Episode 8 — Aptitude and Reasoning / 8.8 — Average

8.8.b Tips, Tricks, and Shortcuts

1. The Deviation Method (Assumed Mean Method)

This is the single most powerful shortcut for average calculations. Instead of adding up all the numbers and dividing, you pick a reference value and work with deviations from it.

How It Works

Step 1: Choose an assumed mean (A) -- pick a number close to the middle.
Step 2: Calculate the deviation of each number from A.
Step 3: Find the average of the deviations.
Step 4: True Average = A + (average of deviations)

Formula:

Average = Assumed Mean + (Sum of deviations / n)

Worked Example

Find the average of: 78, 84, 82, 80, 76

Traditional method:

(78 + 84 + 82 + 80 + 76) / 5 = 400 / 5 = 80

Deviation method (assume A = 80):

Deviations: -2, +4, +2, 0, -4
Sum of deviations = -2 + 4 + 2 + 0 + (-4) = 0
Average = 80 + 0/5 = 80

Much faster -- no large addition needed!

Another Example

Find the average of: 235, 242, 238, 245, 230

Assume A = 240:

Deviations: -5, +2, -2, +5, -10
Sum of deviations = -5 + 2 - 2 + 5 - 10 = -10
Average = 240 + (-10)/5 = 240 - 2 = 238

When to Use It

Numbers are clustered closely together
Numbers are large (saves mental arithmetic)
You need to calculate quickly in exams

2. Shortcut for Consecutive Numbers

2.1 Average = Middle Number

For any set of consecutive numbers (or any AP), the average equals the middle term.

Odd count: Average = the exact middle number
Even count: Average = mean of the two middle numbers

Example:

Average of 4, 5, 6, 7, 8 = 6  (middle number)
Average of 3, 5, 7, 9 = (5 + 7)/2 = 6  (mean of two middle terms)

2.2 Average = (First + Last) / 2

This works for ALL arithmetic progressions.

Average of 10, 15, 20, 25, 30 = (10 + 30)/2 = 20

2.3 Quick Count Formula

Count of numbers from a to b (inclusive) = b - a + 1
Count of even numbers from a to b = (b - a)/2 + 1    (if both a, b are even)
Count of odd numbers from a to b  = (b - a)/2 + 1    (if both a, b are odd)
Count of multiples of k from a to b = floor(b/k) - ceil(a/k) + 1

3. Weighted Average Shortcuts

3.1 The Alligation / Ratio Method

When mixing two groups, the ratio of their sizes is inversely proportional to their distances from the combined average.

n1 / n2 = (A2 - Ac) / (Ac - A1)

Where:
  A1 = average of group 1
  A2 = average of group 2
  Ac = combined average
  n1 = size of group 1
  n2 = size of group 2

Visual representation (Alligation Cross):

       A1                A2
         \              /
          \            /
           Ac (combined)
          /            \
         /              \
    (A2 - Ac)      (Ac - A1)
    
    n1 : n2 = (A2 - Ac) : (Ac - A1)

Worked Example:

Two classes have averages 60 and 75. The combined average is 66. Find the ratio of students.

n1/n2 = (75 - 66) / (66 - 60) = 9/6 = 3/2

The classes are in the ratio 3:2.

3.2 Finding Combined Average Using Ratio

If you know the ratio n1:n2 and individual averages A1 and A2:

Combined Average = A1 + (A2 - A1) x n2 / (n1 + n2)

Or equivalently:

Combined Average = A1 + d x n2/(n1+n2)   where d = A2 - A1

Worked Example:

Boys and girls in a class are in ratio 3:2. Boys' average is 60, girls' average is 80. Find combined average.

