Quantities `a`, `2`, `10` and `b` are in continued proportion, find the values of `a` and `b`.

Answer

To find the values of \( a \) and \( b \) when the quantities \( a \), \( 2 \), \( 10 \), and \( b \) are in continued proportion, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Continued Proportion**:
- If \( a \), \( 2 \), \( 10 \), and \( b \) are in continued proportion, then the following relationship holds:
\[ \frac{a}{2} = \frac{2}{10} = \frac{10}{b} \]

2. **Setting Up the First Equation**:
- From the first part of the proportion, we can write:
\[ \frac{a}{2} = \frac{2}{10} \]

3. **Cross-Multiplying to Solve for \( a \)**:
- Cross-multiplying gives us:
\[ a \cdot 10 = 2 \cdot 2 \]
- Simplifying this, we find:
\[ 10a = 4 \]
- Dividing both sides by 10:
\[ a = \frac{4}{10} = \frac{2}{5} \]

4. **Setting Up the Second Equation**:
- Now, using the second part of the proportion:
\[ \frac{2}{10} = \frac{10}{b} \]

5. **Cross-Multiplying to Solve for \( b \)**:
- Cross-multiplying gives us:
\[ 2b = 10 \cdot 10 \]
- Simplifying this, we find:
\[ 2b = 100 \]
- Dividing both sides by 2:
\[ b = \frac{100}{2} = 50 \]

6. **Final Values**:
- Therefore, the values are:
\[ a = \frac{2}{5}, \quad b = 50 \]

### Summary of Results:
- The value of \( a \) is \( \frac{2}{5} \).
- The value of \( b \) is \( 50 \).