If AM and GM of two positive numbers a and b are 10 and 8 respectively

1.	If AM and GM of two positive numbers a and b are 10 and 8 respectively, find the numbers.
Answer» ⇒ AM = a+b/2 ⇒ GM = √ab Given AM = 10, GM = 8. ⇒ a+b/2 = 10 ⇒ a + b = 20 ⇒ a = 20 – b ⇒ \(\sqrt{(20-b)b}\) = 8 ⇒ 20b – b² = 64 ⇒ b² – 20b + 64 = 0 ⇒ b² – 16b – 4b + 64 = 0 ⇒ b(b – 16) – 4(b – 16) = 0 ⇒ b = 4 or b = 16 ⇒ If b = 4 then a = 16 ⇒ If b = 16 then a = 4. Hence, the numbers are 4 and 16.

If AM and GM of two positive numbers a and b are 10 and 8 respectively, find the numbers.

Answer»

⇒ AM = a+b/2

⇒ GM = √ab

Given AM = 10, GM = 8.

⇒ a+b/2 = 10

⇒ a + b = 20

⇒ a = 20 – b

⇒ \(\sqrt{(20-b)b}\) = 8

⇒ 20b – b² = 64

⇒ b² – 20b + 64 = 0

⇒ b² – 16b – 4b + 64 = 0

⇒ b(b – 16) – 4(b – 16) = 0

⇒ b = 4 or b = 16

⇒ If b = 4 then a = 16

⇒ If b = 16 then a = 4.

Hence, the numbers are 4 and 16.