Two numbers are in the ratio 3 : 5. If 6 is added to each of the numbers, the new ratio becomes 2 : 3. What are the original two numbers?

Introduction:
This is a classic ratio and proportion question where you are given an initial ratio and a transformation (adding the same number to both terms) that produces a new ratio. The task is to find the original numbers that satisfy both conditions.

Given Data / Assumptions:

• The original ratio of the two numbers is 3 : 5.
• When 6 is added to each number, the ratio becomes 2 : 3.
• The numbers are positive real numbers, usually treated as integers in such problems.
• We need to determine the original two numbers.

Concept / Approach:
When numbers are in the ratio 3 : 5, we can represent them as 3k and 5k for some positive constant k. After adding 6 to each number, we are told that the new ratio is 2 : 3. We set up an equation based on this new ratio and solve for k, then compute the original numbers.

Step-by-Step Solution:
Step 1: Represent the numbers.Let the numbers be 3k and 5k.Step 2: Add 6 to each.New numbers: 3k + 6 and 5k + 6.Step 3: Use the new ratio 2 : 3.(3k + 6) / (5k + 6) = 2 / 3Step 4: Cross-multiply.3 * (3k + 6) = 2 * (5k + 6)9k + 18 = 10k + 12Step 5: Solve for k.9k + 18 = 10k + 12 ⇒ k = 6Step 6: Find the original numbers.First number = 3k = 3 * 6 = 18Second number = 5k = 5 * 6 = 30

Verification / Alternative check:
Check with the transformed ratio: 18 + 6 = 24 and 30 + 6 = 36. Now 24 : 36 simplifies to 2 : 3 by dividing both terms by 12. This confirms that 18 and 30 satisfy the condition.

Why Other Options Are Wrong:
21 and 35: Adding 6 gives 27 and 41; the ratio 27 : 41 is not 2 : 3.
30 and 50: Adding 6 gives 36 : 56, which simplifies to 9 : 14, not 2 : 3.
24 and 40: Adding 6 gives 30 : 46, not in the ratio 2 : 3.
12 and 20: Adding 6 gives 18 : 26, simplifies to 9 : 13, not 2 : 3.

Common Pitfalls:
Students sometimes incorrectly assume that the ratio difference directly gives the added number, or they try trial and error with options. A systematic algebraic approach using k makes the solution clean and error free.

Final Answer:
The original numbers are 18 and 30.