Multiplication Identities - STD 9 - Page 80.4 EM

Two Odd Numbers - After and Before

Find the two odd numbers

Problem

The product of two odd numbers just after each of two odd numbers is 285 and the product of the odd numbers just before each is 165. What are the numbers?

Just after (increased by 2)

(x + 2)(y + 2) = 285

Just before (decreased by 2)

(x - 2)(y - 2) = 165

Step-by-Step Solution (Using Identities Only)

Step 1: Expand both equations

(x + 2)(y + 2) = xy + 2x + 2y + 4 = 285
(x - 2)(y - 2) = xy - 2x - 2y + 4 = 165

Step 2: Let P = xy and S = x + y

From first: P + 2S + 4 = 285 → P + 2S = 281
From second: P - 2S + 4 = 165 → P - 2S = 161

Step 3: Find P and S

Add both: 2P = 442 → P = 221 (Product)
Subtract: 4S = 120 → S = 30 (Sum)

Step 4: Find the numbers using identities

We know: (x + y)² - 4xy = (x - y)²
S² - 4P = (x - y)²
30² - 4×221 = 900 - 884 = 16 = 4²
x - y = 4

Now solve:
x + y = 30
x - y = 4
x = 17, y = 13

Product of the numbers

221

Sum of the numbers

30

The numbers

17 and 13