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Shortcut Tricks for Average

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  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 = ...
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Problems on Averages

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  1. Basic Problems: Finding the Average of Given Numbers The average (arithmetic mean) of a set of numbers is calculated using the formula: Average = Sum of all values Total number of values Average = Total number of values Sum of all values ​ Example: Find the average of  12, 18, 24, and 30 . Solution: Sum of numbers =  12 + 18 + 24 + 30 = 84 Total numbers =  4 Average = 84 4 = 21 Average = 4 84 ​ = 21 So, the  average is 21 . 2. Missing Number Problems: Finding the Missing Value When Average is Given When the average of a group is known, and one number is missing, we use the formula: Total Sum = Average × Total numbers Total Sum = Average × Total numbers Example: The average of  4 numbers  is  20 . Three of the numbers are  18, 22, and 19 . Find the missing number. Solution: Total sum =  Average × Total numbers Total Sum = 20 × 4 = 80 Total Sum =...
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Types of Averages

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  Types of Averages Averages are used to represent a central value of a dataset. The three main types of averages are: Arithmetic Mean (Mean) Median Mode 1. Arithmetic Mean (Mean) The  arithmetic mean  is the most common type of average, calculated by dividing the sum of all values by the number of values. Formula: Mean = Sum of all values Total number of values Mean = Total number of values Sum of all values ​ Example: Find the mean of  10, 20, and 30 . Solution: Mean = 10 + 20 + 30 3 = 60 3 = 20 Mean = 3 10 + 20 + 30 ​ = 3 60 ​ = 20 So, the  mean is 20 . 2. Median The  median  is the middle value when numbers are arranged in ascending or descending order. If there is an even number of values, the median is the average of the two middle numbers. Example 1 (Odd set of numbers): Find the median of  3, 5, 7, 9, 11 . Solution: Sorted order:  3, 5, 7, 9, 11 The middle number is  7 ,...
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Types of Averages

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  Types of Averages Averages are used to represent a central value of a dataset. The three main types of averages are: Arithmetic Mean (Mean) Median Mode 1. Arithmetic Mean (Mean) The  arithmetic mean  is the most common type of average, calculated by dividing the sum of all values by the number of values. Formula: Mean = Sum of all values Total number of values Mean = Total number of values Sum of all values ​ Example: Find the mean of  10, 20, and 30 . Solution: Mean = 10 + 20 + 30 3 = 60 3 = 20 Mean = 3 10 + 20 + 30 ​ = 3 60 ​ = 20 So, the  mean is 20 . 2. Median The  median  is the middle value when numbers are arranged in ascending or descending order. If there is an even number of values, the median is the average of the two middle numbers. Example 1 (Odd set of numbers): Find the median of  3, 5, 7, 9, 11 . Solution: Sorted order:  3, 5, 7, 9, 11 The middle number is  7 ,...
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Introduction to Average

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  Definition of Average: An  average  is a value that represents the central or typical value of a set of numbers. It is calculated by dividing the sum of all values in the set by the total number of values. Formula for Average: Average = Sum of all values Total number of values Average = Total number of values Sum of all values ​ Examples of Average: Example 1: Finding the Average of Numbers Find the average of  4, 8, and 12 . Solution: Sum = 4 + 8 + 12 = 24 Sum = 4 + 8 + 12 = 24 Total numbers = 3 Total numbers = 3 Average = 24 3 = 8 Average = 3 24 ​ = 8 So, the average is  8 . Example 2: Finding the Average Speed A person travels  60 km in 2 hours  and then  90 km in 3 hours . Find the average speed. Solution: Total Distance = 60 + 90 = 150  km Total Distance = 60 + 90 = 150  km Total Time = 2 + 3 = 5  hours Total Time = 2 + 3 = 5  hours...
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Special Numbers and Their Properties

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  1. Perfect Numbers Definition A  perfect number  is a  positive integer  that is equal to the sum of its  proper divisors  (excluding itself). Examples of Perfect Numbers 6  → Proper divisors:  1, 2, 3 1 + 2 + 3 = 6  ✅ (Perfect Number) 28  → Proper divisors:  1, 2, 4, 7, 14 1 + 2 + 4 + 7 + 14 = 28  ✅ (Perfect Number) 496  → Proper divisors:  1, 2, 4, 8, 16, 31, 62, 124, 248 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496  ✅ (Perfect Number) MCQs on Perfect Numbers 1. Which of the following is a perfect number? a) 10 b) 15 c) 28 d) 30 ✅  Answer: c) 28 2. The smallest perfect number is: a) 5 b) 6 c) 7 d) 9 ✅  Answer: b) 6 2. Armstrong Numbers Definition An  Armstrong number  (also called a  narcissistic number ) is a number that is equal to the sum of its  digits raised to the power of the number of digits . Examples of Armstrong Numbers 153  → 3-digit number 1³ + 5³ + 3³ = ...
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