(i) AM = 25 and GM = 20

To find: Two positive numbers a and b

Given: AM = 25 and GM = 20

Formula used: (i) Arithmetic mean between

(ii) Geometric mean between

Arithmetic mean of two numbers

⇒ a + b = 50

⇒ b = 50 – a … (i)

Geometric mean of two numbers

Substituting value of b from eqn. (i)

a(50 – a) = 400

⇒ 50a – a² = 400

On rearranging

⇒ a² – 50a + 400 = 0

⇒ a² – 40a – 10a + 400

⇒ a(a – 40) – 10(a – 40) = 0

⇒ (a – 10) (a – 40) = 0

⇒ a = 10, 40

Substituting, a = 10 Or a = 40 in eqn. (i)

b = 40 Or b = 10

Therefore two numbers are 10 and 40

(ii) AM = 10 and GM = 8

To find: Two positive numbers a and b

Given: AM = 10 and GM = 8

Formula used: (i) Arithmetic mean between

(ii) Geometric mean between

Arithmetic mean of two numbers

⇒ a + b = 20

⇒ a = 20 – b … (i)

Geometric mean of two numbers

Substituting value of a from eqn. (i)

b(20 – b) = 64

⇒ 20b – b² = 64

On rearranging

⇒ b² – 20b + 64 = 0

⇒ b² – 16b – 4b + 64

⇒ b(b – 16) – 4(b – 16) = 0

⇒ (b – 16) (b – 4) = 0

⇒ b = 16, 4

Substituting, b = 16 Or b = 4 in eqn. (i)

a = 4 Or b = 16

Therefore two numbers are 16 and 4