Which of the following is the smallest fraction?
`(8)/(25), (7)/(23), (11)/(23), (14)/(53)`

Options

A. `(8)/(25)`
B. `(7)/(23)`
C. `(11)/(23)`
D. `(14)/(53)`

Answer

To determine which of the given fractions is the smallest, we will compare each pair of fractions using cross multiplication. The fractions we need to compare are:

1. \( \frac{8}{25} \)
2. \( \frac{7}{23} \)
3. \( \frac{11}{23} \)
4. \( \frac{14}{53} \)

### Step 1: Compare \( \frac{8}{25} \) and \( \frac{7}{23} \)

We will cross multiply:

\[ 8 \times 23 \quad \text{and} \quad 7 \times 25 \]

Calculating these:

\[ 8 \times 23 = 184 \]
\[ 7 \times 25 = 175 \]

Since \( 175 < 184 \), we conclude that:

\[ \frac{7}{23} < \frac{8}{25} \]

### Step 2: Compare \( \frac{11}{23} \) and \( \frac{14}{53} \)

Next, we will cross multiply these two fractions:

\[ 11 \times 53 \quad \text{and} \quad 14 \times 23 \]

Calculating these:

\[ 11 \times 53 = 583 \]
\[ 14 \times 23 = 322 \]

Since \( 322 < 583 \), we conclude that:

\[ \frac{14}{53} < \frac{11}{23} \]

### Step 3: Compare the results from Step 1 and Step 2

Now we need to compare \( \frac{7}{23} \) and \( \frac{14}{53} \):

We will cross multiply:

\[ 7 \times 53 \quad \text{and} \quad 14 \times 23 \]

Calculating these:

\[ 7 \times 53 = 371 \]
\[ 14 \times 23 = 322 \]

Since \( 322 < 371 \), we conclude that:

\[ \frac{14}{53} < \frac{7}{23} \]

### Conclusion

From our comparisons, we find that:

- \( \frac{7}{23} < \frac{8}{25} \)
- \( \frac{14}{53} < \frac{11}{23} \)
- \( \frac{14}{53} < \frac{7}{23} \)

Thus, the smallest fraction among the four given fractions is:

\[ \frac{14}{53} \]

### Final Answer

The smallest fraction is \( \frac{14}{53} \).

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