📚 Profit & Loss – Comprehensive Study Notes
(Designed for competitive exams – IIT‑JEE, Mains, State TET, etc.)
1️⃣ Introduction
Profit/Loss (P/L) problems test your ability to translate word‑problems into algebraic expressions.
Key ideas:
| Concept | Formula | Interpretation |
|---|---|---|
| Profit | (P = SP - CP) | Amount earned over cost |
| Loss | (L = CP - SP) | Amount lost over cost |
| Profit % | (\displaystyle \frac{SP-CP}{CP}\times 100) | Relative to cost price |
| Loss % | (\displaystyle \frac{CP-SP}{CP}\times 100) | Relative to cost price |
| Margin of profit | (\displaystyle \frac{SP-CP}{SP}\times 100) | Relative to selling price |
Tricky point – All standard percentages are based on cost price unless stated otherwise.
2️⃣ Basic Derivations
2.1 Profit / Loss from CP and SP
Let (CP = x) and (SP = y).
[ \text{Profit } P = y-x,\qquad \text{Loss } L = x-y ]
Profit %:
[ \frac{P}{CP}\times100 = \frac{y-x}{x}\times100 ]
Loss %:
[ \frac{L}{CP}\times100 = \frac{x-y}{x}\times100 ]
2.2 Expressing CP in terms of SP
Suppose the selling price is (p%) higher than the cost price.
[ SP = CP \left(1+\frac{p}{100}\right) \Longrightarrow CP = \frac{SP}{1+\frac{p}{100}} = SP;\frac{100}{100+p} ]
Similarly, if the SP is (q%) lower than CP:
[ SP = CP \left(1-\frac{q}{100}\right) \Longrightarrow CP = \frac{SP}{1-\frac{q}{100}} = SP;\frac{100}{100-q} ]
3️⃣ Combined Profit & Loss (Multiple Items)
3.1 Weighted‑Average Formula
If (n) items are bought with total cost (C_{\text{tot}}) and sold for total selling price (S_{\text{tot}}),
[ \text{Net Profit} = S_{\text{tot}}-C_{\text{tot}} ] [ \text{Net Profit %} = \frac{S_{\text{tot}}-C_{\text{tot}}}{C_{\text{tot}}}\times 100 ]
3.1.1 Using Individual Profit Percentages
Let item (i) be sold at (p_i%) profit on its cost (c_i).
Then selling price of item (i):
[ s_i = c_i\left(1+\frac{p_i}{100}\right) ]
Total SP:
[ S_{\text{tot}} = \sum_{i=1}^{n} s_i ]
Net profit percent:
[ \text{Net %} = \frac{\sum c_i\left(\frac{p_i}{100}\right)}{\sum c_i}\times100 ]
This is a weighted average of the individual profit percentages with weights (c_i).
Tricky point – When some items have a loss, their “loss‑percent” must be treated as negative in the weighted average.