Check whether the following numbers are in proportion or not :
`(1)/(5), (1)/(8),(1)/(4),(1)/(10)`

Answer

To determine whether the numbers \( \frac{1}{5}, \frac{1}{8}, \frac{1}{4}, \frac{1}{10} \) are in proportion, we will use the property that states:

**The numbers \( a, b, c, d \) are in proportion if the product of the extremes is equal to the product of the means.**

### Step-by-Step Solution:

1. **Identify the Numbers:**
The given numbers are:
\[ a = \frac{1}{5}, \quad b = \frac{1}{8}, \quad c = \frac{1}{4}, \quad d = \frac{1}{10} \]

2. **Identify the Extremes and Means:**
- **Extremes:** The first and last numbers:
\[ a = \frac{1}{5}, \quad d = \frac{1}{10} \]
- **Means:** The middle numbers:
\[ b = \frac{1}{8}, \quad c = \frac{1}{4} \]

3. **Calculate the Product of the Extremes:**
\[ \text{Product of extremes} = a \times d = \frac{1}{5} \times \frac{1}{10} = \frac{1 \times 1}{5 \times 10} = \frac{1}{50} \]

4. **Calculate the Product of the Means:**
\[ \text{Product of means} = b \times c = \frac{1}{8} \times \frac{1}{4} = \frac{1 \times 1}{8 \times 4} = \frac{1}{32} \]

5. **Compare the Products:**
Now we compare the two products:
\[ \text{Product of extremes} = \frac{1}{50}, \quad \text{Product of means} = \frac{1}{32} \]
Since \( \frac{1}{50} \neq \frac{1}{32} \), we conclude that the products are not equal.

6. **Conclusion:**
Since the product of the extremes is not equal to the product of the means, the numbers \( \frac{1}{5}, \frac{1}{8}, \frac{1}{4}, \frac{1}{10} \) are **not in proportion**.