One of the two numbers exceeds the other by 9. Four times the smaller added to five times the larger gives 108. Find the numbers.

Let the smaller number be x.

Then the larger number = x + 9.

Four times the smaller added to five times the larger gives 108:

∴ 4(x) + 5(x + 9) = 108

⇒ 4x + 5x + 45 = 108

⇒ 9x + 45 = 108

⇒ 9x = 108 - 45 $\quad$ [Transposing +45 to RHS]

⇒ 9x = 63

⇒ x = $\dfrac{63}{9}$

⇒ x = 7

Smaller number = 7, Larger number = 7 + 9 = 16.

Hence, the required numbers are 16 and 7.