Question

Evaluate the following products without multiplying directly: 

(i) 103 × 107 (ii) 95 × 96 (iii) 104 × 96

Answer 1

Evaluating Products Using Algebraic Identities 

1. Formula Used:

We will use the identity for the product of two numbers close to a base number: \[ (a + b)(a - b) = a^2 - b^2 \] For products near a base number like 100, 1000, etc., this can simplify the calculation.

(i) \( 103 \times 107 \)

Step 1: Notice that \( 103 \) and \( 107 \) are close to 100. We can express them as: \[ 103 = 100 + 3 \quad \text{and} \quad 107 = 100 + 7 \]

Step 2: Apply the formula \( (a + b)(a + c) = a^2 + (b + c)a + bc \): \[ 103 \times 107 = (100 + 3)(100 + 7) \] \[ = 100^2 + (3 + 7) \times 100 + 3 \times 7 \] \[ = 10000 + 1000 + 21 = 11021 \]

(ii) \( 95 \times 96 \)

Step 1: Notice that \( 95 \) and \( 96 \) are close to 100. We can express them as: \[ 95 = 100 - 5 \quad \text{and} \quad 96 = 100 - 4 \]

Step 2: Apply the formula \( (a - b)(a - c) = a^2 - (b + c)a + bc \): \[ 95 \times 96 = (100 - 5)(100 - 4) \] \[ = 100^2 - (5 + 4) \times 100 + (-5) \times (-4) \] \[ = 10000 - 900 + 20 = 9100 \]

(iii) \( 104 \times 96 \)

Step 1: Notice that \( 104 \) and \( 96 \) are close to 100. We can express them as: \[ 104 = 100 + 4 \quad \text{and} \quad 96 = 100 - 4 \]

Step 2: Apply the formula \( (a + b)(a - b) = a^2 - b^2 \): \[ 104 \times 96 = (100 + 4)(100 - 4) \] \[ = 100^2 - 4^2 \] \[ = 10000 - 16 = 9984 \]

Answer Summary:

• (i) \( 103 \times 107 = 11021 \)
• (ii) \( 95 \times 96 = 9100 \)
• (iii) \( 104 \times 96 = 9984 \)