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Question: The arithmetic mean of 100 numbers was completed as 89.05 It was later found that two numbers 92 and...

The arithmetic mean of 100 numbers was completed as 89.05 It was later found that two numbers 92 and 83 have been misread as 192 and 33 respectively. What is the correct arithmetic mean of the numbers?
(a)88.55 (b)87.55 (c)89.55 (D) cannot be calculated from data given

Explanation

Solution

We first find the sum of 100 numbers, subtract wrongly read numbers i.e., 192 & 33 from them and then we add correct numbers 92 & 83 to them and find the arithmetic mean

Complete step by step answer:
Arithmetic mean of (say) 5 numbers a, b, c, d, e is calculated by the expression a+b+c+d+e5=sum of all the numbers No of numbers\dfrac{a+b+c+d+e}{5}=\dfrac{\text{sum}\ \text{of}\ \text{all}\ \text{the}\ \text{numbers}\ }{\text{No}\ \text{of}\ \text{numbers}}
Let the 100 numbers are named as a1, a2, a3,..............., a100{{a}_{1}},\ {{a}_{2}},\ {{a}_{3}},...............,\ {{a}_{100}}
It is given that, arithmetic mean of 100 numbers are 89.05
i.e. a1+a2+..............+a100100=89.05\dfrac{{{a}_{1}}+{{a}_{2}}+..............+{{a}_{100}}}{100}=89.05
∴a1+a2+a3+.............a100=89.05×100=8905\therefore {{a}_{1}}+{{a}_{2}}+{{a}_{3}}+.............{{a}_{100}}=89.05\times 100=8905
It is given that two numbers 92 and 93 have been missed as 192. and 33 respectively
Let a99=192 and a98=33{{a}_{99}}=192\ \text{and}\ {{a}_{98}}=33 (you can take any two numbers as 192. & 33)

Now,
we subtract a99 and a98{{a}_{99}}\ \text{and}\ {{a}_{98}} from total sum of 100 numbers
i.e. sum of remaining 98 numbers are 8905−192−33=86808905-192-33=8680
Sum of remaining 98 numbers are 8680
Now, we add the correct numbers which are unfortunately missed to the sum of 98 numbers and then we find the arithmetic of correct 100 numbers.
i.e. Sum of 100 correct numbers =8680+92+83=8680+92+83
=8855=8855
Correct Arithmetic mean of 100 numbers
= sum of 100 correct numbers /100
=8855/100
= 88.55
Therefore, Option A is correct.

Note: In the given problem, we have to find the correct arithmetic mean of the numbers. We first need to subtract wrongly read numbers from the sum of total 100 numbers and then we have to add the correct numbers which are wrongly read to the sum of 100 numbers and then we have to find the correct arithmetic mean of those 100 numbers. You must be very careful in doing this as you might not subtract wrongly read numbers from the total sum of 100 numbers.