Find the difference:
`4(9)/(12)-3(8)/(14)`

Answer

To solve the problem of finding the difference between the mixed fractions \( 4\frac{9}{12} \) and \( 3\frac{8}{14} \), we will follow these steps:

### Step 1: Convert the mixed fractions to improper fractions
1. For \( 4\frac{9}{12} \):
- Multiply the whole number (4) by the denominator (12): \( 4 \times 12 = 48 \)
- Add the numerator (9): \( 48 + 9 = 57 \)
- So, \( 4\frac{9}{12} = \frac{57}{12} \)

2. For \( 3\frac{8}{14} \):
- Multiply the whole number (3) by the denominator (14): \( 3 \times 14 = 42 \)
- Add the numerator (8): \( 42 + 8 = 50 \)
- So, \( 3\frac{8}{14} = \frac{50}{14} \)

**Result**: We have \( \frac{57}{12} - \frac{50}{14} \).

### Step 2: Find the least common multiple (LCM) of the denominators
- The denominators are 12 and 14.
- The prime factorization of 12 is \( 2^2 \times 3 \) and for 14 is \( 2 \times 7 \).
- The LCM is found by taking the highest power of each prime factor:
- \( 2^2 \) from 12
- \( 3^1 \) from 12
- \( 7^1 \) from 14
- Therefore, \( \text{LCM}(12, 14) = 2^2 \times 3 \times 7 = 84 \).

**Result**: The LCM of 12 and 14 is 84.

### Step 3: Convert the fractions to have a common denominator
1. For \( \frac{57}{12} \):
- Multiply the numerator and denominator by \( \frac{84}{12} = 7 \):
- \( \frac{57 \times 7}{12 \times 7} = \frac{399}{84} \)

2. For \( \frac{50}{14} \):
- Multiply the numerator and denominator by \( \frac{84}{14} = 6 \):
- \( \frac{50 \times 6}{14 \times 6} = \frac{300}{84} \)

**Result**: Now we have \( \frac{399}{84} - \frac{300}{84} \).

### Step 4: Subtract the fractions
- Since both fractions have the same denominator, we can subtract the numerators:
- \( \frac{399 - 300}{84} = \frac{99}{84} \)

**Result**: The difference is \( \frac{99}{84} \).

### Step 5: Simplify the fraction if possible
- The fraction \( \frac{99}{84} \) can be simplified:
- The greatest common divisor (GCD) of 99 and 84 is 3.
- Divide both the numerator and denominator by 3:
- \( \frac{99 \div 3}{84 \div 3} = \frac{33}{28} \)

**Result**: The simplified difference is \( \frac{33}{28} \).

### Step 6: Convert to a mixed number (if needed)
- To convert \( \frac{33}{28} \) to a mixed number:
- Divide 33 by 28, which gives 1 with a remainder of 5.
- So, \( \frac{33}{28} = 1\frac{5}{28} \).

**Final Result**: The difference is \( 1\frac{5}{28} \).