Question

Find the average of all prime numbers between 30 and 50.

Answer 1

Hint: Consider all the numbers between 30 and 50. Find the factors of each number between these numbers, some will have factors of 1 and itself. These are the prime numbers. Then take the average of the prime numbers obtained.

Complete step-by-step answer:
A prime number is a natural number greater than 1 that is not a product of 2 smaller natural numbers other than 1 and itself. A natural number greater than 1 that is not prime is called a composite number.
A prime number’s only factors are 1 and itself.
Now let us find all the prime numbers between 30 and 50. Let us write these numbers.
31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49.
Now let us check each number and see if they have factors or not.
\[\begin{align}
  & 31=1\times 31 \\
 & 32=2\times 8\times 2\times 1 \\
 & 33=3\times 11\times 1 \\
 & 34=2\times 17\times 1 \\
 & 35=1\times 5\times 7 \\
 & 36=1\times 2\times 2\times 3\times 3 \\
 & 37=1\times 37 \\
 & 38=1\times 2\times 19 \\
 & 39=1\times 3\times 13 \\
 & 40=1\times 2\times 2\times 2\times 5 \\
\end{align}\] \[\begin{align}
  & 41=1\times 41 \\
 & 42=1\times 2\times 21 \\
 & 43=1\times 43 \\
 & 44=1\times 4\times 11 \\
 & 45=1\times 5\times 3\times 3 \\
 & 46=1\times 2\times 23 \\
 & 47=1\times 47 \\
 & 48=1\times 2\times 2\times 2\times 6 \\
 & 49=1\times 7\times 7 \\
\end{align}\]
Now we have written all the numbers between 30 and 50, and their factors. Let us select the prime numbers among these numbers, which have factors 1 and itself.
They are 31, 37, 41, 43, 47.
There are a total of 5 numbers that are prime among 19 numbers.
Now we need to find the average of all prime numbers between 30 and 50.
Average \[=\dfrac{Sum}{TotalNos.}\]
Average of prime number \[=\dfrac{SumOf\Pr imeNumbersBetween30and50}{Total\Pr imeNumbersBetween30and50}\]
\[=\dfrac{31+37+41+43+47}{5}=39.8\]

Hence we found the average of all prime numbers between 30 and 50 as 39.8.

Note: Don’t forget to find the average of all prime numbers. The rest of the numbers that we got are composite numbers which have other factors than 1 and the number itself. If we were asked to find the average of composite numbers, then \[=\dfrac{SumOfCompositeNumbers}{14}.\]