Multiplication Identities - STD 9 - Page 80.3 EM
Two Numbers Changed by 1
Find product, sum and the numbers
Problem
When each of two numbers is increased by 1, the product becomes 1271. When each is decreased by 1, the product becomes 1131.
When each increased by 1
(x + 1)(y + 1) = 1271
When each decreased by 1
(x - 1)(y - 1) = 1131
Step-by-Step Solution (Using Identities Only)
Step 1: Expand both equations
(x + 1)(y + 1) = xy + x + y + 1 = 1271
(x - 1)(y - 1) = xy - x - y + 1 = 1131
(x - 1)(y - 1) = xy - x - y + 1 = 1131
Step 2: Let P = xy and S = x + y
From first: P + S + 1 = 1271 → P + S = 1270
From second: P - S + 1 = 1131 → P - S = 1130
From second: P - S + 1 = 1131 → P - S = 1130
Step 3: Find P and S
Add both: 2P = 2400 → P = 1200 (Product)
Subtract: 2S = 140 → S = 70 (Sum)
Subtract: 2S = 140 → S = 70 (Sum)
Step 4: Find the numbers using identities
We know: (x + y)² - 4xy = (x - y)²
S² - 4P = (x - y)²
70² - 4×1200 = 100 = 10²
x - y = 10
Now solve:
x + y = 70
x - y = 10
x = 40, y = 30
S² - 4P = (x - y)²
70² - 4×1200 = 100 = 10²
x - y = 10
Now solve:
x + y = 70
x - y = 10
x = 40, y = 30
Product of the numbers
1200
Sum of the numbers
70
The numbers
40 and 30