Question

The average of five numbers is $40$ and the average of another six numbers is $50$. The average of all numbers taken together is ________________.
A) $44.44$
B)$45.00$
C)$45.45$
D)$90.00$

Answer 1

Hint: First, we shall analyze the given information so that we are able to solve the problem. Here, we are given the average of two sets of numbers. That is, we are given that the average of five numbers is $40$ and the average of another six numbers is $50$. We are asked to calculate the average of all numbers taken together.
    We shall find the sum of the five numbers and the sum of another six numbers. Then we need to use the formula of average to obtain the required answer.
Formula to be used:
    The formula to calculate the average is as follows.
\[Average = \dfrac{{Sum{\text{ }}of{\text{ }}values}}{{Number{\text{ }}of{\text{ }}values}}\]

Complete step by step answer:
It is given that the average of five numbers is $40$and let the five numbers be \[a,b,c,d,e\] .
Now, using the formula\[Average = \dfrac{{Sum{\text{ }}of{\text{ }}values}}{{Number{\text{ }}of{\text{ }}values}}\], we have
\[Average{\text{ }}ofa,b,c,d,e = \dfrac{{a + b + c + d + e}}{5}\]
\[ \Rightarrow 40 = \dfrac{{a + b + c + d + e}}{5}\]
\[ \Rightarrow a + b + c + d + e = 40 \times 5\]
\[ \Rightarrow a + b + c + d + e = 200\] ………..$\left( 1 \right)$
Also, it is given that the average of six numbers is $50$ and let the five numbers be \[p,q,r,s,t,u\] .
Now, using the formula \[Average = \dfrac{{Sum{\text{ }}of{\text{ }}values}}{{Number{\text{ }}of{\text{ }}values}}\], we have
\[Average{\text{ }}ofp,q,r,s,t,u = \dfrac{{p + q + r + s + t + u}}{6}\]
\[ \Rightarrow 50 = \dfrac{{p + q + r + s + t + u}}{6}\]
\[ \Rightarrow p + q + r + s + t + u = 50 \times 6\]
\[ \Rightarrow p + q + r + s + t + u = 300\] ………..$\left( 2 \right)$
Now, we shall calculate the sum of all eleven numbers.
That is we need to add the equations$\left( 1 \right)$and$\left( 2 \right)$.
\[a + b + c + d + e + p + q + r + s + t + u = 200 + 300\]
\[a + b + c + d + e + p + q + r + s + t + u = 500\]
We are asked to calculate the average of all numbers taken together.
Now, using the formula \[Average = \dfrac{{Sum{\text{ }}of{\text{ }}values}}{{Number{\text{ }}of{\text{ }}values}}\] , we have
\[Average{\text{ }}ofa,b,c,d,e,p,q,r,s,t,u = \dfrac{{a + b + c + d + e + p + q + r + s + t + u}}{{11}}\]
Since \[a + b + c + d + e + p + q + r + s + t + u = 500\] , we get
\[ \Rightarrow Average{\text{ }}ofa,b,c,d,e,p,q,r,s,t,u = \dfrac{{500}}{{11}}\]
\[ \Rightarrow Average{\text{ }}ofa,b,c,d,e,p,q,r,s,t,u = 45.45\]
Hence, the average of all numbers is $45.45$.

So, the correct answer is “Option C”.

Note: Here we are given that the average of five numbers is $40$ and the average of another six numbers is $50$. We are asked to calculate the average of all numbers taken together. We have to add the sum of the five numbers with the sum of another six numbers to obtain the desired answer.