Question

Total natural numbers lying between squares of $99$ and $100$

Answer 1

Hint: First, we shall analyze the given information so that we are able to solve the problem. Here in this question, we are given squares of two numbers $99$ and $100$.
We are asked to calculate the total natural number lying between the squares of $99$ and $100$.
We need to apply the formula to calculate the total numbers lying between the squares of two numbers in this problem.
Formula to be used:
The formula to calculate the total natural numbers lying between ${n^2}$ and ${\left( {n + 1} \right)^2}$is as follows.
Total natural numbers lying between ${n^2}$ and ${\left( {n + 1} \right)^2}$$ = 2n$
Where n is the given number.

Complete step by step answer:
Here in this question, we are given squares of two numbers $99$ and $100$.
We are asked to calculate the total natural number lying between the squares of $99$ and $100$.
Now, we shall apply the formula here.
The formula to calculate the natural numbers lying between ${n^2}$ and ${\left( {n + 1} \right)^2}$is as follows.
Total natural numbers lying between ${n^2}$ and ${\left( {n + 1} \right)^2}$$ = 2n$
Where n is the given number.
Now, we have noticed that ${n^2} = {99^2}$ and ${\left( {n + 1} \right)^2} = {100^2}$
That is we get $n = 99$
Hence, total natural numbers lying between ${99^2}$ and ${100^2}$$ = 2 \times 99$
                                                                                                       $ = 198$
Therefore, the total natural numbers lying between ${99^2}$ and ${100^2}$ $ = 198$

Note: We can solve this problem in an alternate method. This alternate method is applicable only for any two numbers \[a\] and \[b\] such that $b > a$. Thus for any two numbers \[a\] and \[b\] such that $b > a$, we apply the formula $b - a - 1$
We are given squares of two numbers $99$ and $100$.
Here ${99^2} = 9801$ and ${100^2} = 10000$
Hence, we get \[a = 9801\] and $b = 10000$
Now, we shall apply the formula $b - a - 1$
$b - a - 1 = 10000 - 9801 - 1$
                    $ = 198$
Therefore, the total natural numbers lying between ${99^2}$ and ${100^2}$$ = 198$