An object is placed at a distance of `1.5 m` from a screen and a convex lens is interposed between them. The magnification produced is `4`. What is the focal length of the lens ?

Explanation

To find the focal length of the convex lens given the object distance, screen distance, and magnification, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values:**
- Distance between object and screen (D) = 1.5 m
- Magnification (m) = -4 (since the image is real and inverted)

2. **Define Variables:**
- Let the object distance from the lens be \( u \).
- Let the image distance from the lens be \( v \).
- The relationship between the object distance, image distance, and the distance between the object and screen is:
\[ u + v = 1.5 \quad (1) \]

3. **Use Magnification Formula:**
- The magnification (m) is given by:
\[ m = \frac{v}{u} \]
- Substituting the value of magnification:
\[ -4 = \frac{v}{u} \quad (2) \]
- Rearranging equation (2):
\[ v = -4u \quad (3) \]

4. **Substitute Equation (3) into Equation (1):**
- Replace \( v \) in equation (1) with the expression from equation (3):
\[ u + (-4u) = 1.5 \]
- Simplifying this gives:
\[ -3u = 1.5 \]
- Solving for \( u \):
\[ u = -0.5 \, \text{m} \quad (4) \]

5. **Find Image Distance \( v \):**
- Substitute \( u \) back into equation (3):
\[ v = -4(-0.5) = 2 \, \text{m} \quad (5) \]

6. **Use Lens Formula to Find Focal Length:**
- The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
- Substituting \( v = 2 \) m and \( u = -0.5 \) m into the lens formula:
\[ \frac{1}{f} = \frac{1}{2} - \frac{1}{-0.5} \]
- Simplifying this gives:
\[ \frac{1}{f} = \frac{1}{2} + 2 = \frac{1}{2} + \frac{4}{2} = \frac{5}{2} \]
- Therefore, the focal length \( f \) is:
\[ f = \frac{2}{5} = 0.4 \, \text{m} \]

### Final Answer:
The focal length of the lens is \( 0.4 \, \text{m} \).