The focal length of a thin biconvex lens is 20cm. When an object is moved from a distance of 25cm in front of it to 50cm, the magnification of its image changes from `m_(25)` to `m_(50)`. The ration `m_(25)//m_(50)` is

Explanation

To solve the problem, we need to find the ratio of magnifications \( \frac{m_{25}}{m_{50}} \) when the object is moved from a distance of 25 cm to 50 cm in front of a biconvex lens with a focal length of 20 cm.

### Step-by-Step Solution:

1. **Identify the given data**:
- Focal length of the lens, \( f = 20 \, \text{cm} \)
- Object distance for the first case, \( u_{25} = -25 \, \text{cm} \) (negative as per sign convention)
- Object distance for the second case, \( u_{50} = -50 \, \text{cm} \)

2. **Calculate the image distance \( v_{25} \) for \( u_{25} \)**:
- Use the lens formula:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
- Rearranging gives:
\[ \frac{1}{v} = \frac{1}{f} + \frac{1}{u} \]
- Substitute the values:
\[ \frac{1}{v_{25}} = \frac{1}{20} + \frac{1}{-25} \]
- Finding a common denominator (100):
\[ \frac{1}{v_{25}} = \frac{5}{100} - \frac{4}{100} = \frac{1}{100} \]
- Therefore, \( v_{25} = 100 \, \text{cm} \).

3. **Calculate the magnification \( m_{25} \)**:
- Magnification is given by:
\[ m = -\frac{v}{u} \]
- Substitute \( v_{25} \) and \( u_{25} \):
\[ m_{25} = -\frac{100}{-25} = 4 \]

4. **Calculate the image distance \( v_{50} \) for \( u_{50} \)**:
- Use the lens formula again:
\[ \frac{1}{v_{50}} = \frac{1}{20} + \frac{1}{-50} \]
- Finding a common denominator (100):
\[ \frac{1}{v_{50}} = \frac{5}{100} - \frac{2}{100} = \frac{3}{100} \]
- Therefore, \( v_{50} = \frac{100}{3} \, \text{cm} \).

5. **Calculate the magnification \( m_{50} \)**:
- Substitute \( v_{50} \) and \( u_{50} \):
\[ m_{50} = -\frac{v_{50}}{u_{50}} = -\frac{\frac{100}{3}}{-50} = \frac{100}{3 \times 50} = \frac{2}{3} \]

6. **Calculate the ratio \( \frac{m_{25}}{m_{50}} \)**:
- Now, we can find the ratio:
\[ \frac{m_{25}}{m_{50}} = \frac{4}{\frac{2}{3}} = 4 \times \frac{3}{2} = 6 \]

### Final Answer:
The ratio \( \frac{m_{25}}{m_{50}} \) is \( 6 \).