Profit & Loss – Advanced Study Notes
(Quantitative Aptitude – UGC NET, MIT‑Level)
1. Introduction
Profit and loss are the most fundamental financial concepts in commerce and economics.
They quantify the difference between the Selling Price (SP) of a good or service and its Cost Price (CP).
In a quantitative aptitude context, the focus is on manipulating algebraic expressions, percentages, and ratios to solve for unknowns.
Notation
- (C) : Cost Price (in rupees)
- (S) : Selling Price (in rupees)
- (P) : Profit (positive) or Loss (negative)
- (p%) : Profit percentage
- (l%) : Loss percentage
2. First‑Principle Derivation
2.1 Absolute Profit / Loss
[ P = S - C ]
- (P>0) → Profit
- (P<0) → Loss
- (P=0) → Break‑even
2.2 Percentage Profit / Loss
[ \text{Profit %} = \frac{P}{C}\times 100 = \frac{S-C}{C}\times 100 ]
[ \text{Loss %} = -\frac{P}{C}\times 100 = \frac{C-S}{C}\times 100 ]
Notice that the denominator is always the cost price.
This convention is historically entrenched and essential for consistency in multiple‑step problems.
2.3 Markup vs Margin
| Concept | Formula | Interpretation |
|---|---|---|
| Markup | (\displaystyle \frac{S-C}{C}\times 100) | % increase over cost |
| Margin | (\displaystyle \frac{S-C}{S}\times 100) | % of selling price that is profit |
Both are often confused; the key difference is the denominator (cost vs. selling price).
3. Weighted Averages in Multi‑Transaction Scenarios
Suppose a trader buys (n) batches of a commodity at different costs and sells them all at the same price (S).
Let the (i^{th}) batch have cost (c_i) and quantity (q_i).
The average cost per unit:
[ C_{\text{avg}} = \frac{\sum_{i=1}^n c_i q_i}{\sum_{i=1}^n q_i} ]
Profit per unit:
[ P_{\text{unit}} = S - C_{\text{avg}} ]
Total profit:
[ P_{\text{total}} = (S - C_{\text{avg}})\sum_{i=1}^n q_i ]
Edge case: If some batches are purchased at a negative cost (e.g., due to rebates or discount vouchers), the average cost can become negative, leading to a negative profit even if (S>0).
4. Advanced Topics
4.1 Profit Function from a Demand–Cost Framework
Consider a firm whose demand function is (q(p)) (units sold per unit price (p)).
Revenue: (R(p)=p,q(p))
Cost: (C(q)=c,q + f) (linear cost: variable cost per unit (c), fixed cost (f))
Profit function:
[ \Pi(p)=R(p)-C(q(p)) = p,q(p)-\bigl(c,q(p)+f\bigr) ]
Profit maximization:
[
\frac{d\Pi}{dp}=0 ;;\Longrightarrow;; q(p)+p\frac{dq}{dp}-c\frac{dq}{dp}=0
]
Assuming (q(p)=a-bp) (linear demand), we get:
[ q(p)=a-bp,;;\frac{dq}{dp}=-b ] [ \frac{d\Pi}{dp}=a-bp - pb + cb = 0 ;;\Longrightarrow;; p^*=\frac{a+c}{2b} ]
This gives the profit‑maximising price (p^).
The associated quantity (q^ = a-bp^) and profit (\Pi^=\Pi(p^*)) can be found directly.
Edge case: If the demand is perfectly elastic ((b=0)), any price leads to the same quantity; profit maximization reduces to simply setting (p) as high as possible.
4.2 Elasticity of Profit
The price elasticity of profit (\varepsilon_{\Pi}) measures responsiveness of profit to price changes