Shortcut Tricks for Average

By Mathlozy March 30, 2025

1. Quick Mental Math Techniques

A. Choosing a Base Value for Faster Calculation

Instead of summing all numbers directly, choose a base value close to the average and find deviations from it.

Example: Find the average of 48, 52, 46, 54, and 50.

Shortcut Trick: Choose 50 as a base value because it is near most numbers.
Now, calculate the deviations:

48 → -2, 52 → +2, 46 → -4, 54 → +4, 50 → 0
Total deviation = (-2 + 2 - 4 + 4 + 0) = 0
New sum = 50 × 5 + 0 = 250
Average = 250 ÷ 5 = 50

✅ Answer: 50 (calculated quickly without full addition).

2. Using the Difference Method for Faster Calculations

This method is useful when numbers are close to a particular reference point.

Example: Find the average of 98, 102, 104, 96, and 100.

Shortcut Trick: Choose 100 as the reference point.

98 → -2, 102 → +2, 104 → +4, 96 → -4, 100 → 0
Total deviation = (-2 + 2 + 4 - 4 + 0) = 0
Average = 100 + (0 ÷ 5) = 100

✅ Answer: 100 (without full addition).

3. Shortcut Formulas for Weighted Average

A weighted average is used when different values contribute differently to the total. The formula is:

Weighted Average = \frac{\sum (W_{i} \times X_{i})}{\sum W_{i}}

Where:

$X_{i}$ = Individual values
$W_{i}$ = Weights assigned to the values

Example 1: Exam Score Calculation

A student scores 80 in Math (weight: 3) and 90 in Science (weight: 2). Find the weighted average.

Weighted Average = \frac{(80 \times 3) + (90 \times 2)}{3 + 2} = \frac{240 + 180}{5} = \frac{420}{5} = 84

✅ Answer: 84

Example 2: Mixing Two Solutions

A 5-liter solution has 10% salt, and a 3-liter solution has 20% salt. Find the average percentage of salt in the mixture.

Weighted Average = \frac{(10 \times 5) + (20 \times 3)}{5 + 3} = \frac{50 + 60}{8} = \frac{110}{8} = 13.75 %

✅ Answer: 13.75%

MCQs on Shortcut Tricks for Average

1. Quick Mental Math

Find the average of 39, 41, 43, 45, 47 using a shortcut.
a) 41
b) 43
c) 45
d) 47

Answer: b) 43

2. Difference Method

Find the average of 195, 205, 200, 210, and 190.
a) 198
b) 200
c) 202
d) 204

Answer: b) 200

3. Weighted Average

A company has two teams: Team A with 20 employees earning ₹50,000 and Team B with 30 employees earning ₹60,000. What is the weighted average salary?
a) ₹55,000
b) ₹56,000
c) ₹57,000
d) ₹58,000

Answer: b) ₹56,000

4. Mixing Solutions

A 6-liter solution has 15% alcohol, and a 4-liter solution has 25% alcohol. Find the average alcohol percentage.
a) 17%
b) 20%
c) 22%
d) 23%

Answer: b) 20%

5. Impact of Adding a Number

The average of 5 numbers is 30. If a new number 40 is added, what is the new average?
a) 31
b) 32
c) 33
d) 34

Answer: b) 32

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