Question

the mean of hundred numbers is 45 the mean of the last 99 numbers is 44 then find the first number

Answer 1

Hello dear

Formula for Mean (Average) = Sum of all numbers / Total number of numbers


Mean of 100 numbers = Sum of all hundred numbers / 100


We don't know the sum of all hundred numbers, so we call it x.
From the question, mean of 100 numbers is 45. Therefore:
45=%28x%2F100%29
Cross Multiply:
4500=x
Therefore, the sum of those hundred numbers is 4,500.
Now,
Mean of 99 numbers = Sum of all ninety-nine numbers / 99
We don't know the sum of all ninety-nine numbers, so we call it y.
Mean of 99 numbers is equal to 44 so,
44=%28y%2F99%29
Cross Multiply:
4356=y
Therefore, the sum of the remaining ninety-nine numbers is 4356.


Do you rememeber that when we had 100 numbers, the sum was 4,500?
When we removed one number, the sum became 4,356. So, what was that number we removed?
4500-4356=144


144 is the first number!

Answer 2

Let the numbers be x₁,x₂,x₃......x₁₀₀
(x₁+x₂+x₃+.....+x₁₀₀)/100=45
⇒x₁+x₂+x₃+.....+x₁₀₀=4500⇒x₂+x₃+x₄+....+x₁₀₀=4500-x₁
(x₂+x₃+x₄+....+x₁₀₀)/99=44
⇒x₂+x₃+x₄+....+x₁₀₀=99×44=4356
4500-x₁=4356
⇒4500=4356+x₁
⇒x₁=4500-4356
=144(Ans.)