Out of the fractions, `( 4)/( 7) , ( 5)/( 13), ( 6)/( 11), ( 3)/( 5)` and `(2)/( 3)`, which is the second smallest fraction?

Options

A. `(4)/(7)`
B. `( 5)/( 13)`
C. `( 6)/( 11)`
D. `( 3)/( 5)`

Answer

To find the second smallest fraction among the given fractions \( \frac{4}{7}, \frac{5}{13}, \frac{6}{11}, \frac{3}{5}, \frac{2}{3} \), we can follow these steps:

### Step 1: List the Fractions
First, we write down the fractions we need to compare:
- \( \frac{4}{7} \)
- \( \frac{5}{13} \)
- \( \frac{6}{11} \)
- \( \frac{3}{5} \)
- \( \frac{2}{3} \)

### Step 2: Find the Least Common Multiple (LCM) of the Denominators
The denominators are 7, 13, 11, 5, and 3. To compare these fractions easily, we need to find the least common multiple (LCM) of these numbers.

- The LCM of 7, 13, 11, 5, and 3 is calculated by multiplying these prime numbers together:
\[ \text{LCM} = 7 \times 13 \times 11 \times 5 \times 3 = 15015 \]

### Step 3: Convert Each Fraction to Have the Same Denominator
Now, we will convert each fraction to have the common denominator of 15015.

1. For \( \frac{4}{7} \):
\[ \frac{4}{7} = \frac{4 \times 2145}{7 \times 2145} = \frac{8580}{15015} \]

2. For \( \frac{5}{13} \):
\[ \frac{5}{13} = \frac{5 \times 1155}{13 \times 1155} = \frac{5775}{15015} \]

3. For \( \frac{6}{11} \):
\[ \frac{6}{11} = \frac{6 \times 1365}{11 \times 1365} = \frac{8190}{15015} \]

4. For \( \frac{3}{5} \):
\[ \frac{3}{5} = \frac{3 \times 3003}{5 \times 3003} = \frac{9009}{15015} \]

5. For \( \frac{2}{3} \):
\[ \frac{2}{3} = \frac{2 \times 5005}{3 \times 5005} = \frac{10010}{15015} \]

### Step 4: Compare the Converted Fractions
Now, we have the following fractions with a common denominator:
- \( \frac{8580}{15015} \) for \( \frac{4}{7} \)
- \( \frac{5775}{15015} \) for \( \frac{5}{13} \)
- \( \frac{8190}{15015} \) for \( \frac{6}{11} \)
- \( \frac{9009}{15015} \) for \( \frac{3}{5} \)
- \( \frac{10010}{15015} \) for \( \frac{2}{3} \)

### Step 5: Identify the Smallest and Second Smallest Fractions
Now we can compare the numerators:
- \( 5775 \) (from \( \frac{5}{13} \)) is the smallest.
- \( 8580 \) (from \( \frac{4}{7} \)) is the second smallest.

### Conclusion
Thus, the second smallest fraction is:
\[ \frac{4}{7} \]