The difference between two numbers is 25, and the smaller number is equal to one-sixth of the larger number. What is the value of the smaller number?

Introduction / Context:
This is a straightforward algebraic word problem involving two numbers and their relationship. Many aptitude tests include such questions to assess basic equation formulation skills. We are told the difference between two numbers, and also that the smaller is a fixed fraction of the larger. Using this information, we can set up a simple system of equations in two variables and solve for the unknown numbers. Here, we are particularly interested in finding the smaller number.

Given Data / Assumptions:

• There are two numbers, one larger and one smaller.
• The difference between the two numbers is 25.
• The smaller number is one-sixth of the larger number.
• We need to determine the smaller number.

Concept / Approach:
Let the larger number be L and the smaller number be S. We are given two conditions: L − S = 25 and S = (1/6) * L. By substituting S from the second equation into the first, we can form a single equation in L. Once we find L, we can use S = L / 6 to find the smaller number. This method is a standard approach for solving linear equations with two variables when we have exactly two relationships.

Step-by-Step Solution:
Step 1: Let the larger number be L and the smaller number be S. Step 2: From the problem, the difference between the numbers is 25, so L − S = 25. Step 3: Also, the smaller number is one-sixth of the larger number, so S = L / 6. Step 4: Substitute S = L / 6 into the equation L − S = 25. Step 5: This gives L − L / 6 = 25. Step 6: Combine like terms: L − L / 6 = (6L / 6 − L / 6) = 5L / 6. Step 7: So 5L / 6 = 25. Step 8: Multiply both sides by 6 to get 5L = 150. Step 9: Divide both sides by 5: L = 30. Step 10: Compute the smaller number S = L / 6 = 30 / 6 = 5.

Verification / Alternative check:
Check the conditions using L = 30 and S = 5. The difference L − S is 30 − 5 = 25, which matches the given difference. Also, S is one-sixth of L because 1/6 of 30 is 5. Since both statements in the problem are satisfied by these values, the solution is correct and the smaller number is 5.

Why Other Options Are Wrong:
Option (a) 10: If S were 10, L would be 60, giving a difference of 50, not 25.
Option (c) 50: This is too large to be the smaller number given the difference and ratio conditions.
Option (d) 25: If S were 25, L would be 150 with a difference of 125, not 25, so it fails both conditions.

Common Pitfalls:
A common mistake is to swap the roles of the larger and smaller numbers or incorrectly set up the equation as S − L = 25. Another frequent error is to mis-handle the fraction one-sixth when substituting into the equation. Carefully defining variables and systematically performing algebraic steps helps avoid such mistakes.

Final Answer:
The value of the smaller number is 5.