Two numbers are respectively 20% and 50% more than a third number. What is the ratio of these two numbers?

Introduction / Context:
This ratio and percentage question tests the link between percentage increase and direct comparison of two numbers. Instead of giving the actual values, the problem describes both numbers relative to a third one. The goal is to express the ratio of the first two numbers in simplest form.

Given Data / Assumptions:
- Let the third number be some base value. - First number is 20 percent more than this base. - Second number is 50 percent more than the same base. - We must find the ratio of the first and second numbers.

Concept / Approach:
Using a convenient assumed value for the third number can simplify calculations. Since we only need a ratio, we can choose any positive value, usually 100 for percentage problems. Once we have concrete values for the first and second numbers, we write their ratio and simplify it by dividing numerator and denominator by their greatest common divisor.

Step-by-Step Solution:
Step 1: Assume the third number is 100 units for convenience. Step 2: First number is 20 percent more than 100, so it is 100 + 20 = 120. Step 3: Second number is 50 percent more than 100, so it is 100 + 50 = 150. Step 4: Write the ratio of the first to the second number as 120 : 150. Step 5: Simplify this ratio by dividing numerator and denominator by 30: 120 / 30 = 4 and 150 / 30 = 5. Step 6: So the simplified ratio is 4 : 5.

Verification / Alternative check:
We can choose a different base for the third number, for example 10 units. Then the first number is 12 and the second number is 15. The ratio 12 : 15 simplifies to 4 : 5 again. This shows that the ratio is independent of the chosen base, as expected, and confirms that 4 : 5 is correct.

Why Other Options Are Wrong:
- 2 : 5 and 3 : 5: These would correspond to much smaller relative increases, not 20 percent and 50 percent. - 5 : 4: This reverses the correct order and suggests that the first number is larger than the second by a bigger margin than actually given.

Common Pitfalls:
A common mistake is to add 20 and 50 and treat 70 percent as the ratio basis, which is not correct. Another error is to compute 120 : 150 correctly but forget to simplify it to lowest terms. In aptitude exams, simplified ratios are usually expected, so always divide by the greatest common divisor at the end.

Final Answer:
The ratio of the two numbers is 4 : 5.