The ratio of two number is 3:2, If 10 and the sum of the two numbers are added to their product, square of sixteen is obtained. What could be the smaller number?

Let the two numbers be 3N and 2N
According to the question.
10 + (3N +2N ) + (3N x 2N) = 16²
⇒ 6N² + 5N - 246 = 0
⇒ 6N² + 41N - 36N - 246 = 0
⇒ N(6N + 41 ) - 6(6N + 41) = 0
⇒ (6N + 41 ) (N - 6) = 0
∴ N = 6 or -41/6 (But -ve value cannot be accepted )
∴ Smaller number = 2N = 2 x 6 = 12 .