6}{7} = \frac{7 + 6}{7} = \frac{13}{7} \]

### Step 2: Rewrite the Expression
Now we can rewrite the expression using the improper fractions:
\[ \frac{27}{14} - \frac{17}{6} + \frac{13}{7} \]

### Step 3: Find the Least Common Multiple (LCM)
To perform the addition and subtraction, we need a common denominator. The denominators are \(14\), \(6\), and \(7\).

1. The prime factorization of each number:
- \(14 = 2 \times 7\)
- \(6 = 2 \times 3\)
- \(7 = 7\)

2. The LCM is found by taking the highest power of each prime:
\[ \text{LCM} = 2^1 \times 3^1 \times 7^1 = 42 \]

### Step 4: Convert Each Fraction to Have the Common Denominator
1. Convert \(\frac{27}{14}\):
\[ \frac{27}{14} = \frac{27 \times 3}{14 \times 3} = \frac{81}{42} \]

2. Convert \(\frac{17}{6}\):
\[ \frac{17}{6} = \frac{17 \times 7}{6 \times 7} = \frac{119}{42} \]

3. Convert \(\frac{13}{7}\):
\[ \frac{13}{7} = \frac{13 \times 6}{7 \times 6} = \frac{78}{42} \]

### Step 5: Rewrite the Expression with Common Denominator
Now the expression is:
\[ \frac{81}{42} - \frac{119}{42} + \frac{78}{42} \]

### Step 6: Combine the Fractions
Combine the fractions:
\[ \frac{81 - 119 + 78}{42} \]

Calculating the numerator:
\[ 81 - 119 + 78 = 81 + 78 - 119 = 159 - 119 = 40 \]

So, we have:
\[ \frac{40}{42} \]

### Step 7: Simplify the Fraction
Now, we simplify \(\frac{40}{42}\):
\[ \frac{40 \div 2}{42 \div 2} = \frac{20}{21} \]

### Final Answer
The simplified result of the expression \(1 \frac{13}{14} - 2 \frac{5}{6} + 1 \frac{6}{7}\) is:
\[ \frac{20}{21} \]

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