Five percent of a number is 25 percent more than another number. The difference between the two numbers is 96. What are the two numbers?

Introduction / Context:
This problem combines percentages with linear equations in two unknown numbers. One number is related to the other through a percentage condition, and the difference between them is given. The question is a typical algebraic word problem found in competitive exams, checking whether you can translate language into equations and solve them systematically.

Given Data / Assumptions:

• Let the first number be x and the second number be y.
• Five percent of x is 25 percent more than y.
• The difference between the two numbers x and y is 96.
• We assume both numbers are positive real numbers and expect integer answers based on the options.

Concept / Approach:
We need to form two equations. The first equation comes from the statement that 5 percent of x is 25 percent more than y. The phrase 25 percent more than y means y plus 25 percent of y, which is 1.25y. The second equation is simply x minus y equals 96. Solving this pair of linear equations yields values for x and y. Then we can match the ordered pair with the given options.

Step-by-Step Solution:
Step 1: Translate the percentage condition. Five percent of x is 0.05x. It is given that 0.05x = 1.25y.Step 2: From 0.05x = 1.25y, divide both sides by 0.05 to express x in terms of y. We get x = (1.25 / 0.05) * y = 25y.Step 3: Use the difference condition x - y = 96.Step 4: Substitute x = 25y into x - y = 96 to get 25y - y = 96.Step 5: Simplify to 24y = 96, so y = 96 / 24 = 4.Step 6: Substitute back to find x: x = 25y = 25 * 4 = 100.

Verification / Alternative check:
Check the percentage condition with x = 100 and y = 4. Five percent of 100 is 5. Now find 25 percent more than y: 25 percent of 4 is 1, so 4 plus 1 equals 5. The condition 0.05x = 1.25y is satisfied. The difference x - y equals 100 - 4 = 96, which matches the given difference. Therefore the pair (100, 4) is fully consistent with both conditions.

Why Other Options Are Wrong:
For 102 and 6, the difference is 96 but 5 percent of 102 is 5.1, which is not 25 percent more than 6. For 99 and 3, the difference is 96 but 5 percent of 99 is 4.95, while 25 percent more than 3 is 3.75. For 104 and 8, the difference is 96, but 5 percent of 104 is 5.2 and 25 percent more than 8 is 10. For 120 and 24, 5 percent of 120 is 6, but 1.25 times 24 is 30. Only 100 and 4 satisfy both conditions exactly.

Common Pitfalls:
Some students misinterpret 25 percent more than y as simply 0.25y instead of y plus 0.25y. Others might incorrectly treat percentages as whole numbers or forget to convert them to decimals. Another common error is forming incorrect equations, such as 0.05x = 0.25y, which would not reflect the phrase 25 percent more than y. Careful attention to wording and algebraic manipulation prevents these mistakes.

Final Answer:
The two numbers are 100 and 4, so the correct option is 100 and 4.