# The average of 5 consecutive numbers is m. If the next three natural numbers are also included, how much more than m will the average of these 8 numbers be?

A. 2

B. 1

C. 1.4

D .1.5

Answer

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Hint- In order to solve this problem, we must choose the formula of Average along with the proper understanding of how we assume consecutive natural numbers.

We know that

An average refers to the arithmetic mean, the sum of the numbers divided by how many numbers are being averaged which is a single number itself taken as representative of a list of numbers.

Average = $\dfrac{{{\text{sum of all terms}}}}{{{\text{total numbers of terms}}}}$

Let the five consecutive natural numbers be x, x + 1, x + 2, x + 3, x + 4

Here x is a general number from which we took consecutive 5 natural number

Average of these 5 consecutive natural number will be

\[\dfrac{{\left( {{\text{x + x + 1 + x + 2 + x + 3 + x + 4}}} \right)}}{5}{\text{ = }}\left( {\dfrac{{{\text{5x + 10}}}}{5}} \right){\text{ = x + 2}}\]

In the question, it is given that Average of these 5 consecutive natural number is m

Hence we can write

⇒ \[{\text{x + 2}}\] = m

⇒ x = m – 2 …………...(1)

If the next three natural numbers are also included,

∴ The 8 consecutive numbers are x, x + 1, x + 2, x + 3, x + 4, x + 5, x + 6, x + 7

Average of these 8 numbers is

\[\dfrac{{\left( {{\text{x + x + 1 + x + 2 + x + 3 + x + 4 + x + 5 + x + 6 + x + 7}}} \right)}}{8} = {\text{ }}\left( {\dfrac{{8{\text{x + 28}}}}{8}} \right)\] ⇒ ${\text{x + }}\dfrac{{28}}{8}{\text{ = x + 3}}{\text{.5}}$

Assume Average of these 8 consecutive natural number is n

Hence we can write

⇒ ${\text{x + 3}}{\text{.5}}$= n …………………... (2)

On putting the value of x from equation (1) in equation (2)

⇒ ${\text{m - 2 + 3}}{\text{.5}}$= n

⇒ ${\text{m + 1}}{\text{.5 = n}}$

⇒ ${\text{m - n = 1}}{\text{.5}}$

⇒ Average of earlier 5 consecutive natural number (m) - Average of 8 consecutive natural number (n) = 1.5

In other words, we can say that m is 1.5 more than the average of 8 numbers or the average of numbers increased by 1.5.

Hence option D is correct.

Note- Whenever we face such type of problems the key concept we have to remember is that always remember the formula of Average which is stated above, then using this formula calculate average. Sometimes in any question you might have used it twice just like in the above question, first for 5 natural numbers then again for 8 natural numbers.

__Complete step-by-step answer:__We know that

An average refers to the arithmetic mean, the sum of the numbers divided by how many numbers are being averaged which is a single number itself taken as representative of a list of numbers.

Average = $\dfrac{{{\text{sum of all terms}}}}{{{\text{total numbers of terms}}}}$

Let the five consecutive natural numbers be x, x + 1, x + 2, x + 3, x + 4

Here x is a general number from which we took consecutive 5 natural number

Average of these 5 consecutive natural number will be

\[\dfrac{{\left( {{\text{x + x + 1 + x + 2 + x + 3 + x + 4}}} \right)}}{5}{\text{ = }}\left( {\dfrac{{{\text{5x + 10}}}}{5}} \right){\text{ = x + 2}}\]

In the question, it is given that Average of these 5 consecutive natural number is m

Hence we can write

⇒ \[{\text{x + 2}}\] = m

⇒ x = m – 2 …………...(1)

If the next three natural numbers are also included,

∴ The 8 consecutive numbers are x, x + 1, x + 2, x + 3, x + 4, x + 5, x + 6, x + 7

Average of these 8 numbers is

\[\dfrac{{\left( {{\text{x + x + 1 + x + 2 + x + 3 + x + 4 + x + 5 + x + 6 + x + 7}}} \right)}}{8} = {\text{ }}\left( {\dfrac{{8{\text{x + 28}}}}{8}} \right)\] ⇒ ${\text{x + }}\dfrac{{28}}{8}{\text{ = x + 3}}{\text{.5}}$

Assume Average of these 8 consecutive natural number is n

Hence we can write

⇒ ${\text{x + 3}}{\text{.5}}$= n …………………... (2)

On putting the value of x from equation (1) in equation (2)

⇒ ${\text{m - 2 + 3}}{\text{.5}}$= n

⇒ ${\text{m + 1}}{\text{.5 = n}}$

⇒ ${\text{m - n = 1}}{\text{.5}}$

⇒ Average of earlier 5 consecutive natural number (m) - Average of 8 consecutive natural number (n) = 1.5

In other words, we can say that m is 1.5 more than the average of 8 numbers or the average of numbers increased by 1.5.

Hence option D is correct.

Note- Whenever we face such type of problems the key concept we have to remember is that always remember the formula of Average which is stated above, then using this formula calculate average. Sometimes in any question you might have used it twice just like in the above question, first for 5 natural numbers then again for 8 natural numbers.

Last updated date: 05th Jun 2023

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