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The average of 5 terms is 50. If the first 4 terms are 45, 42, 119 and 84, then what will be the last term?

Answer
VerifiedVerified
581.7k+ views
Hint: First, we will write down the definition of average and then we will show its conventional formula and then the formula by assuming some set of variable. After that we will write down the given information in the question and put it in the formula of average: $\overline{x}=\dfrac{\sum\limits_{i=1}^{n}{{{x}_{i}}}}{n}$ , we will assume the last number as x and then get the answer.

Complete step by step answer:
First of all let’s understand what is meant by the term average. So the average is defined as the mean value or the central value which is equal to the ratio of sum of number of given set of values to the total number of values present in the set. The average is represented by a bar over the given quantity for example, $\overline{x}$ . it is also denoted by the symbol: $\mu $
The formula to evaluate the average of given set of number is as follows:
$\text{Average}=\dfrac{\text{Sum of numbers}}{\text{Number of units}}$
Suppose, we have been given n number of values such as ${{x}_{1}},{{x}_{2}},{{x}_{3}}........{{x}_{n}}$ , then the average or the mean of the given data will be equal to:
$\overline{x}=\dfrac{\sum\limits_{i=1}^{n}{{{x}_{i}}}}{n}$
Now, let’s take our question, we are given the total number of terms as 5 and the average as 50.
The four numbers given are: 42, 45, 119 and 84. Let the fifth number be $x$ . So, we will put these values in the formula for average seen above:
\[\Rightarrow \overline{x}=\dfrac{\sum\limits_{i=1}^{n}{{{x}_{i}}}}{n}\Rightarrow 50=\dfrac{42+45+119+84+x}{5}\]
Now, we will solve the above equation to find the value of $x$ , so we have with us:
$\Rightarrow \overline{x}=\dfrac{\sum\limits_{i=1}^{n}{{{x}_{i}}}}{n}\Rightarrow 50=\dfrac{42+45+119+84+x}{5}$
We will first take 5 from right hand side to the left hand side and therefore we will get:
$\Rightarrow 250=290+x$
Now, we will again take 290 to the other side to get the value of x :
$\begin{align}
  & \Rightarrow 250-290=x \\
 & \Rightarrow x=-40 \\
\end{align}$
Therefore, the last term will be $-40$ .
Note:
Remember that in questions related to average if you want to find out that whether your obtained number is correct or not, always find average at the end if it is as same as the given average then your answer is right, like in the given question we obtained-40 and the other four terms are: 45, 42, 119, 84. We will now find the average of all the 5 terms:
$\overline{x}=\dfrac{45+42+119+84-40}{5}=\dfrac{250}{5}=50$
As we see that the average is same as the given average in the question our answer is correct.