Which of the fraction is largest among `(2)/( 5) , ( 5)/( 11), ( 8)/( 17), ( 11)/( 23) ` ?

Answer

To determine which of the fractions \( \frac{2}{5}, \frac{5}{11}, \frac{8}{17}, \frac{11}{23} \) is the largest, we can follow these steps:

### Step 1: Find a common denominator
To compare the fractions, we need a common denominator. The denominators are 5, 11, 17, and 23. Since these numbers are coprime (no common factors other than 1), we can find the least common multiple (LCM) by multiplying them together.

\[ \text{LCM} = 5 \times 11 \times 17 \times 23 \]

### Step 2: Calculate the LCM
Calculating the LCM step-by-step:

1. \( 5 \times 11 = 55 \)
2. \( 55 \times 17 = 935 \)
3. \( 935 \times 23 = 21405 \)

So, the common denominator is \( 21405 \).

### Step 3: Convert each fraction to have the common denominator
Now we will convert each fraction to have the denominator of \( 21405 \):

1. For \( \frac{2}{5} \):
\[ \frac{2}{5} = \frac{2 \times 4281}{5 \times 4281} = \frac{8562}{21405} \]

2. For \( \frac{5}{11} \):
\[ \frac{5}{11} = \frac{5 \times 1945}{11 \times 1945} = \frac{9725}{21405} \]

3. For \( \frac{8}{17} \):
\[ \frac{8}{17} = \frac{8 \times 1255}{17 \times 1255} = \frac{10040}{21405} \]

4. For \( \frac{11}{23} \):
\[ \frac{11}{23} = \frac{11 \times 930}{23 \times 930} = \frac{10230}{21405} \]

### Step 4: Compare the numerators
Now we have the fractions:

- \( \frac{8562}{21405} \)
- \( \frac{9725}{21405} \)
- \( \frac{10040}{21405} \)
- \( \frac{10230}{21405} \)

Now we can compare the numerators:

- \( 8562 < 9725 < 10040 < 10230 \)

### Conclusion
The largest fraction is \( \frac{11}{23} \).