Which of the following fractions is the smallest?
`1/2,3/7,3/5,4/9`

Options

A. `4/9`
B. `3/5`
C. `3/7`
D. `1/2`

Answer

To find out which of the fractions \( \frac{1}{2}, \frac{3}{7}, \frac{3}{5}, \frac{4}{9} \) is the smallest, we can follow these steps:

### Step 1: Identify the denominators
The denominators of the given fractions are:
- \(2\) from \( \frac{1}{2} \)
- \(7\) from \( \frac{3}{7} \)
- \(5\) from \( \frac{3}{5} \)
- \(9\) from \( \frac{4}{9} \)

### Step 2: Find the Least Common Multiple (LCM)
To compare the fractions easily, we need to find the LCM of the denominators \(2, 5, 7, 9\).

1. **Prime factorization**:
- \(2 = 2^1\)
- \(5 = 5^1\)
- \(7 = 7^1\)
- \(9 = 3^2\)

2. **LCM calculation**:
- Take the highest power of each prime:
- \(2^1\)
- \(3^2\)
- \(5^1\)
- \(7^1\)

Thus, the LCM is:
\[ LCM = 2^1 \times 3^2 \times 5^1 \times 7^1 = 2 \times 9 \times 5 \times 7 \]
- Calculate step by step:
- \(2 \times 9 = 18\)
- \(18 \times 5 = 90\)
- \(90 \times 7 = 630\)

So, the LCM is \(630\).

### Step 3: Convert each fraction to have the same denominator
Now we will convert each fraction to have a denominator of \(630\):

1. **Convert \( \frac{1}{2} \)**:
\[ \frac{1}{2} = \frac{1 \times 315}{2 \times 315} = \frac{315}{630} \]

2. **Convert \( \frac{3}{7} \)**:
\[ \frac{3}{7} = \frac{3 \times 90}{7 \times 90} = \frac{270}{630} \]

3. **Convert \( \frac{3}{5} \)**:
\[ \frac{3}{5} = \frac{3 \times 126}{5 \times 126} = \frac{378}{630} \]

4. **Convert \( \frac{4}{9} \)**:
\[ \frac{4}{9} = \frac{4 \times 70}{9 \times 70} = \frac{280}{630} \]

### Step 4: Compare the numerators
Now we compare the numerators of the fractions:
- \( \frac{315}{630} \)
- \( \frac{270}{630} \)
- \( \frac{378}{630} \)
- \( \frac{280}{630} \)

The numerators are \(315, 270, 378, 280\).

### Step 5: Identify the smallest numerator
The smallest numerator is \(270\), which corresponds to the fraction \( \frac{3}{7} \).

### Conclusion
Thus, the smallest fraction among \( \frac{1}{2}, \frac{3}{7}, \frac{3}{5}, \frac{4}{9} \) is:
\[ \frac{3}{7} \]

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