Multiplication Identities - STD 9 - Page 82.3 EM

Two Numbers - Larger and Smaller Changed

Find product, difference and the numbers

Problem

The product of the larger increased by one and the smaller decreased by one is 540. The product of the larger decreased by one and the smaller increased by one is 560.

Larger +1 and Smaller -1

(x + 1)(y - 1) = 540

Larger -1 and Smaller +1

(x - 1)(y + 1) = 560

Step-by-Step Solution (Using Identities)

Step 1: Expand both equations

(x + 1)(y - 1) = xy - x + y - 1 = 540
(x - 1)(y + 1) = xy + x - y - 1 = 560

Step 2: Let P = xy and D = x - y

From first: P - D - 1 = 540 → P - D = 541
From second: P + D - 1 = 560 → P + D = 561

Step 3: Find P and D

Add both: 2P = 1102 → P = 551 (Product)
Subtract first from second: 2D = 20 → D = 10 (Difference)

Step 4: Find the numbers

We know: (x + y)² = (x - y)² + 4xy
(x + y)² = 10² + 4×551 = 2304 = 48²
x + y = 48

Now solve:
x - y = 10
x + y = 48
x = 29, y = 19

Product of the numbers

551

Difference

10

The numbers

29 and 19