Find the mean proportional between:
(i) 6 and 24
(ii )3 and 27
(iii) 0.4 and 0.9

Answer

To find the mean proportional between two numbers, we can use the property that states if \( a \) and \( b \) are two numbers, then the mean proportional \( x \) can be found using the formula:

\[ x^2 = a \times b \]

This means that \( x \) is the square root of the product of \( a \) and \( b \).

Let's solve the given questions step by step.

### (i) Find the mean proportional between 6 and 24.

1. **Identify the numbers**: Here, \( a = 6 \) and \( b = 24 \).
2. **Use the formula**:
\[ x^2 = 6 \times 24 \]
3. **Calculate the product**:
\[ x^2 = 144 \]
4. **Take the square root**:
\[ x = \sqrt{144} = 12 \]

**Mean Proportional**: The mean proportional between 6 and 24 is **12**.

### (ii) Find the mean proportional between 3 and 27.

1. **Identify the numbers**: Here, \( a = 3 \) and \( b = 27 \).
2. **Use the formula**:
\[ x^2 = 3 \times 27 \]
3. **Calculate the product**:
\[ x^2 = 81 \]
4. **Take the square root**:
\[ x = \sqrt{81} = 9 \]

**Mean Proportional**: The mean proportional between 3 and 27 is **9**.

### (iii) Find the mean proportional between 0.4 and 0.9.

1. **Identify the numbers**: Here, \( a = 0.4 \) and \( b = 0.9 \).
2. **Use the formula**:
\[ x^2 = 0.4 \times 0.9 \]
3. **Calculate the product**:
\[ x^2 = 0.36 \]
4. **Take the square root**:
\[ x = \sqrt{0.36} = 0.6 \]

**Mean Proportional**: The mean proportional between 0.4 and 0.9 is **0.6**.

### Summary of Results:
- Mean proportional between 6 and 24 is **12**.
- Mean proportional between 3 and 27 is **9**.
- Mean proportional between 0.4 and 0.9 is **0.6**.