Question

The mean of 16 numbers is 48. If each number is divided by 4 and diminished by 3, then the new mean is:
A. \[12\]
B. \[48\]
C. \[52\]
D. \[9\]

Answer 1

Hint: Here we will use the concept of mean. The average of the total number of given observations is called arithmetic mean or mean. We will use the fact that dividing each number by 4 is the same as dividing the original mean by 4. Similarly, diminishing each number by 3 is the same as diminishing the whole mean by 3. Using these two facts we will find the new meaning.

Formula used:
We will use the formula of Mean \[ = \] Sum of observations \[ \div \] Total number of observations.

Complete step by step solution:
The mean of 16 numbers \[ = 48\]
Let the numbers be \[{x_{1,}}{x_2},{x_3},.....{x_{16}}\].
Now using the formula of mean, Mean \[ = \] Sum of observations \[ \div \] Total number of observations, we can write above equation as,
\[ \dfrac{{{x_1} + {x_2} + {x_3} + ..... + {x_{16}}}}{{16}} = 48\]
If each number is divided by 4, then the numbers will become \[\dfrac{{{x_1}}}{4},\dfrac{{{x_2}}}{4},\dfrac{{{x_3}}}{4},.....\dfrac{{{x_{16}}}}{4}\].
Therefore, new mean will be:
\[ \Rightarrow \dfrac{{\dfrac{{{x_1}}}{4} + \dfrac{{{x_2}}}{4} + \dfrac{{{x_3}}}{4} + ..... + \dfrac{{{x_{16}}}}{4}}}{{16}} = 48\]
Taking \[\dfrac{1}{4}\] common from the numerator, we get
\[ \Rightarrow \dfrac{{{x_1} + {x_2} + {x_3} + ..... + {x_{16}}}}{{4 \times 16}} = 48\]
Or this can be written as:
Original mean \[ \div \] 4.
Hence, this proves that in a mean, if each number is divided by a constant then we can simply divide the original given mean by the same constant.
Hence, if each number is divided by 4, then,
New mean \[ = \] Original mean \[ \div \] 4
\[ \Rightarrow \] New mean\[ = \dfrac{{48}}{4}\]
Dividing the terms, we get
\[ \Rightarrow \]New mean \[ = 12\]
Now, each number is diminished by 3,
Hence, new numbers will become:
\[\dfrac{{{x_1}}}{4} - 3,\dfrac{{{x_2}}}{4} - 3,\dfrac{{{x_3}}}{4} - 3,.....\dfrac{{{x_{16}}}}{4} - 3\]
Therefore, New mean will be:
\[ \Rightarrow \dfrac{{\dfrac{{{x_1}}}{4} - 3 + \dfrac{{{x_2}}}{4} - 3 + \dfrac{{{x_3}}}{4} - 3 + ..... + \dfrac{{{x_{16}}}}{4} - 3}}{{16}} = 48\]
So, as we earlier took \[\dfrac{1}{4}\] common from the numerator,
Similarly, rather than subtracting each number by \[3\], we can subtract the whole original mean by \[3\].
This would give the same answer.
Hence,
New mean \[ = \] (Original mean \[ \div \] 4)\[ - 3\]
\[ \Rightarrow \]New mean \[ = 12 - 3\]
\[ \Rightarrow \] New mean \[ = 9\]

Hence, the new mean is equal to 9. Therefore, option D is the correct option.

Note: We need to keep in mind the fact that in a mean, if each observation is divided, multiplied, added or subtracted by the same constant, then the mean is mathematically operated by the same constant. We can say that rather than showing its impact on each single observation, we can show its impact on the original mean, then it indirectly affects the mean as a whole. If we know this fact, then we can simply answer the question as:
Original Mean of \[16\] numbers \[ = 48\]
Now, if each number is divided by 4 and diminished by 3,
Then, mean should be divided by 4 and diminished by 3,
New mean \[=\dfrac{{48}}{4} - 3\]
\[ \Rightarrow \] New mean \[ = 12 - 3\]
\[ \Rightarrow \] New mean \[ = 9\]