Let the cost price (CP) of the article be ₹100.

The selling price (SP) at
𝑝
%
p% more than the cost price is:

𝑆
𝑃
=
100
+
𝑝
100
×
100
=
100
+
𝑝
=
(
100
+
𝑝
)
rupees
SP=100+
100
p
​
×100=100+p=(100+p) rupees
If the manufacturer gives a discount of
𝑞
%
q%, the discount amount is:

Discount
=
𝑞
100
×
(
100
+
𝑝
)
Discount=
100
q
​
×(100+p)
Thus, the actual selling price after discount is:

Actual SP
=
(
100
+
𝑝
)
−
𝑞
100
×
(
100
+
𝑝
)
=
(
100
+
𝑝
)
(
1
−
𝑞
100
)
=
(
100
+
𝑝
)
(
100
−
𝑞
100
)
Actual SP=(100+p)−
100
q
​
×(100+p)=(100+p)(1−
100
q
​
)=(100+p)(
100
100−q
​
)
The profit is calculated as:

Profit
=
Actual SP
−
𝐶
𝑃
=
(
100
+
𝑝
)
(
100
−
𝑞
100
)
−
100
Profit=Actual SP−CP=(100+p)(
100
100−q
​
)−100
The profit percentage is:

Profit Percentage
=
Profit
𝐶
𝑃
×
100
=
(
100
+
𝑝
)
(
100
−
𝑞
100
)
−
100
100
×
100
Profit Percentage=
CP
Profit
​
×100=
100
(100+p)(
100
100−q
​
)−100
​
×100
This simplifies to:

Profit Percentage
=
(
100
−
𝑞
)
+
𝑝
(
100
−
𝑞
)
100
−
100
=
𝑝
(
100
−
𝑞
)
100
−
𝑞
Profit Percentage=(100−q)+
100
p(100−q)
​
−100=
100
p(100−q)
​
−q
Thus, the profit percentage is:

Profit Percentage
=
𝑝
(
100
−
𝑞
)
100
−
𝑞
Profit Percentage=
100
p(100−q)
​
−q