Which of the following fractions is the smallest ?
`(8)/(15),(14)/(33),(7)/(13),(11)/(13)`

Options

A. 8/15
B. 4/33
C. 7/33
D. 11/33

Answer

To determine which of the given fractions is the smallest, we will compare each fraction step by step. The fractions we need to compare are:

1. \( \frac{8}{15} \)
2. \( \frac{14}{33} \)
3. \( \frac{7}{13} \)
4. \( \frac{11}{13} \)

### Step 1: Compare \( \frac{8}{15} \) and \( \frac{14}{33} \)

To compare these two fractions, we can cross-multiply:

\[ 8 \times 33 \quad \text{and} \quad 14 \times 15 \]

Calculating these:

- \( 8 \times 33 = 264 \)
- \( 14 \times 15 = 210 \)

Since \( 264 > 210 \), we have:

\[ \frac{8}{15} > \frac{14}{33} \]

### Step 2: Compare \( \frac{14}{33} \) and \( \frac{7}{13} \)

Next, we compare \( \frac{14}{33} \) and \( \frac{7}{13} \) using cross-multiplication again:

\[ 14 \times 13 \quad \text{and} \quad 7 \times 33 \]

Calculating these:

- \( 14 \times 13 = 182 \)
- \( 7 \times 33 = 231 \)

Since \( 182 < 231 \), we have:

\[ \frac{14}{33} < \frac{7}{13} \]

### Step 3: Compare \( \frac{7}{13} \) and \( \frac{11}{13} \)

Now, we compare \( \frac{7}{13} \) and \( \frac{11}{13} \). Since the denominators are the same, we can directly compare the numerators:

\[ 7 < 11 \]

Thus:

\[ \frac{7}{13} < \frac{11}{13} \]

### Step 4: Compare \( \frac{14}{33} \) with \( \frac{11}{13} \)

Now we need to compare \( \frac{14}{33} \) and \( \frac{11}{13} \):

\[ 14 \times 13 \quad \text{and} \quad 11 \times 33 \]

Calculating these:

- \( 14 \times 13 = 182 \)
- \( 11 \times 33 = 363 \)

Since \( 182 < 363 \), we have:

\[ \frac{14}{33} < \frac{11}{13} \]

### Conclusion

From the comparisons, we have:

- \( \frac{8}{15} > \frac{14}{33} \)
- \( \frac{14}{33} < \frac{7}{13} \)
- \( \frac{7}{13} < \frac{11}{13} \)

Thus, the smallest fraction among the given options is:

\[ \frac{14}{33} \]

### Final Answer

The smallest fraction is \( \frac{14}{33} \).

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