Which of the following fraction is the smallest?
`(7)/(6),(7)/(6),(4)/(5),(5)/(7)`

Options

A. `(7)/(6)`
B. `(7)/(9)`
C. `(4)/(5)`
D. `(5)/(7)`

Answer

To determine which of the following fractions is the smallest: \( \frac{7}{6}, \frac{7}{6}, \frac{4}{5}, \frac{5}{7} \), we can follow these steps:

### Step 1: Identify the fractions
We have the following fractions:
1. \( \frac{7}{6} \)
2. \( \frac{7}{6} \) (duplicate of the first)
3. \( \frac{4}{5} \)
4. \( \frac{5}{7} \)

### Step 2: Compare the fractions
Since \( \frac{7}{6} \) appears twice, we can ignore one of them for comparison purposes. Now we need to compare \( \frac{4}{5} \) and \( \frac{5}{7} \) to find which one is smaller.

### Step 3: Cross-multiply to compare \( \frac{4}{5} \) and \( \frac{5}{7} \)
To compare the two fractions, we can cross-multiply:
- For \( \frac{4}{5} \) and \( \frac{5}{7} \):
- Multiply the numerator of the first fraction by the denominator of the second: \( 4 \times 7 = 28 \)
- Multiply the numerator of the second fraction by the denominator of the first: \( 5 \times 5 = 25 \)

### Step 4: Compare the results
Now we compare the results of the cross-multiplication:
- \( 28 \) (from \( 4 \times 7 \)) and \( 25 \) (from \( 5 \times 5 \))
- Since \( 28 > 25 \), it means \( \frac{4}{5} > \frac{5}{7} \)

### Step 5: Conclusion
Thus, \( \frac{5}{7} \) is smaller than \( \frac{4}{5} \). Now we can conclude that among all the fractions, \( \frac{5}{7} \) is the smallest.

### Final Answer
The smallest fraction is \( \frac{5}{7} \).

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