Average

Arithmetic mean, weighted averages

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Average is a fundamental mathematical concept that represents the central value of a set of numbers. In competitive exams, average problems test your ability to manipulate data sets, understand weighted distributions, and handle scenarios involving additions, removals, and replacements of values in a collection.

Key Formulas

Average = Sum of all values / Number of values\text{Average = Sum of all values / Number of values}
Sum of values = Average x Number of values\text{Sum of values = Average x Number of values}
If average of n items is A, and average of m items is B, then average of (n+m) items = (nA + mB) / (n + m)\text{If average of n items is A, and average of m items is B, then average of (n+m) items = (nA + mB) / (n + m)}
If each number is increased by k, new average = Old average + k\text{If each number is increased by k, new average = Old average + k}
If each number is multiplied by k, new average = Old average x k\text{If each number is multiplied by k, new average = Old average x k}
When a person joins a group: New average = (Sum of original values + New person’s value) / New count\text{When a person joins a group: New average = (Sum of original values + New person's value) / New count}
Average of first n natural numbers = (n + 1) / 2\text{Average of first n natural numbers = (n + 1) / 2}
Average of first n even numbers = n + 1\text{Average of first n even numbers = n + 1}
Average of first n odd numbers = n\text{Average of first n odd numbers = n}
Average of squares of first n natural numbers = (n + 1)(2n + 1) / 6\text{Average of squares of first n natural numbers = (n + 1)(2n + 1) / 6}

Key Concepts

Simple Arithmetic Mean

The basic average is calculated by dividing the sum of all observations by the number of observations. This is the most common form of average and applies to situations where all values have equal importance. For a set {a , a , ..., an}, the average = (a + a + ... + an) / n.

Weighted Average

When different values have different weights or importance, use weighted average. Formula: Weighted Average = (w x + w x + ... + wnxn) / (w + w + ... + wn), where w represents weights and x represents values. Common applications include calculating average marks with different subject weightings, average prices of mixed quantities, and portfolio returns.

Average of Consecutive Numbers

For any set of consecutive numbers (arithmetic progression), the average equals the middle term. If there are an even number of terms, the average is the average of the two middle terms. This property is useful for sequences like consecutive integers, even numbers, odd numbers, or multiples of a constant.

Adding/Removing Members

When a new member joins: New Sum = Old Average x Old Count + New Member's Value; New Average = New Sum / New Count. When a member leaves: New Sum = Old Average x Old Count - Departing Member's Value. The change in average can be found by: Change = (New Member's Value - Old Average) / New Count.

Replacement Scenarios

When one member is replaced by another: Effect on Sum = New Value - Old Value. Effect on Average = (New Value - Old Value) / Total Count. If the average increases by X after replacement, then: New Value = Old Value + (X x Total Count).

Age Problems with Averages

In age-related average problems, remember that everyone ages by the same amount over time. If the average age of a group is A today, it will be (A + n) after n years, assuming no members join or leave. When solving problems about past or future averages, track how each individual's age changes and recalculate accordingly.

Solved Examples

Problem 1:

The average weight of 8 persons increases by 2.5 kg when a new person comes in place of one of them weighing 65 kg. What might be the weight of the new person?

Solution:

Step 1: Understand that total increase in weight = 8 persons x 2.5 kg = 20 kg
Step 2: This total increase equals (Weight of new person - Weight of replaced person)
Step 3: Let weight of new person = W
Step 4: W - 65 = 20
Step 5: W = 65 + 20 = 85 kg
Answer: The new person weighs 85 kg.

Problem 2:

A batsman in his 17th innings makes a score of 85 and thereby increases his average by 3. What is his average after 17 innings?

Solution:

Step 1: Let average after 16 innings = A
Step 2: Total runs after 16 innings = 16A
Step 3: After 17th innings, new average = A + 3
Step 4: Total runs after 17 innings = 16A + 85
Step 5: Also, total after 17 innings = 17 x (A + 3) = 17A + 51
Step 6: Setting equal: 16A + 85 = 17A + 51
Step 7: 85 - 51 = 17A - 16A
Step 8: 34 = A
Step 9: Average after 17 innings = 34 + 3 = 37
Answer: 37 runs.

Problem 3:

The average of 11 results is 60. If the average of first six results is 58 and that of last six is 63, find the sixth result.

Solution:

Step 1: Sum of all 11 results = 11 x 60 = 660
Step 2: Sum of first 6 results = 6 x 58 = 348
Step 3: Sum of last 6 results = 6 x 63 = 378
Step 4: Notice the 6th result is counted in both first 6 and last 6
Step 5: Sum of first 6 + Sum of last 6 = 348 + 378 = 726
Step 6: This counts the 6th result twice and all others once
Step 7: So: 726 = Sum of all 11 + 6th result = 660 + 6th result
Step 8: 6th result = 726 - 660 = 66
Answer: The sixth result is 66.

Problem 4:

The average monthly salary of 12 employees is Rs. 1540. If the manager's salary is added, the average becomes Rs. 1650. What is the manager's salary?

Solution:

Step 1: Total salary of 12 employees = 12 x 1540 = Rs. 18,480
Step 2: Let manager's salary = M
Step 3: New total for 13 people = 18,480 + M
Step 4: New average = 1650, so: (18,480 + M) / 13 = 1650
Step 5: 18,480 + M = 1650 x 13 = 21,450
Step 6: M = 21,450 - 18,480 = 2,970
Alternative method:
Step 1: Average increased by 1650 - 1540 = 110
Step 2: Manager's salary must cover the old average + extra for all 13 people
Step 3: Manager's salary = 1650 + (13 x 110) = 1650 + 1430 = 2,970
Answer: Manager's salary is Rs. 2,970.

Tips & Tricks

  • Always verify your answer by working backwards - calculate the total from your answer and check if it matches the given conditions.
  • In replacement problems, remember: Change in Average = (New Value - Old Value) / Number of items. This shortcut saves time.
  • For consecutive number averages, use the middle term property to avoid lengthy calculations.
  • When dealing with overlapping sets (like the 6th result example), use the principle of inclusion-exclusion: Sum of parts = Total + Overlap.
  • In age problems with averages, note that everyone's age increases equally over time, so the average also increases by the same amount.
  • For weighted averages, clearly identify which values are the weights and which are the values being averaged - this is a common point of confusion.

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