The ratio of two numbers is 6 : 11 and their difference is 65. What is the smaller of the two numbers?

Introduction / Context:
This ratio and difference question asks us to find the actual values of two numbers when their ratio and difference are known. It is a standard type in aptitude tests and is solved using a simple linear equation in one variable derived from the ratio representation of the numbers.

Given Data / Assumptions:

• Ratio of two numbers = 6 : 11.
• Difference between the two numbers = 65.
• We need to find the smaller of the two numbers.

Concept / Approach:
We represent the two numbers as 6k and 11k, where k is a positive constant. The difference between them is (11k - 6k) = 5k. We are told this difference equals 65, so we can solve for k. Once k is known, we compute each number and identify the smaller one as 6k.

Step-by-Step Solution:

Verification / Alternative check:
Check the ratio: 78 : 143. Divide both terms by 13 to get 6 : 11, which matches the given ratio. Check the difference: 143 - 78 = 65, which matches the given difference. So the calculations are consistent and confirm the smaller number as 78.

Why Other Options Are Wrong:

• 58, 65 and 52 do not satisfy both the given ratio and difference when paired with any corresponding second number.
• For example, if the smaller number were 58, the larger would be 58 * 11/6 (from the ratio), which is not an integer and does not produce a difference of 65.

Common Pitfalls:
Some students misinterpret the ratio and directly subtract 11 - 6 = 5 and then equate it to the difference, forgetting to multiply by the common factor k. Another error is to mistakenly identify the larger number instead of the smaller one when answering. Keeping track that 6k is the smaller and 11k is the larger is important.

Final Answer:
The smaller number is 78.