- LPU MCA Semester 1
- PEA515
- Unit 1
PEA515 · Unit 1: Number system and averages
Average calculations and inclusion and exclusion
30 practice questions · in the MTE and ETE syllabus
Key points
- Sum = average x count
- Replacing one member shifts the average by the difference divided by n
- Union = A + B - both
- Remember the group that is in none of the sets
Average = sum of values / number of values, so sum = average x count. For an evenly spaced list (such as consecutive numbers) the average is (first + last)/2.
Changing the group: if one member is replaced, the average changes by (new - old)/n. If a member joins and the average rises by d, the newcomer exceeds the old average by d x (n + 1). Combining groups: new average = (n1 x a1 + n2 x a2)/(n1 + n2).
Inclusion and exclusion counts people in overlapping sets without double counting:
- Two sets: |A union B| = |A| + |B| - |A and B|.
- Three sets: |A union B union C| = |A| + |B| + |C| - |A and B| - |B and C| - |A and C| + |A and B and C|.
Example: in a class of 50, 30 take maths and 25 take physics, and 10 take neither. Then 40 take at least one, so both = 30 + 25 - 40 = 15.
Exam trap: forgetting the "neither" group, or adding the pairwise overlaps instead of subtracting them.
Try two questions
Find the average of the following 7 values: 68, 67, 78, 65, 78, 60, 102.
- 68
- 64.75
- 74
- 81
Show the answer
C. Sum = 518, divided by 7 gives 74.
Why not A: Reports the middle value instead of the arithmetic mean.
Why not B: Divides the total by one more than the number of values.
Why not D: Uses the midpoint of the extremes.
The average score of 10 players is 23 runs. When one more player joins, the average becomes 22 runs. What is the score of the new one?
- 22
- 13
- -11
- 12
Show the answer
D. New total = 11 x 22 = 242; old total = 230; difference 12.
Why not A: Reports the new average rather than the new value.
Why not B: Multiplies the change in average by n instead of n + 1.
Why not C: Uses only the total change and ignores the old average.