In this basic arithmetic question, one number is 25% of another number, and the larger number is 12 more than the smaller number. What is the value of the larger number?

Introduction / Context:
This aptitude question tests your understanding of percentages and simple linear equations involving two related numbers. Many exams use this style of problem to check whether you can translate a verbal relationship such as one number being 25 percent of another into algebra, and then combine that with a difference condition to find the exact values.

Given Data / Assumptions:

• One number is 25 percent of another number.
• The larger number is 12 more than the smaller number.
• We must find the value of the larger number.
• Both numbers are assumed to be real positive numbers, typically integers in this type of aptitude question.

Concept / Approach:
The key ideas are percentage conversion and forming simultaneous equations. Saying that one number is 25 percent of another means it is equal to 0.25 times the other number, or one fourth of it. We also know the difference between the larger and smaller number. By naming the numbers with variables, we can write algebraic equations and solve them systematically. Once the smaller number is found, the larger number follows directly from the relationship between them.

Step-by-Step Solution:
Step 1: Let the smaller number be x and the larger number be y. Step 2: The statement "one number is 25 percent of another" becomes x = 0.25 * y, which is the same as x = (1 / 4) * y. Step 3: The difference condition "the larger number is 12 more than the smaller" becomes y = x + 12. Step 4: Substitute x = (1 / 4) * y into y = x + 12 to express everything in terms of y. Step 5: This gives y = (1 / 4) * y + 12. Step 6: Move terms involving y to one side: y - (1 / 4) * y = 12. Step 7: Simplify the left side: (3 / 4) * y = 12. Step 8: Solve for y: y = 12 * (4 / 3) = 16. Step 9: Therefore the larger number is 16 and the smaller number is x = (1 / 4) * 16 = 4, which is consistent with the difference of 12.

Verification / Alternative check:
You can verify the answer quickly without full algebra by trying the options and checking the relationships. If the larger number is 16, then one fourth of 16 is 4, so the smaller number is 4. The difference between 16 and 4 is 12, which matches the condition. Also, 4 is exactly 25 percent of 16. None of the other options will satisfy both the 25 percent relation and the difference of 12 at the same time, so 16 is the only consistent solution.

Why Other Options Are Wrong:
16 is correct, so check the others briefly. If the larger number were 20, then 25 percent of 20 is 5, and the difference 20 - 5 is 15, not 12. If the larger number were 24, then 25 percent of 24 is 6, and the difference is 18, not 12. If it were 12, then 25 percent of 12 is 3, and 12 - 3 is only 9. If it were 8, then 25 percent of 8 is 2, and 8 - 2 is 6. None of these satisfy the stated difference of 12.

Common Pitfalls:
A common mistake is to misread the statement and treat 25 percent incorrectly, either as 0.25 added to the number or as 25 divided by the number. Another error is to assume the smaller number is 25 percent less than the larger number rather than exactly one fourth of it. Students also sometimes mix up which number is larger and which is smaller, leading to reversed equations that do not match the conditions. Carefully defining variables and writing the equations directly from the words helps avoid these issues.

Final Answer:
The larger number that satisfies both conditions is 16.