Step-by-step Solution:

Quick Solution

This problem can be solved quickly by testing the options against the given rules, or more formally with algebra.

Method 1: Testing the Options (Faster)

We just need to find the option that satisfies both conditions given in the problem.

1. Condition 1: The units digit is 3 more than the tens digit.
2. Condition 2: The ratio of the number to the sum of its digits is 4:1.

• A. 24: Fails Condition 1 (4 is not 2 + 3).
• B. 63: Fails Condition 1 (3 is not 6 + 3).
• C. 36:
  • Condition 1: Is 6 equal to 3 + 3? Yes.
  • Condition 2: The sum of digits is 3 + 6 = 9. Is the ratio 36:9 the same as 4:1? Yes, 36 / 9 = 4. Yes.
  This is the correct answer.
• D. 42: Fails Condition 1 (2 is not 4 + 3).

Method 2: Using Algebra

- Let the ten's digit be t and the unit's digit be u. The number can be written as 10t + u.

1. From Condition 1: "the digit in the units place is 3 more than the digit in ten's place".
   Equation: u = t + 3
2. From Condition 2: "The ratio between the number and the sum of the digits is 4 : 1".
   Equation: (10t + u) / (t + u) = 4 => 10t + u = 4t + 4u => 6t = 3u => u = 2t
3. Solve the equations: We have two expressions for u: `u = t + 3` and `u = 2t`. Let's set them equal:
   2t = t + 3
   t = 3 (this is the ten's digit).
   Now find u: u = 2 × 3 = 6 (this is the unit's digit).

The number is 36.

Final Answer: The number is 36.