Work out the following :
`1""4/9+3""5/12`

Answer

To solve the problem \(1 \frac{4}{9} + 3 \frac{5}{12}\), we will follow these steps:

### Step 1: Convert Mixed Fractions to Improper Fractions
First, we need to convert the mixed fractions into improper fractions.

For \(1 \frac{4}{9}\):
- Multiply the whole number (1) by the denominator (9): \(1 \times 9 = 9\)
- Add the numerator (4): \(9 + 4 = 13\)
- Therefore, \(1 \frac{4}{9} = \frac{13}{9}\)

For \(3 \frac{5}{12}\):
- Multiply the whole number (3) by the denominator (12): \(3 \times 12 = 36\)
- Add the numerator (5): \(36 + 5 = 41\)
- Therefore, \(3 \frac{5}{12} = \frac{41}{12}\)

### Step 2: Add the Improper Fractions
Now we add the two improper fractions:
\[ \frac{13}{9} + \frac{41}{12} \]

### Step 3: Find the Least Common Multiple (LCM) of the Denominators
The denominators are 9 and 12. We need to find the LCM of these two numbers.

- The prime factorization of 9 is \(3^2\).
- The prime factorization of 12 is \(2^2 \times 3\).
- The LCM is found by taking the highest power of each prime factor:
- For 2: \(2^2\)
- For 3: \(3^2\)

Thus, the LCM of 9 and 12 is:
\[ LCM = 2^2 \times 3^2 = 4 \times 9 = 36 \]

### Step 4: Convert Each Fraction to Have the Same Denominator
Now we convert each fraction to have the common denominator of 36.

For \(\frac{13}{9}\):
- Multiply the numerator and denominator by 4:
\[ \frac{13 \times 4}{9 \times 4} = \frac{52}{36} \]

For \(\frac{41}{12}\):
- Multiply the numerator and denominator by 3:
\[ \frac{41 \times 3}{12 \times 3} = \frac{123}{36} \]

### Step 5: Add the Fractions
Now that we have a common denominator, we can add the fractions:
\[ \frac{52}{36} + \frac{123}{36} = \frac{52 + 123}{36} = \frac{175}{36} \]

### Step 6: Convert Back to a Mixed Fraction
Finally, we convert \(\frac{175}{36}\) back to a mixed fraction:
- Divide 175 by 36, which gives us 4 with a remainder:
\[ 175 \div 36 = 4 \quad \text{(whole number)} \]
- The remainder is:
\[ 175 - (36 \times 4) = 175 - 144 = 31 \]
Thus, \(\frac{175}{36} = 4 \frac{31}{36}\).

### Final Answer
The final answer is:
\[ 1 \frac{4}{9} + 3 \frac{5}{12} = 4 \frac{31}{36} \]

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