When 18 is included in a list of natural numbers, the average of these numbers…
2023
When 18 is included in a list of natural numbers, the average of these numbers is increased by 2. When 8 is
included in this new list of natural numbers, the average of these numbers in the new list remains the same. If
there are N natural numbers in the original list of numbers and their average is X, find the value of N and X,
respectively.
Answer: A. 5, 6 — ConceptFor N numbers with average X, the total is NX. After adding a value a, the new average is (NX + a)/(N + 1). If another value is added without changing…
- A.
5, 6
- B.
6, 8
- C.
8, 6
- D.
6, 5
Show answer & explanation
Correct answer: A
Concept
For N numbers with average X, the total is NX. After adding a value a, the new average is (NX + a)/(N + 1). If another value is added without changing an average, that added value must equal the unchanged average.
Application
Let the original sum be NX. Adding 18 raises the average by 2, so (NX + 18)/(N + 1) = X + 2.
Multiply by N + 1: NX + 18 = (N + 1)(X + 2) = NX + X + 2N + 2. Hence X + 2N = 16.
After 18 is added, the average is X + 2. Adding 8 leaves this average unchanged, so the newly added value equals that average: X + 2 = 8. Thus X = 6.
Substitute X = 6 into X + 2N = 16: 6 + 2N = 16, so 2N = 10 and N = 5.
Cross-check
The original total is 5 × 6 = 30. With 18 added, the average is 48/6 = 8, which is 2 more than 6. With 8 then added, the average is 56/7 = 8, so it remains unchanged. Therefore N = 5 and X = 6.