Evaluate: b. `5 (6)/(7) - 2 (5)/(6) - 1 (2)/(3)`

Answer

To evaluate the expression \(5 \frac{6}{7} - 2 \frac{5}{6} - 1 \frac{2}{3}\), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we need to convert each mixed number into an improper fraction.

1. For \(5 \frac{6}{7}\):
\[ 5 \frac{6}{7} = \frac{5 \times 7 + 6}{7} = \frac{35 + 6}{7} = \frac{41}{7} \]

2. For \(2 \frac{5}{6}\):
\[ 2 \frac{5}{6} = \frac{2 \times 6 + 5}{6} = \frac{12 + 5}{6} = \frac{17}{6} \]

3. For \(1 \frac{2}{3}\):
\[ 1 \frac{2}{3} = \frac{1 \times 3 + 2}{3} = \frac{3 + 2}{3} = \frac{5}{3} \]

Now we rewrite the expression:
\[ \frac{41}{7} - \frac{17}{6} - \frac{5}{3} \]

### Step 2: Find the Least Common Multiple (LCM)
Next, we need to find the LCM of the denominators \(7\), \(6\), and \(3\).

- The prime factorization of \(7\) is \(7^1\).
- The prime factorization of \(6\) is \(2^1 \times 3^1\).
- The prime factorization of \(3\) is \(3^1\).

The LCM is calculated as follows:
\[ \text{LCM} = 7^1 \times 2^1 \times 3^1 = 42 \]

### Step 3: Convert Each Fraction to Have the Same Denominator
Now we convert each fraction to have the denominator of \(42\).

1. For \(\frac{41}{7}\):
\[ \frac{41}{7} = \frac{41 \times 6}{7 \times 6} = \frac{246}{42} \]

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

3. For \(\frac{5}{3}\):
\[ \frac{5}{3} = \frac{5 \times 14}{3 \times 14} = \frac{70}{42} \]

### Step 4: Combine the Fractions
Now we can combine the fractions:
\[ \frac{246}{42} - \frac{119}{42} - \frac{70}{42} = \frac{246 - 119 - 70}{42} \]

Calculating the numerator:
\[ 246 - 119 = 127 \]
\[ 127 - 70 = 57 \]

So we have:
\[ \frac{57}{42} \]

### Step 5: Simplify the Fraction
Now we simplify \(\frac{57}{42}\):
- The greatest common divisor (GCD) of \(57\) and \(42\) is \(3\).
- Dividing both the numerator and the denominator by \(3\):
\[ \frac{57 \div 3}{42 \div 3} = \frac{19}{14} \]

### Final Answer
Thus, the final answer is:
\[ \frac{19}{14} \text{ or } 1 \frac{5}{14} \]

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