Averages

Average is defined as the ratio of sum of the quantities to the number of quantities. Average or mean is said to be a measure of central tendency.

For example, average of the first five natural numbers 1, 2, 3, 4 and 5.

Sum = 1 + 2 + 3 + 4 + 5 = 15

Average = 15/5 = 3

Basic Formula

Average = Sum of quantities / Number of quantities

Sum of quantities =Average × Number of quantities

Number of quantities = Sum of quantities / Average

Example 1: If a person with age 45 joins a group of 5 persons with an average age of 39 years. What will be the new average age of the group?

Total age is 45 + 5 × 39 = 240. There are 6 persons now.

So, the average = 240/6 = 40

Since 45 is 6 more than 39, by joining the new person, the total will increase by 6 and so the average will increase by 1.

So, the average is 39 + 1 = 40.

Example 2: Two students with marks 50 and 54 leave class VIII A and move to class VIII B. As a result the average marks of the class VIII A fall from 48 to 46. How many students were there initially in the class VIII A?

The average of all the students of class VIII A is 46, excluding these two students. They have 4 and 8 marks more than 46. So with the addition of these two students, 12 marks are adding more, and hence the average is increasing 2. There should be 6 students in that class including these two. This is the initial number of students.

Example 3: Total temperature for the month of September is 840°C. If the average temperature of that month is 28°C, find how many days is the month of September.

Number of days in the month of September = Total temperature / Average temperature

= 840/28

= 30 days

Weighted Mean

If some body asks you to calculate the combined average marks of both the sections of class X A and X B, when both sections have 60% and 70% average marks respectively? Then, your answer will be 65% but this is wrong as you do not know the total number of students in each sections. So to calculate weighted average you have to know the number of students in both the sections.

Let N₁, N₂, be the weights attached to variable values X₁, X₂, respectively, then the weighted arithmetic mean is

Weighted Mean = (N₁X₁ + N₂X₂)/(N₁ + N₂)

The weighted average is just like a see-saw. More the ratio of a quantity more will be the inclination of the average from mid value towards the value with more ratios.

Example 4: The average marks of 30 students in a section of class X are 20 while that of 20 students of second section is 30. Find the average marks for the entire class X?

You can do the question by using both the Simple average and weighted average method.

Simple average = Sum of marks of all students / Total number of students

= (20 × 30 + 30 × 20)/(30 + 20) = 24

By the weighted mean method, Average = 3/5 × 20 + 2/5 × 30

= 12 + 12 = 24

Facts About Average

If each number is increased or decreased by a certain quantity n, then the mean also increases or decreases by the same quantity.
If each number is multiplied or divided by a certain quantity n, then the mean also gets multiplied or divided by the same quantity.
If the same value is added to half of the quantities and same value is subtracted from other half quantities then there will not be any change in the final value of the average.

Average Speed

1. Average Speed = Total distance covered / Total time taken

2. When distance is equal

Suppose a man covers a certain distance at x kmph and an equal distance at y kmph, then the average speed during the whole journey is (2xy/x+y) kmph. This is the harmonic mean of the two velocities.

3. When time is equal

The average speed is the algebraic mean of two velocities.

Average speed = (x + y)/2

Example 5: The average of 10 consecutive numbers starting from 21 is

The average is simply the middle number, which is the average of 5th and 6th numbers i.e, 25 & 26 i.e. 25.5.

Example 6: There are two classes A and B, each has 20 students. The average weight of class A is 38 and that of class B is 40. X and Y are two students of classes A and B respectively. If they interchange their classes, then the average weight of both the classes will be equal. If weight of X is 30 kg, what is the weight of Y?

Total weight of class A = 38 × 20, and class B = 40 × 20, if X & Y are interchanged, then the total weights of both the classes are equal.

⇒ 38 × 20 – x + y = 40 × 20 – y + x

⇒ 2(y – x) = 2 × 20

⇒ y = x + 20 = 50

Alternatively, since both the classes have same number of students, after interchange, the average of each class will be 39. Since the average of class A is increasing by 1, the total should increase by 20.

So, x must be replaced by ‘y’, who must be 20 years elder to ‘x’. So, y must be 50 years old.

Example 7: The average weight of 10 apples is 0.4 kg. If the heaviest and lightest apples are taken out, the average is 0.41 kg. If the lightest apple weights 0.2 kg, what is the weight of heaviest apple?

Total weight of the apples is 0.4 × 10 = 4 kg.

Weight of apples except heaviest and lightest = 0.41 × 8 = 3.28 kg

∴ Heaviest + lightest = 4 – 3.28 = 0.72 kg

It is given lightest = 0.2 kg

∴ Heaviest is 0.72 – 0.2 = 0.52 kg

Example 8: While finding the average of ‘9’ consecutive numbers starting from X; a student interchanged the digits of second number by mistake and got the average which is 8 more than the actual. What is X?

Since the average is 8 more than the actual, the second no will increase by 72 (9 × 8) by interchanging the digits.

If ab is the second number, then

10a + b + 72 = 10b + a

⇒ 9(b – a) = 72

∴ b – a = 8

The possible number ab is 19. Since the second number is 19.

The first number is 18.

∴ X = 18

Example 9: There are 30 consecutive numbers. What is the difference between the averages of first and last 10 numbers?

The average of first 10 numbers is the average of 5th & 6th numbers. Whereas the average of last 10 numbers is the average of 25th & 26th numbers.

Since all are consecutive numbers, 25th number is 20 more than the 5th number. We can say that the average of last 10 numbers is 20 more than the average of first 10 numbers. So, the required answer is 20.