Question

If the arithmetic mean of x, x+2, x+5, x+9 and x + 10 is 10, then find x...

Plz tell ASAP​

Answer 1

$\frac{(x) + (x+2) + (x+5) + (x+9 ) + (x + 10)}{5} = 10//= >\frac{(x) + (x+2) + (x+5) + (x+9 ) + (x + 10)}{5} = 10 \\ \\ = > \frac{5x + 26}{5} = 10 \\ \\ = > 5x + 26 = 50 \\ \\ = > 5x = 24 \\ \\ = > x = \frac{24}{5} \\ \\ = > x = 4.8$

Here I have divides whole expression by 5 because there are total 5 terms here..

Now solve for x, you will get the answer...

Answer 2

$\\\;\underbrace{\underline{\sf{Understanding\;the\;Question\;:-}}}$

Here the Concept of Mean of the numbers has been used. We know that Arithmetic Mean is the ratio of sum of the terms by number of terms. Using this, firstly we can apply the values in the equation and then find the answer.

Let's do it !!

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★ Formula Used :-

$\\\;\boxed{\sf{\pink{Arithmetic\;Mean\;=\;\bf{\dfrac{Sum\;of\;Terms}{Number\;of\;Terms}}}}}$

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★ Solution :-

Given,

» Arithmetic Mean of the terms = A.M. = 10

» First Term = a₁ = x

» Second Term = a₂ = x + 2

» Third Term = a₃ = x + 5

» Fourth Term = a₄ = x + 9

» Fifth Term = a₅ = x + 10

» Number of Terms = n = 5

Now using the formula of Arithmetic Mean, we get,

$\\\;\displaystyle{\sf{:\rightarrow\;\;Arithmetic\;Mean\;=\;\bf{\gray{\dfrac{Sum\;of\;Terms}{Number\;of\;Terms}}}}}$

$\\\;\displaystyle{\sf{:\Longrightarrow\;\;A.\:M.\;=\;\bf{\dfrac{\blue{a_{1}\;+\;a_{2}\;+\;a_{3}\;+\;a_{4}\;+\;a_{5}}}{\orange{5}}}}}$

By applying values, we get,

$\\\;\displaystyle{\sf{:\Longrightarrow\;\;10\;=\;\bf{\dfrac{x\:+\:(x\:+\:2)\:+\:(x\:+\:5)\:+\:(x\:+\:9)\:+\:(x\:+\:10)}{5}}}}$

$\\\;\displaystyle{\sf{:\Longrightarrow\;\;10\;=\;\bf{\dfrac{x\:+\:x\:+\:2\:+\:x\:+\:5\:+\:x\:+\:9\:+\:x\:+\:10}{5}}}}$

$\\\;\displaystyle{\sf{:\Longrightarrow\;\;10\;=\;\bf{\dfrac{5x\;+\;26}{5}}}}$

$\\\;\displaystyle{\sf{:\Longrightarrow\;\;5x\;+\;26\;=\;\bf{5\;\times\;10}}}$

$\\\;\displaystyle{\sf{:\Longrightarrow\;\;5x\;+\;26\;=\;\bf{50}}}$

$\\\;\displaystyle{\sf{:\Longrightarrow\;\;5x\;=\;\bf{50\;-\;26}}}$

$\\\;\displaystyle{\sf{:\Longrightarrow\;\;5x\;=\;\bf{24}}}$

$\\\;\displaystyle{\sf{:\Longrightarrow\;\;x\;=\;\bf{\dfrac{24}{5}}}}$

$\\\;\displaystyle{\bf{:\Longrightarrow\;\;x\;=\;\bf{\red{4.8}}}}$

$\\\;\underline{\boxed{\tt{Hence,\;\;x\;\;=\;\bf{\purple{4.8}}}}}$

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★ Verification :-

In order to verify our answer, we simply need to apply the value we got into the equation. Then,

$\\\;\tt{:\leadsto\;\;Arithmetic\;Mean\;=\;\dfrac{Sum\;of\;Terms}{Number\;of\;Terms}}$

$\\\;\tt{:\leadsto\;\;A.\:M.\;=\;\dfrac{a_{1}\;+\;a_{2}\;+\;a_{3}\;+\;a_{4}\;+\;a_{5}}{5}}$

$\\\;\tt{:\leadsto\;\;\dfrac{x\:+\:x\:+\:2\:+\:x\:+\:5\:+\:x\:+\:9\:+\:x\:+\:10}{5}\;=\;A.M.}$

$\\\;\tt{:\leadsto\;\;\dfrac{4.8\:+\:4.8\:+\:2\:+\:4.8\:+\:5\:+\:4.8\:+\:9\:+\:4.8\:+\:10}{5}\;=\;A.M.}$

$\\\;\tt{:\leadsto\;\;\dfrac{50}{5}\;=\;A.M.}$

$\\\;\tt{\green{:\leadsto\;\;10\;=\;A.M.}}$

And, we know that given A.M. = 10.

This means our answer is correct.

Hence, Verified.

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★ More Formulas to know :-

$\\\;\sf{\mapsto\;\;Geometric\;Mean\;=\;\sqrt{ab}}$

$\\\;\sf{\mapsto\;\;Harmonic\;Mean\;=\;\dfrac{n}{\bigg(\dfrac{1}{x_{1}}\bigg)\;+\;\bigg(\dfrac{1}{x_{2}}\bigg)\;+\;\bigg(\dfrac{1}{x_{3}}\bigg)\;+\;.....\;+\;\bigg(\dfrac{1}{x_{n}}\bigg)}}$