Combined = 60 + (80 - 60) x 2/(3+2) = 60 + 20 x 2/5 = 60 + 8 = 68

4. Average Speed Trap (Harmonic Mean)

4.1 The Trap

The most common mistake in average speed problems is taking the arithmetic mean. Remember:

WRONG: Average Speed = (S1 + S2) / 2         (for same distance)
RIGHT: Average Speed = 2*S1*S2 / (S1 + S2)   (for same distance)

Quick rule: The average speed for equal distances is ALWAYS less than the arithmetic mean.

4.2 Quick Fraction Trick for Average Speed

For two speeds S1 and S2 with equal distance:

Average Speed = 2*S1*S2 / (S1 + S2)

If the speeds are simple numbers, use this mental shortcut:

Product of speeds = S1 x S2
Sum of speeds     = S1 + S2
Average speed     = 2 x Product / Sum

Example: Speeds 30 and 70

Average Speed = 2 x 30 x 70 / (30 + 70) = 4200 / 100 = 42

4.3 Three Equal Distances

Average Speed = 3*S1*S2*S3 / (S1*S2 + S2*S3 + S1*S3)

4.4 Ratio Shortcut

If speeds are in ratio a:b (and distances are equal):

Average Speed = 2ab / (a+b) x (unit speed)

Example: Speeds in ratio 2:3, actual speeds 40 and 60:

Average Speed = 2 x 2 x 3 / (2 + 3) x 20 = 12/5 x 20 = 48

5. The "New Average" Shortcut

5.1 When a New Person Joins

Instead of calculating old sum and new sum, use this direct formula:

New element = New Average + n x (Increase in average)

Where n = original count (before adding)

Worked Example:

Average of 5 numbers is 12. After adding a number, the average becomes 14. Find the new number.

New number = 14 + 5 x (14 - 12) = 14 + 5 x 2 = 14 + 10 = 24

Verification: Old sum = 60, new sum = 60 + 24 = 84, new average = 84/6 = 14. Correct.

5.2 When a Person Leaves

Removed element = Old Average - n_new x (Increase in average)

Or more generally:

Removed element = Old Average - (n-1) x (New Average - Old Average)

5.3 When One Element Is Replaced

If the average changes by d when element x is replaced by y:

y - x = n x d

Where:
  n = total count (unchanged)
  d = change in average (positive if increased)

Worked Example:

The average of 10 numbers is 80. One number is replaced and the average becomes 82. If the replaced number was 50, find the new number.

y - 50 = 10 x (82 - 80)
y - 50 = 20
y = 70

6. Age Problem Shortcuts

6.1 Current Average After t Years

New average = Old average + t

This works because EVERY person ages by t years.

6.2 New Member Doesn't Change Average

If a new person joins a group and the average remains the same:

Age of new person = Current average of the group

6.3 Replacement Keeping Average Same

If one person is replaced by another and the average doesn't change:

Age of new person = Age of replaced person

6.4 Marriage Problems

A common pattern:

Average age of a couple at the time of marriage was X years. After Y years, a baby is born. What is the average age when the baby is Z years old?

At the time of birth (Y years after marriage):
  Sum of couple's ages = 2X + 2Y
  Baby's age = 0
  Sum = 2X + 2Y
  Average = (2X + 2Y) / 3

After Z more years (baby is Z years old):
  Sum = 2X + 2Y + 2Z + Z = 2X + 2Y + 3Z
  Average = (2X + 2Y + 3Z) / 3

7. Exam Pattern Recognition

7.1 Pattern: "Average increases/decreases by x when one number is added/removed"

Use the "New Average" shortcut (Section 5).

7.2 Pattern: "Find the missing number given the average"

Missing number = Total sum (from average) - Sum of known numbers

7.3 Pattern: "Average of first k and last k numbers"

Often, you can find the middle elements using:
  Sum of all = Average x total count
  Sum of first k = Avg_first_k x k
  Sum of last k = Avg_last_k x k
  
  If overlapping: Sum of middle elements can be found
  If not overlapping: Sum of excluded elements = Total - First_k_sum - Last_k_sum

7.4 Pattern: "One wrong entry -- find correct average"

Correct sum = Wrong sum - Wrong value + Correct value
Correct average = Correct sum / n

Worked Example:

A teacher calculated the average of 20 students as 62. Later she found that one student's marks were read as 74 instead of 47. Find the correct average.

