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Average: In this, we are going to discuss about basic average and weighted average which is an important concept

Definition: An average or arithmetic mean of given data is the sum of the given observations divided by the number of observations.

Average (A) = (Sum of given observation(S)) / (Number of observation(N))

• The average is also called the mean.

E.g. Find the average of 10, 15, 20, 25, and 30.

Step-1 First calculate the sum of all the quantities (S)

⇒ 10 + 15 + 20 + 25 + 30 = 100

Step-2 Now count the number of quantities (N)

That is five in this case

Step-3 According to formula divide the sum by number of quantities

A = S / N

A = 100 / 5 = 20

So we got 20 as the average of all the five quantities.

Let’s first deal with some properties of averages to get some clarity of thought on averages

• The average of a given data is less than the greatest observation and greater than the smallest observation of the given data.

We saw in our last example that average (20) is between 10 and 30.

• If all the numbers get increased by “a”, then their average must be increased by “a”.
• Similarly, If all the numbers get decreased by “a”, then their average must be decreased by “a”.
• If all the numbers are multiplied by “a”, then their average must be multiplied by “a”.
• Similarly, If all the numbers are divided by “a”, then their average must be divided by “a”.

E.g. The weight of A is 60 kg, Weight of B is 45 kg, and the average weight of A, B, and C is 53 kg, find the weight of C.

Let the weight of C be “x”

Step-1 the sum of all the weights is

⇒  60 + 45 + x

Step-2 the number of quantities is 3

and as given the average of them is 53

Step-3 so by the formula of averages

⇒  53 = (60 + 45 + x) / 3

⇒  53 × 3 = 60 + 45 + x

⇒  159 = 105 + x

⇒  x = 159 – 105

⇒  x = 54 kg(Ans.)

E.g. A, B, C, D, and E are the five electronic shops in the Naza market. Which have 20,30,60,80 and 50 T.V. sets with them respectively if each of them has imported 12 new T.V. sets then the average of T.V. sets is?

First, find out the average without considering the imports of T.V.

According to formula

⇒  A = S / N

⇒  A = (20 + 30 + 60 + 80 + 50) / 5

⇒  A = 240 / 5

⇒  A = 48

As we learned in the properties of the average new average would be = 48 + 12 = 60 Ans. (Ans.)

E.g. The average of four numbers is 80. The first number is 1 / 3 of the sum of three numbers. What would be the first number?

By rearranging the formula of averages we found that

Sum of all the observations = Average × number of observations

⇒  S = A × N

So according to formula

⇒  S = 80 × 4

⇒  320

Now let the sum of three numbers excluding the first number be “g”

So sum of all the four numbers = g + g / 3

⇒  g + g / 3 = 320

⇒  4g / 3 = 320

⇒  g = 320 × 3 / 4

⇒  g = 240

So the first number would be “g / 3″ = 240 / 3 = 80(Ans.)

Weighted Average: When the average of groups or sets, instead of individuals, having different numbers of elements is being calculated then it is calculated then it is called weighted average.

Let’s discuss the weighted average for two groups:

Group 1           Group 2           Group (1 + 2)

No. of items                m         n          m + n

Average                       a          b          A

Sum of all items                      ma       nb        ma + nb

So by the formula of averages = (Sum of all items of the group (1 + 2)) / (No.of items of the group (1 + 2))

⇒  A = (ma + nb) / (m + n)

E.g. The average weight of 17 girls is 20kg and the average weight of 23 boys is 22kg. Find the average weight of the class.

So there are two groups of people boys and girls that’s why we will use weighted average concept,

By putting the values in the formula

⇒  A = (17 × 20 + 22 × 23) / (17 + 23)

⇒  A = (340 + 506) / 40

⇒  A = 846 / 40

⇒  A = 21.15kg (Ans.)

E.g. In Plutarch enterprise, 70% of the employees are marketers, 20% are engineers, and the rest are the managers. Marketers make an average salary of 50,000rs. a year and engineers make an average of 80,000rs. what is the average salary of managers if the average for all employees is also 80,000rs.?

Let the average salary of managers be “x”

Multiply each group average by the percent expressed as a decimal:

Marketers = 0.70 × 50,000 = 35,000

Engineers = 0.20 × 80,000 = 16,000

Managers = 0.10 × x = 0.1x

These should add up to the average for all employees

⇒  35,000 + 16,000 + 0.1x = 80,000

⇒  0.1x = 80,000 – 51,000

⇒  0.1x = 29,000

⇒  x = 290,000 rs.(Ans.)

replacement and number system-based questions on averages.

Let’s first discuss a replacement.

Addition & removal of item & change in average:

As we know the average = (Sum of given observation(S)) / (Number of observation(N))

So when the number of items increases or decreases, the average would change.

