Multiplication Identities - STD 9 - Page 82.2 EM
Two Numbers - Product Increase Identity
The increase is always constant
Problem
The product of two numbers is 5 more than the product of the larger increased by one and the smaller decreased by one. How much is the increase in the product if the larger is decreased by one and the smaller increased by one?
Given
xy = (x + 1)(y - 1) + 5
Step-by-Step Solution (Using Identities)
Step 1: Simplify the given equation
xy = (x + 1)(y - 1) + 5
xy = xy - x + y - 1 + 5
0 = -x + y + 4
x - y = 4
xy = xy - x + y - 1 + 5
0 = -x + y + 4
x - y = 4
Step 2: Find the increase when larger is decreased by 1 and smaller increased by 1
New product = (x - 1)(y + 1)
Increase = (x - 1)(y + 1) - xy
= xy + x - y - 1 - xy
= x - y - 1
= 4 - 1 = 3
Increase = (x - 1)(y + 1) - xy
= xy + x - y - 1 - xy
= x - y - 1
= 4 - 1 = 3
The increase in the product is always
3