# Problems Based on Average | Free & Printable Average Problems with Solutions PDF Worksheet

Problems Based on Average Worksheet PDF aid students to solve all types of easy to complex average problems with ease. Practicing from average word problems & solutions pdf can excel in the concepts of Statistics like mean, arithmetic mean, median, mode, etc.

Also, you can simply rely on the answers solved here and check out the mistakes that you have made while practicing. Access and download the Average practice questions worksheet with solutions PDF easily without any charge and prepare any time anywhere.

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## How To Solve Average Word Problems?

The calculation of the Average or Arithmetic Mean or Mean of a number of quantities can be done by using the formula ie., the Average of a number of quantities of the same units is equal to their sum divided by the number of those quantities.

Arithmetic average is utilized for all averages such as Average income, average profit, average age, average marks, etc. However, to solve the sum of observations, they must be in the same unit. Check out the below sections to know some quick tips & tricks to solve and practice with the questions based on average.

### Easy Average Problems Tricks | Common Average Aptitude Formulas

The list of easy tricks to solve average problems pdf is given here for your reference. Simply memorize them daily and practice well from average questions for competitive exams with solutions pdf.

1. Average = $$\frac { Sum of quantities }{ Number of quantities }$$
2. Sum of quantities = Average * Number of quantities
3. The average of first n natural numbers is $$\frac { (n +1) }{ 2 }$$
4. The average of the squares of first n natural numbers is $$\frac { (n +1)(2n+1 ) }{ 6 }$$
5. The average of cubes of first n natural numbers is $$\frac { n(n+1)2 }{ 4 }$$
6. The average of first n odd numbers is given by $$\frac { (last odd number +1) }{ 2 }$$
7. The average of first n even numbers is given by $$\frac { (last even number + 2) }{ 2 }$$
8. The average of squares of first n consecutive even numbers is $$\frac { 2(n+1)(2n+1) }{ 3 }$$
9. The average of squares of consecutive even numbers till n is $$\frac { (n+1)(n+2) }{ 3 }$$
10. The average of squares of consecutive odd numbers till n is $$\frac { n(n+2) }{ 3 }$$
11. If the average of n consecutive numbers is m, then the difference between the smallest and the largest number is 2(m-1)
12. If the number of quantities in two groups be n1 and n2 and their average is x and y respectively, the combined average is $$\frac { (n1x+n2y) }{ (n1+ n2) }$$
13. The average of n quantities is equal to x. When a quantity is removed, the average becomes y. The value of the removed quantity is n(x-y) + y
14. The average of n quantities is equal to x. When a quantity is added, the average becomes y. The value of the new quantity is n(y-x) + y

### Word Problems on Average | Average Practice Questions and Answers

Example 1:
The average mark of 50 students in a class is 70. Out of these 50 students, if the average mark of 25 students is 65, what is the average mark of the remaining 25 students?
Solution:
Average of 50 students = 70
Therefore, total marks of all 50 students = 50 x 70 =  3500
Average of 25 students = 65
Therefore, total marks of 25 students = 25 x 65 = 1625
So, the total marks of remaining 25 students = 3500 – 1625 = 1875
Hence, the average of remaining 25 students out of 50 students = 1875 / 30 = 62.

Example 2:
The mean of 15 observations is 45. If the mean of the first observations is 10 and that of the last 15 observations is 45, find the 15th observation.
Solution:
Given that,
Mean of the first 15 observations = 10
Then, Sum of the first 15 observations = 10 x 15 = 150
Mean of the last 15 observations = 45
Then, Sum of the last 15 observations = 45 x 15 = 675
Mean of the 15 observations = 45
Then, Sum of all 15 observations = 45 x 15 = 675
Now, find the 15th observation
= 150 + 675 – 675
= 150
Therefore, the 15th observation is 150.

Example 3:
The mean of 7 numbers is 35. If one of the numbers is excluded, the mean gets reduced by 5. Find the excluded number.
Solution:
Given Mean of 7 numbers = 35
Sum of the 7 numbers = 35 x 7 = 245
Mean of remaining 6 numbers = 35-5 = 30
Sum of remaining 6 numbers = 30 x 6 = 180
Now, find the excluded number to do that;
Excluded number = (sum of the given 7 numbers) – (sum of the remaining 6 numbers)
= 245 – 180
= 65
Hence, the excluded number is 65.

Example 4:
Find the average of 2, 4, 6, 8, 10, 12?
Solution:
Here, you will know that the given numbers are even in numbers ie., a total of 6 numbers. These kinds of questions are solved by looking at the difference in numbers (the constant difference is 2).

If the number were given odd in numbers means 2, 4, 6, 8, 10, then the answers will always be the middle term ie., 6. However, if the given numbers are in even count ie, six numbers like 2, 4, 6, 8, 10, 12 then the answer is always Sum of middle terms/Sum of opposite terms divided by 2 or Sum of quantities divided by total quantities.
ie., (2+12)/2 or (4+10)/2 or (6+8)/2. (the average is 7)
The answer is the same always.
As per the average formula = Sum of quantities divided by total quantities
= 2+4+6+8+10+12 / 6
= 42 / 6
= 7
Hence, the average of 2, 4, 6, 8, 10, 12 is 7.

### Some More Example Questions & Problems Based on Average

Example 5:
The average height of a group of four boys is 5.6 inches. The individual height (in inches) of three of them are 5.2, 5.4, 5.5. What is the height of the fourth boy?
Solution:
Given Average height of 4 boys = 5.6 inches
Total height of 4 boys = (5.6 x 4) in = 22.4 in
Total height of 3 boys = (5.2 + 5.4 + 5.5) in = 16.1 in
Height of the 4th boy = (total height of 4 boys) – (total height of 3 boys)
= (22.4 – 16.1) in
= 6.3 in
Hence, the height of the fourth boy is 6.3 in

Example 6:
The average of 5 numbers is 30. If each number is increased by 2, what will the new average be?
Solution:
Given that the average of 5 numbers = 30
Sum of the 5 numbers = 30 x 5 = 150
If each number is increased by 2, the total increase = 2 x 5 = 10
Then, the new sum = 150+10 = 160 and new Average = 160/10 = 16
Hence, the new average is 16.

Example 7:
The average age of four boys is 25 years and their ages are in proportion 2:4:6:8. What is the age in years of the youngest boy?
Solution:
Given, Average age of four boys = 25 years
Sum of four boys = 25 x 4 = 100
Ages = 2x + 4x + 6x + 8x
2x + 4x + 6x + 8x = 100
20x = 100
x = 100/20
x = 5 years
Age of the youngest boy = 2x years = 2(5) years = 10 years.