Question

If each of ‘n’ readings is increased by 5, then the arithmetic mean of new ‘n’ readings is equal to 10. If each of the original readings is multiplied by 3, then the arithmetic mean of new readings will be
(A)5
(B)10
(C)15
(D)20

Answer 1

Hint: Here for answering question of this type the arithmetic mean of $n$ numbers is found by using the mathematical formula given by$\dfrac{\sum\limits_{i=1}^{n}{{{a}_{i}}}}{n}$ . We will increase each of ‘n’ readings by 5, and we will use the given value of the arithmetic mean of new ‘n’ readings equal to 10 and find the original arithmetic mean. After that we will multiply each of the original readings by 3, and then we will get the arithmetic mean of the new readings using the derived arithmetic mean of the original readings.

Complete step-by-step answer:
It is given in the question that we have ‘n’ readings and when we increase each reading by 5 the arithmetic mean of new readings is 10. Let us assume the ‘n’ original readings as ${{a}_{1}},{{a}_{2}},..............,{{a}_{n}}$ and the arithmetic mean of original readings as $a$ this can be mathematically given as ${{a}_{x}}=\dfrac{\sum\limits_{i=1}^{n}{\left( {{a}_{i}}+5 \right)}}{n}\Rightarrow {{a}_{x}}=a=\dfrac{\sum\limits_{i=1}^{n}{{{a}_{i}}}}{n}$ .
Now let us add 5 for each reading ${{a}_{x}}=\dfrac{\left( {{a}_{1}}+5 \right)+\left( {{a}_{2}}+5 \right)+............+\left( {{a}_{n}}+5 \right)}{n}\Rightarrow {{a}_{x}}=\dfrac{\sum\limits_{i=1}^{n}{\left( {{a}_{i}}+5 \right)}}{n}$
By simplifying this we will have ${{a}_{x}}=\dfrac{\sum\limits_{i=1}^{n}{{{a}_{i}}}+\left( 5n \right)}{n}$ .
By further simplifying this we will have${{a}_{x}}=\dfrac{\sum\limits_{i=1}^{n}{{{a}_{i}}}}{n}+5$ $\Rightarrow {{a}_{x}}=a+5$ .
This is given as 10 in the question this can be mathematically given as ${{a}_{x}}=10$.
Now by simplifying we will have the value of $a$. For this we will simplify ${{a}_{x}}=a+5\Rightarrow 10=a+5\Rightarrow 5=a$ .
We have $a=5$ . The arithmetic mean of the original readings is 5.
Now we need to find the arithmetic mean when we multiply the original readings by 3.
This is mathematically given as ${{a}_{y}}=\dfrac{3{{a}_{1}}+3{{a}_{2}}+........+3{{a}_{n}}}{n}\Rightarrow {{a}_{y}}=\dfrac{\sum\limits_{i=1}^{n}{3{{a}_{i}}}}{n}$ .
By taking 3 as common in the summation we will have ${{a}_{y}}=3\dfrac{\sum\limits_{i=1}^{n}{{{a}_{i}}}}{n}$ .
By performing the simplifications we have ${{a}_{y}}=3a$ . Since we know the value of $a$ as 10 by substituting it we will have ${{a}_{y}}=3\left( 5 \right)\Rightarrow {{a}_{y}}=15$ .
So, we have the new arithmetic mean as 15.

So, the correct answer is “Option C”.

Note: While answering this type of question we should perform the calculations carefully. If we perform a wrong calculation such as ${{a}_{y}}=3\left( 5 \right)\Rightarrow {{a}_{y}}=10$ .We will end up with a wrong conclusion that the value of the arithmetic mean of the new readings as 10. Also, take care to read the given question properly and then form the relations. There is a possibility that we might misinterpret the questions as numbers being multiplied by 5 and terms being added by 3.