Question

The units digit of a 2-digit number exceeds the tens digit by 3, and the product of the given number and the sum of its digits is 324. The number lies between∶ 
1. 30 and 35
2. 45 and 50
3. 35 and 40
4. 40 and 45

Answer 1

Correct Answer - Option 3 : 35 and 40

Given:

The units digit of a 2-digit number exceeds the tens digit by 3

The product of the given number and the sum of its digits = 324

Concept Used:

Principle of Place Values (Units digit and Ten's Digit) of a number 

Calculation:

Let the 2 digit number be (10x + y)

From the given conditions, we get the following equations:

y – x = 3      ----(i)

(10x + y) × (x + y) = 324      ----(ii)

From the equation (i), we get:

y = x + 3      ----(iii)

On substituting the value of equation (iii) into equation (ii), we get:

(10x + x + 3) × (x + x + 3) = 324      

⇒ (11x + 3) × (2x + 3) = 324      

⇒ 22x² + 33x + 6x + 9 = 324

⇒ 22x2 + 39x – 315 = 0

⇒ [x + (105/22)](x – 3) = 0

As we cannot have a negative value of x,

⇒ x = 3      ----(iv)

Also, from equation (iii), we get:

y = 3 + 3 = 6      ----(v)

On combining equations (iv) and (v), we obtain the 2-digit number as:

(10 × 3) + 6 = 36

∴ The given two-digit number lies between 35 and 40