Given:
- Focal length of convex mirror (f) = 30 cm
- Size of the image (h') is one-fourth (1/4) the size of the object (h)

To find:
- Distance of the object from the mirror (d)

Formula:
The mirror formula for a convex mirror is given by:
1/f = 1/v - 1/u

Where:
- f is the focal length of the mirror
- v is the image distance from the mirror
- u is the object distance from the mirror

Solution:

Step 1: Find the magnification (m) of the mirror using the given information
- Magnification (m) = -(v/u) = -(h'/h)

Given that the size of the image (h') is one-fourth (1/4) the size of the object (h), we can say:
- m = -(1/4)

Step 2: Use the magnification formula to find the object distance (u)
- m = -(v/u)
- -(1/4) = -(v/u)

Since the magnification is negative, it indicates that the image formed by the convex mirror is virtual and upright.

Step 3: Substitute the focal length (f) and magnification (m) into the mirror formula to find the object distance (u)
- 1/f = 1/v - 1/u
- 1/30 = 1/v - 1/u

Since the mirror is a convex mirror, the focal length is positive, and the image formed is always virtual and upright.

Step 4: Simplify the mirror formula using the given information
- 1/30 = 1/v - 1/u
- 1/v - 1/u = 1/30

Step 5: Substitute the magnification formula into the simplified mirror formula
- 1/v + 1/(4u) = 1/30

Step 6: Solve the equation to find the object distance (u)
- Multiply through by 30u to eliminate the denominators:
30u/v + 7.5 = u
- Multiply through by v to eliminate the fraction:
30u + 7.5v = uv
- Substitute the magnification formula:
30u + 7.5v = -uv/4

Step 7: Substitute the value of m into the equation
- 30u + 7.5v = -u(1/4)
- 30u + 7.5v = -u/4

Step 8: Rearrange the equation to isolate the variable u
- Multiply through by 4 to eliminate the fraction:
120u + 30v = -u
- Rearrange the equation:
120u + u = -30v
121u = -30v
u = -30v/121

Step 9: Substitute the value of u into the magnification formula to find v
- m = -(v/u)
- -(1/4) = -(v/(-30v/121))
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