Find:
`3(1)/(8)+1(5)/(12)`

Options

A. ` 4(13)/(24)`
B. ` 4(15)/(24)`
C. ` 4(17)/(24)`
D. ` 4(19)/(24)`

Answer

To solve the problem \( 3\frac{1}{8} + 1\frac{5}{12} \), we will follow these steps:

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

For \( 3\frac{1}{8} \):
- Multiply the whole number (3) by the denominator (8):
\[ 3 \times 8 = 24 \]
- Add the numerator (1) to this product:
\[ 24 + 1 = 25 \]
- So, \( 3\frac{1}{8} \) becomes:
\[ \frac{25}{8} \]

For \( 1\frac{5}{12} \):
- Multiply the whole number (1) by the denominator (12):
\[ 1 \times 12 = 12 \]
- Add the numerator (5) to this product:
\[ 12 + 5 = 17 \]
- So, \( 1\frac{5}{12} \) becomes:
\[ \frac{17}{12} \]

### Step 2: Add the Improper Fractions
Now we need to add \( \frac{25}{8} + \frac{17}{12} \). To do this, we need a common denominator.

### Step 3: Find the Least Common Multiple (LCM)
The denominators are 8 and 12. We need to find the LCM of these two numbers.

- The prime factorization of 8 is \( 2^3 \).
- The prime factorization of 12 is \( 2^2 \times 3^1 \).

The LCM will take the highest power of each prime:
- For 2, the highest power is \( 2^3 \).
- For 3, the highest power is \( 3^1 \).

Thus,
\[ \text{LCM} = 2^3 \times 3^1 = 8 \times 3 = 24 \]

### Step 4: Convert Fractions to Have a Common Denominator
Now we convert both fractions to have the common denominator of 24.

For \( \frac{25}{8} \):
- Multiply the numerator and denominator by 3:
\[ \frac{25 \times 3}{8 \times 3} = \frac{75}{24} \]

For \( \frac{17}{12} \):
- Multiply the numerator and denominator by 2:
\[ \frac{17 \times 2}{12 \times 2} = \frac{34}{24} \]

### Step 5: Add the Converted Fractions
Now we can add the two fractions:
\[ \frac{75}{24} + \frac{34}{24} = \frac{75 + 34}{24} = \frac{109}{24} \]

### Step 6: Simplify the Result
The fraction \( \frac{109}{24} \) is already in its simplest form.

### Final Answer
Thus, the final answer is:
\[ \frac{109}{24} \]

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