Evaluate: f. `12 (1)/(2) -3 (9)/(14) + 4 (5)/(7)`

Answer

To evaluate the expression \( 12 \frac{1}{2} - 3 \frac{9}{14} + 4 \frac{5}{7} \), we will follow these steps:

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

1. \( 12 \frac{1}{2} = \frac{12 \times 2 + 1}{2} = \frac{24 + 1}{2} = \frac{25}{2} \)
2. \( 3 \frac{9}{14} = \frac{3 \times 14 + 9}{14} = \frac{42 + 9}{14} = \frac{51}{14} \)
3. \( 4 \frac{5}{7} = \frac{4 \times 7 + 5}{7} = \frac{28 + 5}{7} = \frac{33}{7} \)

Now our expression looks like this:
\[ \frac{25}{2} - \frac{51}{14} + \frac{33}{7} \]

### Step 2: Find a Common Denominator
The denominators are 2, 14, and 7. The least common multiple (LCM) of these numbers is 14.

### Step 3: Convert Each Fraction to Have the Common Denominator
Now we convert each fraction to have a denominator of 14.

1. For \( \frac{25}{2} \):
\[ \frac{25}{2} = \frac{25 \times 7}{2 \times 7} = \frac{175}{14} \]

2. For \( \frac{51}{14} \), it remains the same:
\[ \frac{51}{14} = \frac{51}{14} \]

3. For \( \frac{33}{7} \):
\[ \frac{33}{7} = \frac{33 \times 2}{7 \times 2} = \frac{66}{14} \]

Now our expression is:
\[ \frac{175}{14} - \frac{51}{14} + \frac{66}{14} \]

### Step 4: Combine the Fractions
Now we can combine the fractions since they have the same denominator:
\[ \frac{175 - 51 + 66}{14} \]

Calculating the numerator:
\[ 175 - 51 = 124 \]
\[ 124 + 66 = 190 \]

So we have:
\[ \frac{190}{14} \]

### Step 5: Simplify the Fraction
Now we simplify \( \frac{190}{14} \):
Both 190 and 14 can be divided by 2:
\[ \frac{190 \div 2}{14 \div 2} = \frac{95}{7} \]

### Final Answer
Thus, the final answer is:
\[ \frac{95}{7} \]

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