Of the two numbers, 4 times the smaller one is less than 3 times the larger one by 6. Also, the sum of the numbers is larger than 6 times their difference by 5. Find the numbers.

Let two numbers be x and y, where x > y.

Given,

4 times the smaller one is less than 3 times the larger one by 6.

⇒ 3x - 4y = 6

⇒ 4y = 3x - 6

⇒ y = $\Big(\dfrac{3x - 6}{4}\Big)$     ………….(1)

Given,

Sum of the numbers is larger than 6 times their difference by 5.

⇒ x + y - 6(x - y) = 5

⇒ x + y - 6x + 6y = 5

⇒ x - 6x + y + 6y = 5

⇒ 7y - 5x = 5     ………..(2)

Substituting value of y from equation (1) in (2), we get :

$\Rightarrow 7\Big(\dfrac{3x - 6}{4}\Big) - 5x = 5 \\[1em] \Rightarrow \Big(\dfrac{21x - 42}{4}\Big) - 5x = 5 \\[1em] \Rightarrow \Big(\dfrac{21x - 42 - 20x}{4}\Big) = 5 \\[1em] \Rightarrow \dfrac{x - 42}{4} = 5 \\[1em] \Rightarrow x - 42 = 20 \\[1em] \Rightarrow x = 20 + 42 \\[1em] \Rightarrow x = 62.$

Substituting value of x in equation (1), we get :

$\Rightarrow y = \Big(\dfrac{3x - 6}{4}\Big) \\[1em] \Rightarrow y = \Big(\dfrac{3 \times 62 - 6}{4}\Big) \\[1em] \Rightarrow y = \Big(\dfrac{186 - 6}{4}\Big) \\[1em] \Rightarrow y = \Big(\dfrac{180}{4}\Big) = 45.$

Hence, the numbers are 62 and 45.