Question

The mean of a certain number of items is 20. If an observation 25 is added to the data, the mean becomes 21. Find the no. of items in original data​

Answer 1

Given :-

The mean of a certain number of items is 20.

If an observation 25 is added to the data, then the mean becomes 21.

To find :-

The number of items in the original data.

Solution :-

Given that

Let the number of items in the original data be X

The mean of the items = 20

We know that

Mean = Sum of observations/Number of observations

=> 20 = Sum of observations / X

=> Sum of observations = 20×X

Sum of the observations = 20X

Added observation = 25

Then the number of observations = X+1

Sum of the observations = 20X+25

The new mean of observations = 21

=> (20X+25)/(X+1) = 21

=> 20X+25 = 21(X+1)

=> 20X+25 = 21X+21

=> 25-21 = 21X-20X

=> 4 = X

Therefore , X = 4

Answer :-

The number of items in the original data is 4

Check :-

The number of items = 4

Mean = 20

Sum of 4 items = 20×4 = 80

Added item = 25

Sum of all items = 80+25 = 105

Number of items = 4+1 = 5

New mean = 105/5 = 21

Verified the given relations in the given problem.

Used formulae:-

→ Mean = Sum of observations/Number of observations

Answer 2

Answer:

Given :-

• The mean of a certain number of items is 20.
• If an observation 25 is added to the data, the mean becomes 21.

To Find :-

• What is the number of items in original data.

Formula Used :-

$\clubsuit$ Mean Formula :

$\footnotesize \bigstar \: \: \sf\boxed{\bold{\pink{Mean =\: \dfrac{Sum\: Of\: Observations}{Total\: Number\: Of\: Observations}}}}\: \: \: \bigstar$

Solution :-

Let,

$\small \mapsto \bf Number\: of\: items\: in\: original\: data =\: x$

First, we have to find the sum of observation :

Given :

• Number of Observations = x
• Mean = 20

According to the question by using the formula we get,

$\footnotesize \implies \bf Mean =\: \dfrac{Sum\: Of\: Observations}{Total\: Number\: Of\: Observations}$

$\implies \sf 20 =\: \dfrac{Sum\: Of\: Observations}{x}$

By doing cross multiplication we get,

$\implies \sf Sum\: Of\: Observations =\: 20(x)$

$\implies \sf Sum\: Of\: Observations =\: 20 \times x$

$\implies \sf\bold{\purple{Sum\: Of\: Observations =\: 20x}}$

Now,

$\bigstar$ If 25 is added to the data, the mean becomes 21.

So, the total sum of observations is :

$\small \leadsto \sf\bold{\blue{Total\: Sum\: Of\: Observation =\: 20x + 25}}$

Now, again we have to find the number of items in original data :

Given :

• Sum Of Observations = 20x + 25
• Mean = 21
• Number Of Observations = x + 1

According to the question by using the formula we get,

$\footnotesize \dashrightarrow \bf Mean =\: \dfrac{Sum\: Of\: Observations}{Total\: Number\: Of\: Observations}$

$\dashrightarrow \sf 21 =\: \dfrac{20x + 25}{x + 1}$

By doing cross multiplication we get,

$\dashrightarrow \sf 21(x + 1) =\: 20x + 25$

$\dashrightarrow \sf 21x + 21 =\: 20x + 25$

$\dashrightarrow \sf 21x - 20x =\: 25 - 21$

$\dashrightarrow \sf\bold{\red{x =\: 4}}$

$\therefore$ The number of items in original data is 4 .