The number 96 is divided into two positive parts such that one-seventh of the first part is equal to one-ninth of the second part. What is the smaller of the two parts?

Introduction / Context:
This question is a classic example of a linear equation word problem involving the division of a total quantity into two parts with a relation between them. Such problems commonly appear in aptitude exams and help test whether candidates can convert a verbal condition into algebraic equations and solve them systematically. Here, the total is 96, and we are told that certain fractions of each part are equal. We must determine the smaller part using basic algebra.

Given Data / Assumptions:

• Total sum of two parts is 96.
• Let the two parts be x and y, both positive.
• x + y = 96.
• One-seventh of the first part equals one-ninth of the second part: x/7 = y/9.
• We are asked to find the smaller part.

Concept / Approach:
We will model the problem using two equations: one from the total (x + y = 96) and another from the fractional relationship (x/7 = y/9). The second equation allows us to express one variable in terms of the other. Substituting this expression into the first equation results in a single-variable linear equation that we can solve. Once x and y are found, we simply pick the smaller value.

Step-by-Step Solution:
Step 1: Let the two parts be x and y. Step 2: The total is 96, so x + y = 96. Step 3: From the given relation x/7 = y/9, cross multiply to obtain 9x = 7y. Step 4: Express y in terms of x: y = 9x / 7. Step 5: Substitute y = 9x / 7 into x + y = 96. Step 6: This gives x + 9x / 7 = 96. Step 7: Combine like terms: (7x + 9x) / 7 = 96, so 16x / 7 = 96. Step 8: Multiply both sides by 7: 16x = 96 * 7. Step 9: Compute 96 * 7 = 672, hence 16x = 672. Step 10: Divide both sides by 16 to get x = 672 / 16 = 42. Step 11: Find y from x + y = 96: y = 96 - 42 = 54. Step 12: The smaller part is x = 42.

Verification / Alternative check:
Check the fractional condition: one-seventh of the first part is 42 / 7 = 6, and one-ninth of the second part is 54 / 9 = 6. Since both are equal, the relationship is satisfied. Also, their sum is 42 + 54 = 96, matching the given total. This confirms that the values are correct and the smaller part is indeed 42.

Why Other Options Are Wrong:
Option (b) 54: This is the larger part, not the smaller one, although it still fits the equations when paired with 42.
Option (c) 46: Using 46 as one part does not give integer values that satisfy both conditions.
Option (d) 58: Similarly, 58 cannot produce consistent values for the fractional relationship and the total of 96.

Common Pitfalls:
Learners sometimes confuse which part is smaller or make errors while cross multiplying the equation x/7 = y/9. Another frequent mistake is mishandling the substitution into the total equation and performing arithmetic incorrectly. Carefully writing each step and rechecking the operations helps to avoid such issues.

Final Answer:
The smaller of the two parts is 42.