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.

SSC CGL Tier 1 Maths Short Tricks and Formulas

Here are some powerful short tricks for solving “Average” problems in Mathematics, useful for competitive exams like SSC, Banking, Railways, NEET, JEE (for general aptitude), and school-level exams:

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What is Average?

Formula:

Average=Sum of all observationsNumber of observations\text{Average} = \frac{\text{Sum of all observations}}{\text{Number of observations}}Average=Number of observationsSum of all observations​

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Basic Tricks and Shortcuts for Averages

1. Finding a New Average When a Value is Added or Removed

Trick:

New Average=Old Average+(NewValue−OldAverage)n+1\text{New Average} = \text{Old Average} + \frac{(New Value – Old Average)}{n+1}New Average=Old Average+n+1(NewValue−OldAverage)​

(where n is the number of items before addition)

Example:

Average of 5 numbers is 10. A new number 25 is added. Find the new average.
Solution:

New Average=10+(25−10)6=10+156=10+2.5=12.5\text{New Average} = 10 + \frac{(25 – 10)}{6} = 10 + \frac{15}{6} = 10 + 2.5 = 12.5New Average=10+6(25−10)​=10+615​=10+2.5=12.5

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2. Shortcut for Average Speed

If a person travels a certain distance at speed x and returns at speed y, then:

Average Speed=2xyx+y\text{Average Speed} = \frac{2xy}{x + y}Average Speed=x+y2xy​

Example:

A car travels to a city at 60 km/h and returns at 40 km/h. Find average speed.

Average Speed=2×60×4060+40=4800100=48 km/h\text{Average Speed} = \frac{2 \times 60 \times 40}{60 + 40} = \frac{4800}{100} = 48 \text{ km/h}Average Speed=60+402×60×40​=1004800​=48 km/h

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3. Shortcut for Finding a Missing Value

Trick:

$$\text{Missing value}=(\text{Average}\times \text{Total items})-(\text{Sum of known values})$$

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4. When Average of Subgroups is Known

Trick:
If group A (avg = x, size = m) and group B (avg = y, size = n), then:

Combined Average=(m×x)+(n×y)m+n\text{Combined Average} = \frac{(m \times x) + (n \times y)}{m + n}Combined Average=m+n(m×x)+(n×y)​

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5. Average of First n Natural Numbers

Average=n+12\text{Average} = \frac{n+1}{2}Average=2n+1​

6. Average of Squares of First n Natural Numbers

Average=(n+1)(2n+1)6\text{Average} = \frac{(n+1)(2n+1)}{6}Average=6(n+1)(2n+1)​

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Memory Tricks / Tips

• “When one goes up, the average may go up — depending on how far it is from the mean.”

• Keep shortcuts ready for average speed and number replacement.

• Always check whether the new value added is above or below the current average.

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Recommended Videos (Hindi + English Medium)

1. Maths Average Tricks by Abhinay Maths (Hindi)

2. Shortcuts for Averages by Dear Sir (Hindi)

3. Maths Tricks for Competitive Exams – Average

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If you’d like, I can create a printable PDF of these tricks or give you practice questions with answers. Would you like that?