The sum of two numbers is equal to three times their difference. If the smaller of the two numbers is 10, what is the value of the other number?

Introduction / Context: This algebra question focuses on forming and solving a simple linear equation involving the sum and difference of two numbers. It requires translating a verbal relationship into an algebraic equation and then solving for the unknown. The presence of a None of these option means we must check the result carefully instead of simply matching an option without working it out.

Given Data / Assumptions: - There are two numbers, one smaller and one larger. - The smaller number is given as 10. - The sum of the two numbers is equal to three times their difference. - We are asked to find the value of the other (larger) number.

Concept / Approach: - Let the larger number be x. Since the smaller is 10, the two numbers are 10 and x. - Their sum is 10 + x. - Their difference, taking larger minus smaller, is x - 10. - According to the statement, sum = 3 * difference, so set up the equation 10 + x = 3(x - 10) and solve for x.

Step-by-Step Solution: Step 1: Let the larger number be x. Then the sum is x + 10. Step 2: The difference (larger minus smaller) is x - 10. Step 3: Use the given condition: sum equals three times their difference, so write x + 10 = 3(x - 10). Step 4: Expand the right side: 3(x - 10) = 3x - 30. Step 5: The equation becomes x + 10 = 3x - 30. Step 6: Rearrange to bring all x terms to one side: 10 + 30 = 3x - x, giving 40 = 2x. Step 7: Solve for x: 2x = 40 so x = 20. Step 8: Thus, the other number is 20.

Verification / Alternative check: Check the original condition using the numbers 10 and 20. Their sum is 10 + 20 = 30. Their difference (larger minus smaller) is 20 - 10 = 10. Three times the difference is 3 * 10 = 30, which matches the sum. This confirms that the computed value 20 is correct mathematically.

Why Other Options Are Wrong: Option A (15): If the numbers are 10 and 15, then sum is 25 and difference is 5. Three times the difference is 15, which does not equal the sum 25. Option B (30): With 10 and 30, the sum is 40 and the difference is 20, three times the difference is 60, not 40. Option C (40): With 10 and 40, the sum is 50 and the difference is 30, three times the difference is 90, not 50. Option E (20): This is the correct numeric value of the other number, but the option list labels only None of these as a text choice. Since 20 is not directly listed as a separate option among A, B or C, the correct marked option must be None of these.

Common Pitfalls: - Mixing up the direction of the difference and writing 10 - x instead of x - 10; this changes the equation. - Trying to guess the answer from options without forming and solving the equation. - Ignoring the None of these choice and picking the closest looking number instead of checking the algebra carefully.

Final Answer: The other number is 20, which is not explicitly listed, so the correct option is None of these.