Which is the smallest fraction among the following ?

Options

A. `(11)/(13)`
B. `(17)/(19)`
C. `(19)/(24)`
D. `(21)/(25)`

Answer

To determine the smallest fraction among the given options, we will follow these steps:

### Step 1: Identify the fractions
Let's say we have the following fractions:
1. \( \frac{11}{13} \)
2. \( \frac{17}{19} \)
3. \( \frac{19}{24} \)
4. \( \frac{21}{25} \)

### Step 2: Find the LCM of the denominators
To compare the fractions easily, we need to find the Least Common Multiple (LCM) of the denominators (13, 19, 24, and 25).

- The prime factorization of each number is:
- 13: \( 13^1 \)
- 19: \( 19^1 \)
- 24: \( 2^3 \times 3^1 \)
- 25: \( 5^2 \)

- The LCM is obtained by taking the highest power of each prime:
- LCM = \( 13^1 \times 19^1 \times 2^3 \times 3^1 \times 5^2 = 148200 \)

### Step 3: Convert each fraction to have the same denominator
Now we will convert each fraction to have a denominator of 148200.

1. For \( \frac{11}{13} \):
\[ \frac{11}{13} = \frac{11 \times 11400}{13 \times 11400} = \frac{125400}{148200} \]

2. For \( \frac{17}{19} \):
\[ \frac{17}{19} = \frac{17 \times 7800}{19 \times 7800} = \frac{132600}{148200} \]

3. For \( \frac{19}{24} \):
\[ \frac{19}{24} = \frac{19 \times 6175}{24 \times 6175} = \frac{117325}{148200} \]

4. For \( \frac{21}{25} \):
\[ \frac{21}{25} = \frac{21 \times 5928}{25 \times 5928} = \frac{124488}{148200} \]

### Step 4: Compare the numerators
Now we compare the numerators of the fractions:
- \( 125400 \) (from \( \frac{11}{13} \))
- \( 132600 \) (from \( \frac{17}{19} \))
- \( 117325 \) (from \( \frac{19}{24} \))
- \( 124488 \) (from \( \frac{21}{25} \))

The smallest numerator is \( 117325 \), which corresponds to the fraction \( \frac{19}{24} \).

### Conclusion
Thus, the smallest fraction among the given options is:
\[ \frac{19}{24} \]

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