The difference between two numbers is 11 and one fifth of their sum is 9. What are the two numbers?

Introduction:
This is a classic linear equations problem involving two numbers. You are given their difference and a relation involving their sum. Solving such problems efficiently is crucial for many aptitude and reasoning exams.

Given Data / Assumptions:

• Let the two numbers be x and y, with x > y.
• Their difference is x − y = 11.
• One fifth of their sum is 9, so (x + y)/5 = 9.
• We must find the values of x and y.

Concept / Approach:
We are given two linear equations in two unknowns. The standard method is to solve this system by addition or substitution. First, we determine the sum from the second condition, then use the sum and difference to solve for each number.

Step-by-Step Solution:
Step 1: Use the sum condition.(x + y)/5 = 9 ⇒ x + y = 45.Step 2: Use the difference condition.x − y = 11.Step 3: Add the two equations.(x + y) + (x − y) = 45 + 11.2x = 56 ⇒ x = 28.Step 4: Substitute x back into one equation to find y.x + y = 45 ⇒ 28 + y = 45 ⇒ y = 17.

Verification / Alternative check:
Check both conditions: Difference = 28 − 17 = 11 (correct). Sum = 28 + 17 = 45; one fifth of 45 is 45/5 = 9 (also correct). Thus both original statements are satisfied by the pair (28, 17).

Why Other Options Are Wrong:
Each other option pair either does not have a difference of 11, or if it does, its sum does not give (sum)/5 = 9. For example, 31 and 20 have sum 51, and one fifth of 51 is 10.2, not 9.

Common Pitfalls:
Students sometimes mis-handle the fraction and set the sum equal to 9 instead of 45. Others may mix up sum and difference while solving. Writing equations clearly and proceeding stepwise prevents these errors.

Final Answer:
The two numbers are 28 and 17.