Question

The mean $8$ of numbers is $35$. If a number is excluded then the mean is reduced by $3$. Find the excluded number.

Answer 1

Here, we need to calculate the total of these numbers. And, then we will consider the total of the rest of the numbers by excluding one number. Then we need to subtract these two totals to find out the excluded number.

Formula used: Average of ${x_1},{x_2},{x_3},......,{x_n} = \dfrac{{{x_1} + {x_2} + {x_3} + ..... + {x_n}}}{n}$ .

Complete step-by-step solution:
It is given that the question stated as, mean of $8$ numbers is $35$.
So, we can write it as the total of these $8$ numbers is $=$ $(8 \times 35) = 280.$
Again it is given that the question stated as, if one number is excluded then the mean is reduced by $3$.
Then the mean becomes $=$ $(35 - 3) = 32.$
Also, the number of numbers in the given set becomes = $$$$(8 - 1) = 7.
So, after the exclusion of that number, the total would be = $$$$(7 \times 32) = 224.
Let us consider, ${x_1},{x_2},{x_3},{x_4},{x_5},{x_6},{x_7},{x_8}$ are the $8$ numbers and we have excluded ${x_8}$, so that the mean of the set is reduced by $3$.
So, we can derive the following equations from the above information:
$(x{}_1 + {x_2} + {x_3} + {x_4} + {x_5} + {x_6} + {x_7} + {x_8}) = 280.$
Again, $({x_1} + {x_2} + {x_3} + {x_4} + {x_5} + {x_6} + {x_7}) = 224.$
So, the excluded number is ${x_8}$.
Here we have to subtract the given information and we write it as,
${x_8} = ({x_1} + {x_2} + {x_3} + {x_4} + {x_5} + {x_6} + {x_7} + {x_8}) - ({x_1} + {x_2} + {x_3} + {x_4} + {x_5} + {x_6} + {x_7})$
Putting the values we get,
$= (280 - 224)$
Let us subtract we get,
$= 56.$

$\therefore$ The excluded number is $56$.

Note: Following points should be remember
Find the total of the number set.
Find the reduced mean (average).
Find the total of the reduced number set.
Subtract the smaller total from the bigger total to find out the excluded number of the given numbers.
Another approach we can apply by taking the sets of $8$ numbers and $7$ numbers. Then we can calculate totals of these two separate sets. If we subtract these two then we can get the excluded number.