Question

The mean of 20 items of data is 5 and if each item is multiplied by 3 then the mean will be
(a) 5
(b) 10
(c) 15
(d) 20

Answer 1

Hint: To solve this question, we will first of all calculate the sum of all the terms by using the number of observations multiplied by the mean. Finally, we will calculate the new sum by multiplying the old sum by 3 and finally calculate the mean using the formula, \[\text{Mean}=\dfrac{\text{New Sum}}{\text{Number of terms}}.\]

Complete step by step answer:
The mean of 20 items of the given data is 5. Now each item is multiplied by 3. We are given the items as 20. Then the sum of all 20 terms can be obtained by multiplying 20 by mean 5. Hence, the sum of all 20 terms will be \[20\times 5=100.\] Let it be S.
\[S=100\]
Now, that all the terms are multiplied by 3. The new sum can be obtained by multiplying 3 to the old sum as all the numbers or terms are multiplied by 3. Let the new sum be S’.
\[\Rightarrow {{S}^{'}}=\text{new sum}=3\times 100=300\]
Now, the formula to calculate the mean in such case can be given as
\[\text{Mean}=\dfrac{\text{New Sum}}{\text{Number of terms}}\]
Now, the new sum S’ = 300 and the number of terms = 20. Substituting these values in the above given formula to calculate the mean.
\[\Rightarrow \text{Mean}=\dfrac{{{S}^{'}}}{20}\]
\[\Rightarrow \text{Mean}=\dfrac{300}{20}\]
\[\therefore \text{Mean}=15\]
Hence, the mean is given by 15.

Hence, the option (c) is the right answer.

Note:
If the question was given that the number 3 is added to each term of the given data, then the new sum would be obtained by multiplying 3 with the number of terms. Here, if 3 was added then, \[3\times 20=60\] and then the new mean would be \[\text{mean}=\dfrac{\text{new sum}}{20},\] the new sum = old sum + 60.