Question

What is the average of all 4 digit even numbers, which are divisible by 5?

Answer 1

The numbers are 5,15,25,35

so, the average value of all 4 digit,

= sum of all digits ÷ 4

= (5+15+25+35)÷4

= 80÷4

= 20 is answer

Which is the average value of all 4 digit even numbers divisible by 5.

Answer 2

Given,

4 digit even numbers divisible by 5.

To find,

Average of all such numbers.

Solution,

We can solve this problem using the formula for the sum of an A. P. So, we need to find the A. P. first.

As we can see that the 4 digit numbers start from 1000, and 1000 is an even number as well as divisible by 5.

Thus, the 1st number is 1000. Similarly, a few of the next even numbers divisible by 5 will be 1010, 1020, 1030, and so on.

Now, it can be observed that the last term will be 9990 because the next even number divisible by 5 (that is, 10000) will be of 5 digits.

This way, we can see that the A. P. formed is,

1000, 1010, 1020, 1030,.............., 9990,

having a = 1000, d = 10, and, $a_n=9990$.

Now, we can find the number of terms using

$a_n=a+(n-1)d$

Substituting the values,

9990 = 1000 + (n-1)·10

⇒ 8990 = 10(n-1)

Rearranging and simplifying,

n-1 = 899

⇒ n = 900

So, the number of terms will be 900.

Now, using the sum formula for an A. P. that is,

$S_n=\frac{n}{2} [2a+(n-1)d]$

⇒ $S_n=\frac{900}{2} [2(1000)+(900-1)10]$

⇒ $S_n=450 [2000+8990]$

⇒ $S_n=450 [10990]$

⇒ $S_n=4945500$

Now, the average can be found using

$Average=\frac{S_n}{n}$

⇒ $Average = \frac{4945500}{900}$

⇒ Average = 5495.

Therefore, the average of all 4 digit even numbers divisible by 5 will be 5495.