1. Basic Average

What is the question?

These questions usually ask:

“The average of some numbers is given. Find the total.”

Or:

“The total is given. Find the average.”

Idea

Average means:

Average = Total / Number of values

Therefore:

Total = Average × Number of values

Example

Question: The average marks of 8 students is 65. Find their total marks.

Solution

Average = 65
Number  = 8

Total = 65 × 8
      = 520

Answer: 520

Remember

Average × Number = Total

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2. A Person Joins or Leaves a Group

What is the question?

These questions say:

“The average age/marks/weight of a group changes when one person joins or leaves. Find that person’s value.”

Idea

Convert both averages into totals.

Old total = Old average × Old number

New total = New average × New number

Then find the difference.

Example

Question: The average age of 5 people is 30 years. One person joins the group and the average becomes 28 years. Find the age of the new person.

Step 1: Find the old total

Old total = 5 × 30
          = 150

Step 2: Find the new total

There are now 6 people:

New total = 6 × 28
          = 168

Step 3: Find the new person’s age

168 - 150 = 18

Answer: 18 years

Visual

BEFORE                    AFTER

5 people                  6 people
Average = 30              Average = 28

Total = 150               Total = 168
                              |
                              ↓
                         New person
                            18

Shortcut

If n people have average A, and one person joins:

New person's value
= (n + 1)(New average) - n(Old average)

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3. A Person Leaves a Group

What is the question?

“The average of a group is known. One person leaves and the average changes. Find the age/weight/marks of the person who left.”

Idea

Again, find the two totals.

Person who left
= Old total - New total

Example

Question: The average age of 6 people is 40 years. One person leaves and the average of the remaining 5 people becomes 38 years. Find the age of the person who left.

Step 1: Old total

6 × 40 = 240

Step 2: New total

5 × 38 = 190

Step 3: Person who left

240 - 190 = 50

Answer: 50 years

Remember

Person joins  → New total - Old total
Person leaves → Old total - New total

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4. Replacement of One Person

What is the question?

These questions say:

“One person is replaced by another and the average changes. Find the value of the new person.”

Idea

When the number of people does not change, use:

Change in total
= Number of people × Change in average

Then apply that change to the person being replaced.

Example

Question: The average weight of 10 students is 45 kg. A student weighing 50 kg is replaced by another student. The new average is 44 kg. Find the new student’s weight.

Step 1: Find the change in total

Old average = 45
New average = 44

Change = 44 - 45
       = -1

For 10 students:

Change in total = 10 × (-1)
                = -10 kg

So the total weight decreased by 10 kg.

Step 2: Find the new student’s weight

The old student weighed 50 kg.

New student = 50 - 10
            = 40 kg

Answer: 40 kg

Shortcut

New person's value
= Old person's value
  + Number × (New average - Old average)

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5. Correcting a Wrongly Read Number

What is the question?

These questions say:

“The average was calculated using an incorrect value. Find the correct average.”

Idea

First find the wrong total, correct it, then calculate the average again.

Example

Question: The average of 25 numbers was calculated as 60. Later it was discovered that 86 was entered instead of 56. Find the correct average.

Step 1: Find the incorrect total

Wrong total = 25 × 60
            = 1500

Step 2: Correct the total

86 was wrongly added, but the correct value is 56.

Correct total
= 1500 - 86 + 56
= 1470

Step 3: Find the correct average

1470 ÷ 25 = 58.8

Answer: 58.8

Visual

Wrong total
   1500
    |
    | remove wrong value 86
    ↓
   1414
    |
    | add correct value 56
    ↓
   1470
    |
    ÷ 25
    ↓
   58.8

Remember

Remove the wrong value → add the correct value → divide.

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6. Combined Average

What is the question?

These questions say:

“Two classes/groups have different numbers of people and different averages. Find the average of the combined group.”

Idea

Do not simply average the two averages unless the groups have equal sizes.

First find each group’s total.

Total 1 = Number 1 × Average 1

Total 2 = Number 2 × Average 2

Then:

Combined average
= (Total 1 + Total 2) / (Number 1 + Number 2)

Example

Question: Class A has 20 students with an average mark of 75. Class B has 30 students with an average mark of 65. Find the combined average.

Step 1: Find Class A’s total

20 × 75 = 1500

Step 2: Find Class B’s total

30 × 65 = 1950

Step 3: Combine

Total marks = 1500 + 1950
            = 3450

Total students = 20 + 30
               = 50

Therefore:

Average = 3450 ÷ 50
        = 69

Answer: 69

Important trap

Do not do:

(75 + 65) / 2 = 70

because the two classes have different numbers of students.

