Profit, Loss & Discount
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Profit, Loss and CP–SP Basics
Profit / loss per cent
\text{P\%} = \frac{SP - CP}{CP} \times 100
Selling price
SP = CP\left(1 \pm \frac{x}{100}\right)
Cost price from SP
CP = \frac{SP}{1 \pm \frac{x}{100}} = \frac{100 \cdot SP}{100 \pm x}
No profit, no loss
SP = CP
Successive Changes & Equivalent Single Change
Two successive changes
\text{net} = a + b + \frac{ab}{100}
Same change twice (increase)
2x + \frac{x^2}{100}
Same change twice (decrease)
2x - \frac{x^2}{100}
Equivalent single discount
D = d_1 + d_2 - \frac{d_1 d_2}{100}
Marked Price, Discount and the CP–MP–SP Chain
Discount per cent
\text{D\%} = \frac{MP - SP}{MP} \times 100
Marked price for target gain
MP = CP \cdot \frac{100 + g}{100 - d}
SP through the chain
SP = CP\left(1 + \frac{m}{100}\right)\left(1 - \frac{d}{100}\right)
Multi-level markups
\text{Final} = \text{Base} \times \prod \left(1 + \frac{x_k}{100}\right)
Dishonest Dealer, False Weights & Claims
False weight gain
\text{gain\%} = \frac{\text{true weight} - \text{used weight}}{\text{used weight}} \times 100
Claimed loss + short weight
\text{multiplier} = \frac{100 - L}{100 - c}
Cheat at both ends
\text{multiplier} = \frac{100 + b}{100 - s}
True gain from multiplier
\text{gain\%} = (\text{multiplier} - 1) \times 100
Same Selling Price: One Profit, One Loss
Net loss for equal ±x%
\text{net loss\%} = \frac{x^2}{100}
Reconstructed costs
CP_1 = \frac{S}{1 + x/100}, \quad CP_2 = \frac{S}{1 - x/100}
Overall position
\text{Loss} = CP_1 + CP_2 - 2S