Question

Evaluate $ 55 \times 45 $ using suitable identity.

Answer 1

Hint: Here, we can observe that the numbers in multiplication differ by a common difference of 5 from the number 50. So, we will use this observation and a suitable algebraic identity to find the given product.
Formula Used:
We will use the identity for the difference of two squares, $ {a^2} - {b^2} = (a + b)(a - b) $ .

Complete step-by-step answer:
The question asks us to find the given product using a suitable algebraic identity. We know the three algebraic identities where only squares are involved: $ {(a + b)^2} = {a^2} + {b^2} + 2ab $ , $ {(a - b)^2} = {a^2} + {b^2} - 2ab $ and $ {a^2} - {b^2} = (a + b)(a - b) $ . We will use one of these to find the product.
Observe that the numbers in multiplication differ by a common difference of 5 from the number 50. That is, the given product can be written as $ 55 \times 45 = (50 + 5)(50 - 5) $ . Now on comparing this with the known algebraic identities we can see that the identity $ {a^2} - {b^2} = (a + b)(a - b) $ matches with the given product. So, we will use this identity to simplify the expression.
On comparing $ 55 \times 45 = (50 + 5)(50 - 5) $ with $ (a + b)(a - b) $ we get a to be 50 and b to be 5 and the squares of both of these numbers are easy to find.
So, on using the identity $ {a^2} - {b^2} = (a + b)(a - b) $ we get $ (50 + 5)(50 - 5) = {50^2} - {5^2} $
 $ \Rightarrow 55 \times 45 = (50 + 5)(50 - 50) = {50^2} - {5^2} $
 $ \Rightarrow 55 \times 45 = {50^2} - {5^2} $
 $ \Rightarrow 55 \times 45 = 2500 - 25 $
 $ \Rightarrow 55 \times 45 = 2475 $
Hence the value of the product $ 55 \times 45 $ is $ 2475 $ .
So, the correct answer is “2475.

Note: When using identities to solve a question like $ {48^2} $ you can express it as $ {(45 + 3)^2} $ or as $ {(40 + 8)^2} $ or even as $ {(50 - 2)^2} $ . But you have to use the simplification where the squares of the numbers are easier to find. It is difficult to know the square of 45 so we can go with either of the other versions where the simplest being $ {(50 - 2)^2} $ .