Question

Evaluate 101 × 99 without multiplication

Answer 1

Answer :

Given :

101 x 99

$\rule{400}{4}$

Required to find :

1. Product of the two numbers

$\rule{400}{4}$

Mentioned Condition :

• Find the product without performing actual multiplication

$\rule{400}{4}$

Identity used :

$\tt{\boxed{\Large{(a + b)(a - b) = {a}^{2} - {b}^{2}}}}$

$\rule{400}{4}$

Explanation :

In the question it is given that ;

Find the product of 101 x 99 .

The condition he mentioned is we should not perform actual division .

So, to Solve this question.

we have to use brackets .

Before using brackets ;

we should split the number such a way that the first number should be the nearest number which should be divisible by 10 and second number should be the remaining number .

Between the two numbers we need to keep an one of the two mathematical functions .

That is ;

• Addition
• Subtraction

If we add first number and second number we get the actual number .

For example :

Split the number 102 using brackets .

The number 102 can be splited as ;

( 100 + 2)

Here 100 which is the first number is divisble by 10 .

Similarly, 2 is the left over value which should be added in order to get the actual number .

In the above question we were given with two numbers .

So, we have to split the 2 numbers with their respective brackets .

In this question we have to use the identity .

$\blue{\longrightarrow{\tt{(a + b)(a - b) = {a}^{2} - {b}^{2}}}}$

The way in which we will split this number will actually represents this identity .

So, Now let's crack the above question .

$\rule{400}{4}$

Solution :

Given Number ;

$\tt{\rightarrow{101 \times 99}}$

The way in which we can split this number is ;

101 can be splited as ;

$\red{\Rightarrow{\tt{(100 + 1)}}}$

Similarly,

99 can be splited as ;

$\red{\Rightarrow{\tt{(100 - 1)}}}$

Now, integrate the 2 numbers which had been splited .

So,

$\tt{(100 + 1) \times (100 - 1)}$

This actually represents an identity is ;

The identity is ;

$\green{\tt{(a + b)(a - b) = {a}^{2} - {b}^{2}}}$

So, using this let's find their product ,

Here,

$\longrightarrow{\tt{a = 100 }}$

$\longrightarrow{\tt{b = 1 }}$

Now substitute this values in place a and b .

$\longrightarrow{\tt{{(100)}^{2} - {(1)}^{2}}}$

$\longrightarrow{\tt{ 10000 - 1 }}$

$\red{\tt{\implies{ 9,999}}}$

$\green{\tt{\boxed{\therefore{101 \times 99 = 9,999}}}}$

$\rule{400}{4}$

✅ Hence Solved .