Mathematics Short Tricks/ MATHEMATICS AVERAGE TRICKS.

Mathematics Short Tricks/ MATHEMATICS AVERAGE TRICKS.

Mathematics Short Tricks for Averages

What is an Average?

The average (or mean) of a set of numbers is the sum of the values divided by the number of values.

Formula:

Average=Sum of all valuesNumber of values\text{Average} = \frac{\text{Sum of all values}}{\text{Number of values}}Average=Number of valuesSum of all values

Shortcut Tricks for Solving Average Problems:

1. Shortcut for Finding the New Average:

If the average of $n$ numbers is A, and a new value x is added, the new average becomes:

New Average=A×n+xn+1\text{New Average} = \frac{A \times n + x}{n + 1}New Average=n+1A×n+x​

Example:
Average of 5 numbers is 20. A new number 30 is added. What is the new average?

New Average=20×5+305+1=100+306=1306=21.67\text{New Average} = \frac{20 \times 5 + 30}{5 + 1} = \frac{100 + 30}{6} = \frac{130}{6} = 21.67New Average=5+120×5+30​=6100+30​=6130​=21.67

2. Shortcut for Removing an Element:

If a value x is removed from the set:

New Average=A×n−xn−1\text{New Average} = \frac{A \times n – x}{n – 1}New Average=n−1A×n−x

3. Shortcut for Consecutive Numbers:

For consecutive numbers like $1,2,3,4,5$, the average is the middle number.

Example:
Average of $4,5,6,7,8$ is 6 (middle number).

4. Shortcut for Evenly Spaced Numbers:

For evenly spaced numbers, the average is the average of the first and last numbers.

Example:
Average of $10,20,30,40$:

Average=10+402=25\text{Average} = \frac{10 + 40}{2} = 25Average=210+40​=25

5. Shortcut for Weighted Averages:

When different groups have different averages and sizes:

Weighted Average=(A1×n1)+(A2×n2)n1+n2\text{Weighted Average} = \frac{(A_1 \times n_1) + (A_2 \times n_2)}{n_1 + n_2}Weighted Average=n1​+n2​(A1​×n1​)+(A2​×n2​)​

Example:
Class A has an average of 60 for 20 students. Class B has an average of 70 for 30 students.

Weighted Average=(60×20)+(70×30)20+30=1200+210050=66\text{Weighted Average} = \frac{(60 \times 20) + (70 \times 30)}{20 + 30} = \frac{1200 + 2100}{50} = 66Weighted Average=20+30(60×20)+(70×30)​=501200+2100​=66

Quick Tips:

1. If all values increase or decrease by the same amount, the average also increases or decreases by that amount.
2. If the same value is added or removed from all elements, the average remains unchanged.

Practice Questions:

1. The average of 6 numbers is 15. If a new number 21 is added, what is the new average?
2. The average of 5 consecutive even numbers is 16. Find the numbers.
3. The average of 10 numbers is 25. If one number, 30, is excluded, what is the new average?

If you need help solving these or more tricks, feel free to ask!

Mathematics Short Tricks/ MATHEMATICS AVERAGE TRICKS.

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