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
Always on CP.
Selling price
SP = CP \times \frac{100 \pm x}{100}
Plus for profit, minus for loss.
Cost price from SP
CP = SP \times \frac{100}{100 \pm x}
Divide by the chip.
Equal profit and loss at two prices
CP = \frac{S_1 + S_2}{2}
When profit at S1 equals loss at S2.
Successive Changes & Equivalent Single Change
Two successive changes
\text{net} = a + b + \frac{ab}{100}
Use negative b for a fall.
Same change twice (rise)
2x + \frac{x^2}{100}
Same change twice (fall)
2x - \frac{x^2}{100}
Equivalent single discount
D = d_1 + d_2 - \frac{d_1 d_2}{100}
For two discounts.
Kept fraction
\text{kept} = \prod \frac{100 - d_i}{100}
Any number of discounts; customer pays this share.
Marked Price, Discount and the CP–MP–SP Chain
Discount per cent
\text{D\%} = \frac{MP - SP}{MP} \times 100
On the marked price.
Marked price for a target gain
MP = CP \times \frac{100 + g}{100 - d}
g = gain%, d = discount%.
SP through the chain
SP = CP\left(1 + \frac{m}{100}\right)\left(1 - \frac{d}{100}\right)
m = markup%.
Markup per cent
\text{markup\%} = \left(\frac{100+g}{100-d} - 1\right) \times 100
Multi-level markups
\text{final} = \text{cost} \times \prod \frac{100 + x_k}{100}
One chip per trader.
Free-item discount
\text{discount\%} = \frac{\text{free}}{\text{received}} \times 100
Dishonest Dealer, False Weights & Claims
False weight gain
\text{gain\%} = \frac{W - w}{w} \times 100
W = true weight, w = weight used; divide by w.
Weight from a given gain
w = W \times \frac{100}{100 + \text{gain\%}}
Claimed loss with short weight
\text{multiplier} = \frac{100 - L}{100 - c}
L = claimed loss%, c = short weight%.
Cheat at both ends
\text{multiplier} = \frac{100 + b}{100 - s}
b = extra taken while buying, s = shortfall while selling.
True gain from multiplier
\text{gain\%} = (\text{multiplier} - 1) \times 100
Same Selling Price: One Profit, One Loss
Net loss for equal plus-minus x%
\text{net loss\%} = \frac{x^2}{100}
Per cent of total CP.
Reconstructed costs
CP_1 = \frac{S}{1 + x/100}, \quad CP_2 = \frac{S}{1 - x/100}
S = common selling price.
Loss in rupees
\text{loss} = CP_1 + CP_2 - 2S
x from the loss per cent
x = 10\sqrt{\text{loss\%}}