When the distance of an object …

Magnification = Image distance / Object distance

In this case :

Let the magnification in the first case be x.

The magnification in the second case will be 3x.

The mirror formula is :

1/u + 1/v = 1/f

The first case

Magnification = x

Object distance = 25 cm

Image distance = Object distance × Magnification

= 15x cm

Now :

1/15 + 1/15 x = 1/f

Lcm = 15x

Solving we have :

(1 + x)/15x = 1/f

f = 15x/(1 + x)

Second case

Object distance = 9 cm

magnification = 3x

Image distance = 27x cm

1/9 + 1/27x = 1/f

Lcm = 27x

(3x + 1)/27x = 1/f

f = 27x/(3x + 1)

Since we have two values of f we can equate them as follows:

27x/(3x + 1) = 15x/(x + 1)

Solving for x we have :

27x(x + 1) = 15x(3x + 1)

27x² + 27x = 45x² + 15x

Collecting like terms together

45x² - 27x² = 27x - 15x

18x² = 12x

Divide through by x

18x = 12

x = 12/18

x = 2/3

Substitute to get f:

f = (27 × 2/3)/(3 × 2/3 + 1)

f = 18/3

= 6 cm