Question

Find 103x97 by formula

Answer 1

Answer:

Hint: We will consider variables a and b and we will take a+b=103 and a-b=97 and then solve the above question using the formula (a+b)(a−b)=a2−b2(a+b)(a−b)=a2−b2.

Complete step-by-step answer:

According to the above question we have to evaluate the value of 103×97103×97.

Let us assume that the value of 103×97103×97 is (a+b)(a−b)(a+b)(a−b).

Now we will compare 103 with a+b and 97 with a-b to obtain the values of a and b and then we will proceed to do multiplication using an algebraic formula.

i.e. a+b=103 let it be equation (1) and a-b=97 and let this be equation (2).

So, a+b=103….(1) and a-b=97….(2)

Now we will solve the equations (1) and (2) and then we will obtain the values of a and b.

Adding (1) and (2) ⇒⇒

We will get 2a=200, that means the value of a is 100, i.e. a=100.

We found the value of a. Now we have to find the value of b by substituting the value of a in either in equation(1) or in equation(2).

We will substitute the value of a in equation(1), we will get 100+b=103.

Then b=103-100 which is equals to 3.

So, b=3.

The value of a= 100 and the value of b=3.

Now we will express 103×97103×97in the form of (a+b)(a−b)(a+b)(a−b).

⇒103×97=(100+3)(100−3)⇒103×97=(100+3)(100−3), we substituted the values of a and b in (a+b)(a−b)(a+b)(a−b).

We know that the formula, (a+b)(a−b)=a2−b2(a+b)(a−b)=a2−b2, now we will express (100+3)(100−3)(100+3)(100−3) as

1002−321002−32

.

⇒103×97=1002−32⇒103×97=1002−32 and 1002=100001002=10000 , 32=932=9

⇒10000−9⇒9991⇒10000−9⇒9991

Hence, the value of 103×97=9991103×97=9991.

Note: We can obtain the value of 103×97103×97 directly by multiplying it but this procedure requires time, rather if we use the algebraic formula to find the multiplication we can save the time. If this question is asked as an objective question then look after the last digit as it should be 1 because when the last digit of 103 is 3 and the last digit of 97 is 7 when we multiply both of them we will get 21 the last digit of 21 is 1. So, finally the last digit will be 1 when we multiply 103 with 97. So, that we can mark the correct answer fast.

Step-by-step explanation:

Answer 2

Step-by-step explanation:

103=100+3

97=100-3

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

(100+3)(100-3)

10000-9

9991

thus the 103×94=9991