A lens of focal length `12 cm` forms an upright image three times the size of a real object. Find the distance in cm between the object and image.

Explanation

To solve the problem step by step, we will use the lens formula and the magnification formula.

### Step 1: Identify the given values
- Focal length of the lens, \( f = 12 \, \text{cm} \)
- The magnification, \( m = 3 \) (since the image is three times the size of the object)

### Step 2: Understand the type of image formed
Since the image is upright and magnified, it indicates that the image is virtual. For a virtual image formed by a convex lens, the magnification is given by:
\[ m = -\frac{v}{u} \]
Where:
- \( v \) = image distance
- \( u \) = object distance

### Step 3: Set up the relationships
Given that the image is three times the size of the object, we can express this as:
\[ m = 3 = -\frac{v}{u} \]
From this, we can express \( v \) in terms of \( u \):
\[ v = -3u \]

### Step 4: Use the lens formula
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
Substituting the values we have:
\[ \frac{1}{12} = \frac{1}{-3u} - \frac{1}{u} \]

### Step 5: Simplify the equation
To simplify the right side, we need a common denominator:
\[ \frac{1}{12} = -\frac{1}{3u} - \frac{1}{u} = -\frac{1 + 3}{3u} = -\frac{4}{3u} \]
Thus, we can rewrite the equation as:
\[ \frac{1}{12} = -\frac{4}{3u} \]

### Step 6: Cross-multiply to solve for \( u \)
Cross-multiplying gives:
\[ 3u = -48 \]
Thus,
\[ u = -16 \, \text{cm} \]

### Step 7: Find \( v \)
Using the relationship \( v = -3u \):
\[ v = -3(-16) = 48 \, \text{cm} \]

### Step 8: Calculate the distance between the object and the image
The distance between the object and the image is given by:
\[ \text{Distance} = |v - u| = |48 - (-16)| = |48 + 16| = 64 \, \text{cm} \]

### Final Answer:
The distance between the object and the image is \( 64 \, \text{cm} \).

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