A person wants to use `+20 D` lens as simple microscope. If the person uses `+1.5 D` glasses to have normal vision from 25cm onwards. What maximum magnification can the person achieve if he uses microscope without the glasses ?
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Explanation

To solve the problem step by step, we need to calculate the maximum magnification that can be achieved using a +20 D lens as a simple microscope without the glasses.

### Step 1: Understand the given data
- The power of the lens (P) = +20 D
- The power of the glasses (Pg) = +1.5 D
- The near point for normal vision (D) = 25 cm

### Step 2: Calculate the focal length of the glasses
The focal length (f) of a lens can be calculated using the formula:
\[ f = \frac{100}{P} \]
For the glasses:
\[ f_g = \frac{100}{1.5} = \frac{100 \times 2}{3} = \frac{200}{3} \text{ cm} \approx 66.67 \text{ cm} \]

### Step 3: Determine the effective near point without glasses
The glasses allow the person to see clearly at 25 cm. Without the glasses, the person can see at a distance determined by the focal length of the glasses. The distance (u) at which the person can see without glasses can be calculated using the lens formula:
\[ \frac{1}{f_g} = \frac{1}{v} - \frac{1}{u} \]
Where:
- \( v = -25 \) cm (the image distance for normal vision)
- \( f_g = \frac{200}{3} \) cm

Rearranging gives:
\[ \frac{1}{u} = \frac{1}{v} - \frac{1}{f_g} \]
Substituting the values:
\[ \frac{1}{u} = \frac{1}{-25} - \frac{3}{200} \]
Finding a common denominator (200):
\[ \frac{1}{u} = \frac{-8}{200} - \frac{3}{200} = \frac{-11}{200} \]
Thus,
\[ u = -\frac{200}{11} \approx -18.18 \text{ cm} \]

### Step 4: Calculate the focal length of the simple microscope
The focal length (f) of the +20 D lens is:
\[ f_m = \frac{100}{20} = 5 \text{ cm} \]

### Step 5: Calculate the maximum magnification
The magnification (M) of a simple microscope is given by:
\[ M = 1 + \frac{D}{f} \]
Where D is the distance at which the person can see without glasses (in cm) and f is the focal length of the microscope (in cm).
Substituting the values:
\[ M = 1 + \frac{40}{5} = 1 + 8 = 9 \]

### Conclusion
The maximum magnification that the person can achieve using the +20 D lens as a simple microscope without the glasses is **9**.

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