Three numbers are in continued proportion, whose mean proportional is

1.	Three numbers are in continued proportion, whose mean proportional is 12 and the sum of the remaining two numbers is 26, then find these numbers.
Answer» Let the first number be x. ∴ Third number = 26 – x 12 is the mean proportional of x and (26 – x). ∴ \(\frac{x}{12}\) = \(\frac{12}{26-x}\) ∴ x(26 – x) = 12 x 12 ∴ 26x – x² = 144 ∴ x² – 26x + 144 = 0 ∴ x² – 18x – 8x + 144 = 0 ∴ x(x – 18) – 8(x – 18) = 0 ∴ (x – 18) (x – 8) = 0 ∴ x = 18 or x = 8 ∴ Third number = 26 – x = 26 – 18 = 8 or 26 – x = 26 – 8 = 18 ∴ The numbers are 18, 12, 8 or 8, 12, 18.

Three numbers are in continued proportion, whose mean proportional is 12 and the sum of the remaining two numbers is 26, then find these numbers.

Answer»

Let the first number be x.

∴ Third number = 26 – x

12 is the mean proportional of x and (26 – x).

∴ \(\frac{x}{12}\) = \(\frac{12}{26-x}\)

∴ x(26 – x) = 12 x 12

∴ 26x – x² = 144

∴ x² – 26x + 144 = 0

∴ x² – 18x – 8x + 144 = 0

∴ x(x – 18) – 8(x – 18) = 0

∴ (x – 18) (x – 8) = 0

∴ x = 18 or x = 8

∴ Third number = 26 – x = 26 – 18 = 8 or 26 – x

= 26 – 8 = 18

∴ The numbers are 18, 12, 8 or 8, 12, 18.