Tips And Tricks And Shortcuts for Averages

Tips and Tricks and Shortcuts to Calculate average of a Series:

• Average  of ‘n’ consecutive Natural Numbers =
  \mathbf{\frac{n+1}{2}}
• Average of the square of consecutive n natural numbers =
  \mathbf{\frac{(n+1)(2n+1)}{6}}
• Average of cubes of consecutive n natural numbers =
  \mathbf{\frac{n\times (n+1)^{2}}{4}}
• Average of n consecutive even numbers = (n+1)
• Average of consecutive even numbers till n =
  \mathbf{\frac{n}{2}+1}
• Average of n consecutive odd numbers = n
• Average of consecutive odd numbers till n =
  \mathbf{\frac{n+1}{2}}
• Sum of 1st n even consecutive natural numbers is =  n(n + 1)
• Sum of 1st n odd consecutive natural numbers is = \mathbf{n^{2}}

Type 1: Tips to Solve Problems on Weights and Ages Average

Question 1: When a new man joins a group of 5 people after Replacing a man, their average age increases by 2 kg. If he replaces a man weighing 40 kg, how much does he weigh?

Solution :  Increased weight = (5 x 2) = 10

Weight of the new man = (40 +10) = 50 kg

Question 2: The average age of 4 monkeys is 20 years. The youngest monkey is eight years old. When he was born, the average age of the remaining monkeys was N years. Calculate the average age of the monkeys excluding the youngest monkey?

Solution:     The average age of monkey = 20 years.

Sum of all their ages = 20 × 4 = 80 year’s

Sum of their ages excluding the youngest monkey = 80 – 8 = 72 year’s

The average age of the remaining monkey = \frac{72}{3} = 24 years

Type 2: Tips to Solve Problems on Average Marks and Scores

Question 1 : If the Marks of Each student is 70 and it is increased by 20, then the average of the class will be

Solution:  Average of the class = 70+20 = 90

Solution: Average of n consecutive natural numbers is = \frac{(n+1)}{2}

7 = \frac{(n+1)}{2}

n+1= 14

n = 13