Ratio shift after equal increments: Two numbers are in the ratio 3 : 4. If 3 is added to each number, the new ratio becomes 4 : 5. Find the difference between the original numbers and show your algebra clearly.

Introduction / Context:
Changing ratios after adding equal amounts to each term can be solved by expressing the original numbers with a common multiplier and then imposing the new ratio. This leads to a simple linear equation in the multiplier.

Given Data / Assumptions:

• Original numbers: 3x and 4x.
• Add 3 to each → (3x + 3) and (4x + 3).
• New ratio = 4 : 5.

Concept / Approach:
Set (3x + 3)/(4x + 3) = 4/5 and solve for x. Then compute the original difference 4x − 3x = x.

Step-by-Step Solution:

Verification / Alternative check:
Original numbers 9 and 12; after adding 3 → 12 and 15, ratio 12 : 15 simplifies to 4 : 5, as required.

Why Other Options Are Wrong:
2, 5, 7, 9 result from arithmetic slips when solving the proportion or misreading the original ratio.

Common Pitfalls:
Cross-multiplying incorrectly or computing the difference after the increment rather than for the original pair.

Final Answer:
3