The sum of two numbers is three times their difference. What is the ratio of the two numbers?

Introduction / Context:
This question tests basic algebraic manipulation and understanding of ratios. When a relationship is given between the sum and the difference of two numbers, we can express both conditions in terms of the unknowns and derive their ratio. Problems of this type appear frequently in aptitude tests because they check the ability to form and solve simple linear equations and to interpret the result as a ratio rather than as individual numerical values.

Given Data / Assumptions:

• Let the two numbers be A and B.
• We can assume A is greater than B for simplicity (A > B).
• The sum of the numbers is A + B.
• The difference of the numbers is A − B.
• Given condition: sum is three times the difference, that is A + B = 3(A − B).

Concept / Approach:
We use algebra to translate the word relation into an equation and then solve for the ratio A:B. Ratios are scale free, so we do not need actual values of A and B. Instead, we simplify the equation to find A/B. Once this fraction is simplified, it directly gives the ratio in its simplest form. This method is common in questions about number relationships, ages and mixtures.

Step-by-Step Solution:
Step 1: Let the two numbers be A and B with A > B. Step 2: Given that sum is three times the difference, write A + B = 3(A − B). Step 3: Expand the right side: A + B = 3A − 3B. Step 4: Bring all terms involving A and B to one side. Move A and B from left to right or vice versa. Step 5: Subtract A from both sides: B = 2A − 3B. Step 6: Add 3B to both sides: B + 3B = 2A, so 4B = 2A. Step 7: Divide both sides by 2: 2B = A, or A/B = 2/1. Step 8: Therefore, the ratio A:B = 2:1.

Verification / Alternative check:
Choose concrete numbers that follow the ratio 2:1, for example A = 20 and B = 10. Their sum is 20 + 10 = 30. Their difference is 20 − 10 = 10. The sum 30 is exactly three times the difference 10. This verifies that the ratio 2:1 satisfies the condition and confirms the algebraic result.

Why Other Options Are Wrong:

• 1:2 would give numbers where the smaller number comes first, and it does not satisfy the equation A + B = 3(A − B).
• 3:1 leads to a different relationship where the sum would not be three times the difference; checking with sample numbers quickly disproves it.
• 1:3 is the extreme opposite and clearly fails the condition since the larger number must be at least as large as the sum relation implies.

Common Pitfalls:
A common mistake is to mishandle signs when expanding and rearranging the equation, especially when moving B terms across the equality. Some students also confuse the ratio A:B with B:A and may choose 1:2 instead of 2:1. Another error is to try random trial numbers without first ensuring they satisfy the algebraic condition. Writing each algebraic step carefully helps avoid these issues.

Final Answer:
The ratio of the two numbers is 2:1.