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7. Average of Consecutive Numbers

What is the question?

These questions involve numbers such as:

10, 11, 12, 13, 14

or:

20, 22, 24, 26, 28

and ask for their average.

Idea

For equally spaced numbers:

Average = (First + Last) / 2

Example

Question: Find the average of:

12, 15, 18, 21, 24

First number:

12

Last number:

24

Therefore:

Average = (12 + 24) / 2
        = 36 / 2
        = 18

Answer: 18

Visual

12   15   18   21   24
|--------------------|
First              Last

Average = (First + Last) / 2

This is a very useful shortcut in aptitude exams.

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8. Average of Consecutive Integers

What is the question?

“Find the average of the first 20 natural numbers.”

Or:

“The average of 7 consecutive integers is 24. Find the integers.”

Idea

For consecutive numbers, the average is the middle number when there is a middle number.

Example

Question: The average of 5 consecutive integers is 32. Find the integers.

Since 5 numbers have one middle number:

30, 31, 32, 33, 34

Answer: 30, 31, 32, 33, 34

Shortcut

If there are n consecutive numbers:

Average = (First + Last) / 2

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9. Average Increase / Decrease

What is the question?

“The average of a group increases by 3 when a new score is added. Find the new score.”

Idea

When one value is added, the increase in the total is:

Number of old values × Increase in average

Then use the old total.

Example

Question: The average marks of 10 students is 60. A new student joins and the average becomes 62. Find the new student’s marks.

Step 1: Old total

10 × 60 = 600

Step 2: New total

There are now 11 students:

11 × 62 = 682

Step 3: New student’s marks

682 - 600 = 82

Answer: 82

Shortcut

New value
= New total - Old total

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10. Average of Two Groups — Missing Average

What is the question?

“The combined average is known. The average and size of one group are known. Find the average of the other group.”

Idea

Work with totals, not averages.

Example

Question: 20 students have an average of 60 marks. Another 30 students join, and the average of all 50 students becomes 66. Find the average of the 30 new students.

Step 1: First group’s total

20 × 60 = 1200

Step 2: Combined total

50 × 66 = 3300

Step 3: New group’s total

3300 - 1200 = 2100

Step 4: New group’s average

2100 ÷ 30 = 70

Answer: 70 marks

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11. Batting / Sports Average

What is the question?

Usually:

“A batsman has an average of X after N innings. How many runs must he score to increase his average to Y?”

Idea

Convert the average into total runs.

Current runs = Current average × Current innings

Then find the target total.

Target runs = Target average × New innings

Finally:

Required runs = Target total - Current total

Example

Question: A batsman has an average of 40 after 15 innings. How many runs must he score in the next innings to make his average 45?

Step 1: Current total

15 × 40 = 600

Step 2: Target total

After the next innings:

16 × 45 = 720

Step 3: Required runs

720 - 600 = 120

Answer: 120 runs

Remember

Average × innings = total runs.

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12. Average Speed — Important Related Variant

What is the question?

These questions ask:

“A vehicle travels at different speeds for different distances/times. Find the average speed.”

Idea

Average speed is not usually the simple average of speeds.

Use:

Average speed = Total distance / Total time

Example

Question: A car travels 60 km at 30 km/h and another 60 km at 60 km/h. Find the average speed.

Step 1: Find the time for each part

Time = Distance / Speed

First part:

60 / 30 = 2 hours

Second part:

60 / 60 = 1 hour

Step 2: Find total distance

60 + 60 = 120 km

Step 3: Find total time

2 + 1 = 3 hours

Therefore:

Average speed = 120 / 3
              = 40 km/h

Answer: 40 km/h

Important

Do not simply do:

(30 + 60) / 2 = 45

That would be wrong here.

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13. Advanced: Average of Averages / Weighted Average

This is useful when a question gives several groups with different sizes.

What is the question?

“Three classes have different numbers of students and different average marks. Find the overall average.”

Idea

Repeat the same process:

Each group's total
= Number × Average

Then:

Overall average
= Total of all groups / Total number of people

Example

Question:

Class A: 10 students, average 50
Class B: 20 students, average 60
Class C: 30 students, average 70

Find the overall average.

Solution

A total = 10 × 50 = 500
B total = 20 × 60 = 1200
C total = 30 × 70 = 2100

Total:

500 + 1200 + 2100 = 3800

Students:

10 + 20 + 30 = 60

Therefore:

Overall average = 3800 / 60
                = 63⅓

Answer: 63⅓

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