Question

The average of 20 numbers is zero. Of them, at the most, how many may be greater than zero ?

Answer 1

Hint: Average is the sum of all the values divided by total number of values for a given data.

Complete step by step answer:
As we know we can conclude that the average of any given data is the total sum of given data by number of data. Here, 20 numbers are given and the sum of that 20 numbers is 0 .
To calculate average , formula is given below
Average $ = \dfrac{{sum\,of\,numbers}}{{total\,number}}$
For this question, let us denote this 20 numbers by ${x_1},{x_2}.................{x_{20}}$ .
Sum of this 20 numbers will be ${{x{}_1 + {x_2} + ........................ + {x_{20}}}}$
We know total number is 20
Putting the values in the formula of average.
Average = $\dfrac{{x{}_1 + {x_2} + ........................ + {x_{20}}}}{{20}}$
 average is given as 0 , putting the value 0
0 = $\dfrac{{x{}_1 + {x_2} + ........................ + {x_{20}}}}{{20}}$
Cross multiply the numbers , that is
$20 \times 0 = {x_1} + {x_2} + ........... + {x_{20}}$
Solve the above,
$0 = {x_1} + {x_2} + ........... + {x_{20}}$
This means sum = 0
If you transpose one of the numbers to left hand side you will get
${x_1} + {x_2} + ........... + {x_{19}} = - {x_{20}}$
Now for the sum of 20 numbers to be equal to zero, there may be all 19 numbers of them greater than 0 and only one number which is negative of all the 19 numbers sum.

Thus, there can be all 19 numbers of them greater than 0 and only one number which is negative of all the 19 numbers sum.

Additional information:
For consecutive series , the sum of numbers may be greater than zero at most will be exactly half of the total no of numbers. To calculate the sum or average of consecutive series you can use the arithmetic progressions formula.

Note:
For random numbers taken in the series, to get an average zero there can be at most 19 positive numbers and one negative number or one positive number and 19 negative numbers.