Which of the following the fractions is the smallest?
`(7)/(6),(7)/(9),(4)/(5),(5)/(7)`

Options

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

Answer

To determine which of the given fractions is the smallest, we can use cross multiplication to compare them. The fractions we need to compare are:

1. \( \frac{7}{6} \)
2. \( \frac{7}{9} \)
3. \( \frac{4}{5} \)
4. \( \frac{5}{7} \)

### Step 1: Compare \( \frac{7}{6} \) and \( \frac{7}{9} \)

We will cross multiply:

\[ 7 \times 9 = 63 \]
\[ 7 \times 6 = 42 \]

Since \( 63 > 42 \), we conclude that:

\[ \frac{7}{9} < \frac{7}{6} \]

### Step 2: Compare \( \frac{7}{9} \) and \( \frac{4}{5} \)

Now we compare \( \frac{7}{9} \) and \( \frac{4}{5} \) using cross multiplication:

\[ 7 \times 5 = 35 \]
\[ 4 \times 9 = 36 \]

Since \( 36 > 35 \), we conclude that:

\[ \frac{4}{5} > \frac{7}{9} \]

### Step 3: Compare \( \frac{7}{9} \) and \( \frac{5}{7} \)

Next, we compare \( \frac{7}{9} \) and \( \frac{5}{7} \):

\[ 7 \times 7 = 49 \]
\[ 9 \times 5 = 45 \]

Since \( 49 > 45 \), we conclude that:

\[ \frac{5}{7} < \frac{7}{9} \]

### Step 4: Summary of Comparisons

From our comparisons, we have:

1. \( \frac{7}{9} < \frac{7}{6} \)
2. \( \frac{7}{9} < \frac{4}{5} \)
3. \( \frac{5}{7} < \frac{7}{9} \)

Thus, the smallest fraction among all is \( \frac{5}{7} \).

### Final Answer

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

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