Evaluate: c. `2 (3)/(14) + 5 (2)/(7) - 4 (3)/(4)`

Answer

To evaluate the expression \( 2 \frac{3}{14} + 5 \frac{2}{7} - 4 \frac{3}{4} \), we will follow these steps:

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

1. **For \( 2 \frac{3}{14} \)**:
\[ 2 \frac{3}{14} = \frac{(2 \times 14) + 3}{14} = \frac{28 + 3}{14} = \frac{31}{14} \]

2. **For \( 5 \frac{2}{7} \)**:
\[ 5 \frac{2}{7} = \frac{(5 \times 7) + 2}{7} = \frac{35 + 2}{7} = \frac{37}{7} \]

3. **For \( 4 \frac{3}{4} \)**:
\[ 4 \frac{3}{4} = \frac{(4 \times 4) + 3}{4} = \frac{16 + 3}{4} = \frac{19}{4} \]

Now, we rewrite the expression:
\[ \frac{31}{14} + \frac{37}{7} - \frac{19}{4} \]

### Step 2: Find the LCM of the Denominators
Next, we need to find the least common multiple (LCM) of the denominators \( 14, 7, \) and \( 4 \).

- The prime factorization of \( 14 = 2 \times 7 \)
- The prime factorization of \( 7 = 7 \)
- The prime factorization of \( 4 = 2^2 \)

The LCM will be \( 28 \) (since it includes the highest powers of all prime factors).

### Step 3: Convert Each Fraction to Have the Same Denominator
Now we will convert each fraction to have the denominator of \( 28 \).

1. **Convert \( \frac{31}{14} \)**:
\[ \frac{31}{14} = \frac{31 \times 2}{14 \times 2} = \frac{62}{28} \]

2. **Convert \( \frac{37}{7} \)**:
\[ \frac{37}{7} = \frac{37 \times 4}{7 \times 4} = \frac{148}{28} \]

3. **Convert \( \frac{19}{4} \)**:
\[ \frac{19}{4} = \frac{19 \times 7}{4 \times 7} = \frac{133}{28} \]

### Step 4: Combine the Fractions
Now we can combine the fractions:
\[ \frac{62}{28} + \frac{148}{28} - \frac{133}{28} = \frac{62 + 148 - 133}{28} = \frac{77}{28} \]

### Step 5: Simplify the Result
Finally, we simplify \( \frac{77}{28} \):
\[ \frac{77}{28} = 2 \frac{21}{28} = 2 \frac{3}{4} \]

Thus, the final answer is:
\[ \frac{77}{28} = 2 \frac{21}{28} = 2 \frac{3}{4} \]