The average is a value that represents the middle of a set of numbers. It is calculated by adding all the numbers together and then dividing by how many numbers there are.
Formula: Average = (Sum of quantities) / (Number of quantities)
Problem: If the ages of 4 students are 20 years, 22 years, 18 years, and 24 years, what is the average age?
Solution: Average Age = (20 + 22 + 18 + 24) / 4 = 84 / 4 = 21 years
Tip 1: If the average age of ‘n’ people decreases by ‘x’ years, the total age decreases by (n * x) years.
Tip 2: If the average age of ‘n’ people increases by ‘x’ years, the total age increases by (n * x) years.
Problem: The average age of 6 people increases by 2 years when one person aged 26 is replaced by a new person. What is the age of the new person?
Solution: Total age increased = 6 * 2 = 12 years. Age of new person = 26 + 12 = 38 years.
Average of first ‘n’ natural numbers: (n + 1) / 2
Average of first ‘n’ even numbers: (n + 1)
Average of first ‘n’ odd numbers: n
Average of consecutive numbers: (First Number + Last Number) / 2
Formula: Average Speed = Total Distance / Total Time
If you travel the same distance at two different speeds (x and y), the average speed is:
Average Speed = (2xy) / (x + y)
Problem: A person travels two equal distances at 10 km/hr and 30 km/hr. What is the average speed?
Solution: Average Speed = (2 * 10 * 30) / (10 + 30) = 600 / 40 = 15 km/hr.
Tip 3: For three equal distances at speeds A, B, and C, the average speed is:
Average Speed = 3ABC / (AB + BC + CA)
Tip 4: If you cover parts of the distance at different speeds, use:
Average Speed = xyz / (Pyz + Qxz + Rxy)
Average of squares of first ‘n’ natural numbers: [(n + 1)(2n + 1)] / 6
Average of cubes of first ‘n’ natural numbers: [n(n + 1)2] / 4
Tip 5: Always check if the numbers are equally spaced. If they are, the average is simply the middle number.
Tip 6: For quick calculations, memorize the averages of common number sets (e.g., first 10 natural numbers, first 10 even numbers, etc.).
1. What is the average of the first 10 natural numbers?
Answer: B) 5.5
2. The average of 5 numbers is 20. If one number is removed, the average becomes 18. What is the removed number?
Answer: C) 26
3. A person travels 60 km at 30 km/hr and another 60 km at 60 km/hr. What is the average speed?
Answer: A) 40 km/hr
4. The average of 7 consecutive numbers is 20. What is the largest number?
Answer: B) 23
5. The average of the first 5 even numbers is:
Answer: B) 6
6. If the average of 4 numbers is 15 and one number is 10, what is the average of the remaining 3 numbers?
Answer: B) 16.67
7. The average of the squares of the first 3 natural numbers is:
Answer: B) 4.67
8. The average of the cubes of the first 2 natural numbers is:
Answer: B) 4.5
9. The average age of 5 students is 20 years. If the age of the teacher is included, the average becomes 22 years. What is the age of the teacher?
Answer: B) 32
10. The average of 10 numbers is 50. If each number is multiplied by 2, what is the new average?
Answer: B) 100