Which fraction is largest among `(3)/(13), ( 2)/( 15), ( 4)/( 17)` ?

Answer

To determine which fraction is the largest among \( \frac{3}{13}, \frac{2}{15}, \frac{4}{17} \), we can follow these steps:

### Step 1: Identify the fractions
We have three fractions:
1. \( \frac{3}{13} \)
2. \( \frac{2}{15} \)
3. \( \frac{4}{17} \)

### Step 2: Find the Least Common Multiple (LCM) of the denominators
The denominators are 13, 15, and 17. Since these numbers are co-prime (no common factors other than 1), the LCM is simply the product:
\[ \text{LCM} = 13 \times 15 \times 17 \]

Calculating this gives:
\[ 13 \times 15 = 195 \]
\[ 195 \times 17 = 3315 \]
So, the LCM is 3315.

### Step 3: Convert each fraction to have the same denominator
Now we will convert each fraction to have the denominator of 3315.

#### For \( \frac{3}{13} \):
\[ \frac{3}{13} = \frac{3 \times (15 \times 17)}{13 \times (15 \times 17)} = \frac{3 \times 255}{3315} = \frac{765}{3315} \]

#### For \( \frac{2}{15} \):
\[ \frac{2}{15} = \frac{2 \times (13 \times 17)}{15 \times (13 \times 17)} = \frac{2 \times 221}{3315} = \frac{442}{3315} \]

#### For \( \frac{4}{17} \):
\[ \frac{4}{17} = \frac{4 \times (13 \times 15)}{17 \times (13 \times 15)} = \frac{4 \times 195}{3315} = \frac{780}{3315} \]

### Step 4: Compare the numerators
Now we compare the numerators of the fractions:
- \( 765 \) from \( \frac{3}{13} \)
- \( 442 \) from \( \frac{2}{15} \)
- \( 780 \) from \( \frac{4}{17} \)

### Step 5: Identify the largest fraction
The largest numerator is \( 780 \), which corresponds to the fraction \( \frac{4}{17} \).

### Conclusion
Thus, the largest fraction among \( \frac{3}{13}, \frac{2}{15}, \frac{4}{17} \) is:
\[ \frac{4}{17} \]

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