Wrong sum = 20 x 62 = 1240
Correct sum = 1240 - 74 + 47 = 1213
Correct average = 1213 / 20 = 60.65

7.5 Pattern: "Successive batting average problems"

After n innings: Old average = A
After (n+1) innings: New average = A + x (or A - x)

Runs in (n+1)th innings = (n+1)(A+x) - nA = A + (n+1)x

8. Mental Math Tricks

8.1 Average of Two Numbers

Average of a and b = a + (b - a)/2     (add half the difference to the smaller)

Example: Average of 37 and 53:

37 + (53-37)/2 = 37 + 8 = 45

8.2 Average of Equally Spaced Numbers

Just pick the middle one. No calculation needed.

Average of 5, 10, 15, 20, 25 = 15 (middle number)

8.3 Balancing Out Method

Look at how much each number is above or below a reference. If the "overs" and "unders" balance, the reference IS the average.

Example: Is 50 the average of 45, 48, 52, 53, 52?

Deviations from 50: -5, -2, +2, +3, +2
Sum = -5 - 2 + 2 + 3 + 2 = 0
Yes, 50 is the average.

8.4 Quick Estimation

For rough estimates, round all numbers to the nearest convenient value:

Average of 197, 203, 198, 206, 201
Estimate: all close to 200, so average is approximately 200.
Exact: deviations from 200 are -3, +3, -2, +6, +1 = +5
Average = 200 + 5/5 = 201

9. Common Traps and Mistakes

Trap 1: Average Speed is NOT Arithmetic Mean (for equal distances)

WRONG: Avg speed for 40 km/h and 60 km/h = 50 km/h
RIGHT: Avg speed = 2 x 40 x 60 / (40 + 60) = 48 km/h

Trap 2: Confusing "Average Increases by x" with "New Average is x"

Read the problem carefully:

"Average increases by 2" means New Average = Old Average + 2
"New average is 42" means the average is now exactly 42

Trap 3: Weighted vs Simple Average

If groups have DIFFERENT sizes, you MUST use weighted average.

WRONG: Avg of class A (avg 60) and class B (avg 80) = 70
RIGHT: Only 70 if both classes have the SAME number of students

Trap 4: Counting Errors

Numbers from 1 to 50: count = 50 (not 49)
Numbers from 10 to 50: count = 41 (not 40)
Even numbers from 2 to 100: count = 50
Multiples of 3 from 3 to 99: count = 33

Trap 5: Forgetting to Update Count

When adding/removing elements, the count changes!

Adding 1 element to n numbers: new count = n + 1 (NOT n)
Removing 1 element from n numbers: new count = n - 1 (NOT n)

Trap 6: Age Problems -- Time Passes for Everyone

If the problem says "after 5 years," every person in the group ages by 5. The total sum increases by 5n (not just 5).

10. Summary of Key Shortcuts

| Situation                        | Shortcut                                     |
|----------------------------------|----------------------------------------------|
| Large numbers, clustered         | Deviation method                             |
| Consecutive / AP numbers         | (First + Last) / 2                           |
| Average speed (equal distance)   | 2*S1*S2 / (S1 + S2)                         |
| New member joins, avg changes    | New = New_Avg + n * change_in_avg            |
| Replacement, avg changes         | y - x = n * change_in_avg                   |
| All elements +/- constant        | New avg = Old avg +/- constant               |
| All elements * constant          | New avg = Old avg * constant                 |
| Ratio of group sizes             | Alligation: n1:n2 = (A2-Ac):(Ac-A1)         |
| Wrong entry correction           | Correct_sum = Wrong_sum - wrong + correct    |
| After t years                    | Average increases by t                       |

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