• Addition of item: when items are added then the average also changes, let’s try to find out the change in average;

E.g. The average age of 40 students in a class is 15 years. When 10 new students are admitted, the average increased by 0.2 years. Find the average age of new students.

According to the question, the average age of 40 students is 15

So total age = 40 × 15 = 600 Years

When due to addition the average increases by 0.2

So new average = 15 + 0.2 = 15.2

But the total students also changes due to addition

Total students = 40 + 10 = 50

Now, the sum of ages is = 50 × 15.2

⇒ 760

So the total increment in ages would be 760 – 600 = 160

This is due to those 10 students, sot their average would be

Average of new students added = 160 / 10 = 16 Years (Ans.)

• Removal of item:- when items are removed then the average also changes, let’s try to find out the change in average;

E.g. The average salary of 15 teachers is 4500 rs per month. Three teachers left the school and the average salary of the remaining teachers dropped by 175 rs. Find the total salary of the teachers who left the school.

The average salary of 15 teachers were 4500

So the total salary of all of them would be = 15 × 4500 = 67500

Now 3 teachers left the school so total teachers would be = 15 – 3 = 12

Now the average salary also drops by 175 so the new average would be

⇒ 4500 – 175 = 4325

So the total salary of 12 teachers would be

⇒ 4325 × 12 = 51900

So the difference would be 67500 – 51900 = 15600 rs. (Ans.)

These are bit tedious tasks and time taking in exam so let’s learn about short-tricks.

Let the average of N items = A

Now, ‘n’ new items are added and the average increases or decreases by ‘x’ then

Average of new items added = A ± (1 + N / n)x

Use “+” when average increases and “–” when average decreases.

When n = 1

Average of new items added = A ± (1 + N)x

E.g. The average age of 40 students in a class is 15 years. When 10 new students are admitted, the average increased by 0.2 years. Find the average age of new students.

Here

A = 15

N = 40

n = 10

x = 0.2

So by according to formula

Average of new items added = 15 ± (1 + 40 / 10)0.2

We will use ‘”+”‘ sign here because the average increases

⇒ Average = 15 + (1 + 4)0.2

⇒ Average = 15 + 5 × 0.2

⇒ Average = 15 + 1

⇒ Average = 16 Years (Ans.)

In the case of removing items by replacing -N / n in place of N / n in addition’s case so we got the formula for removal

Average of new items removed = A ± (1 – N / n)x

Use “+” in case of increase on average and use “-” in case of a decrease in average.

E.g. The average salary of 15 teachers is 4500rs per month. Three teachers left the school and the average salary of the remaining teachers dropped by 175rs. Find the total salary of the teachers who left the school.

Here

A = 4500

N = 15

n = 3

x = 175

Average of new items removed = 4500 ± (1 – 15 / 3)175

Use “-” because there is a decrease in average

Average of new items removed = 4500 – (1 – 5)175

Average of new items removed = 4500 + 4 × 175

⇒ 4500 + 700

⇒ 5200

Sum of all the salaries of teachers who left = 5200 × 3 = 15600 rs. (Ans.)

• Replacement: Sometimes, when a number of items are removed and these are replaced by the same amount of different items so the quantity remains unchanged.

The average increases/decreases by “x”, let there be N items then,

Sum of new items are added-Sum of removed items = ± Nx

“-” when average decreases and “+” when average increases.

E.g. when a man weighing 80kg is replaced by another man in a group of five people, the average weight decreases by 3kg, what is the weight of the new man?

Weight of a new man – the weight of the removed man = -Nx

Weight of new man – 80 = -5 × 3

Weight of new man = 80 – 15

⇒ 65kg (Ans.)

E.g. The average weight of 29 students in a class 48kg, if the weight of the teacher is included the average weight rises by 500g, find the weight of the teacher.

Here only one item is added and the increase in average is 0.5kg

A = 48

N = 29

n = 1

x = 0.5

According to our formula

Weight of teacher = A + (N + 1)x

Weight of teacher = 48 + (29 + 1)0.5

Weight of teacher = 48 + 15 = 63kg (Ans.)

We discussed all type of replacement question which are essential for your SBI PO exam point of view, let’s discuss some formula about averages of numbers.

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 squares of first n natural numbers = (n + 1)(2n + 1) / 6

Average of cubes of first n natural numbers = [n(n + 1)2] / 4

E.g. What is the average of numbers from 1 to 100?

According to formula

⇒ A = (n + 1) / 2

⇒ A = (100 + 1) / 2

⇒ A = 101 / 2 = 50.5

E.g. what is the average of first 5 even numbers?

As we know first 5 even numbers are 2,4,6,8 and 10

Their sum is 2 + 4 + 6 + 8 + 10 = 30

Average = 30 / 5 = 6

According to the formula: n + 1

⇒ 5 + 1 = 6 (Ans.)