Profit on One, Loss on One Formula | Arithmetic

I am no fan of formulas, especially the un-intuitive ones but the one we are going to discuss today has proved quite useful. It is for a concept tested on GMAT Prep so it might be worth your while to remember this little formula.

When two items are sold at the same selling price, one at a profit of $x$ % and the other at a loss of $x$ %, there is an overall loss. The loss% = $\frac{x^2}{100}$ %

We will see how this formula is derived but the algebra involved is tedious. You can skip it if you wish.

Say two items are sold at €S each. On one, a profit of $x$ % is made and on the other a loss of $x$ % is made.

Say, cost price of the article on which profit was made = $Ct$

$Ct (1 + \frac{x}{100}) = S$

$Ct = \frac{S}{(1 +\frac{x}{100})}$

Cost Price of the article on which loss was made = $Cs$

$Cs (1 - \frac{x}{100}) = S$

$Cs = \frac{S}{(1 -\frac{x}{100})}$

Total Cost Price of both articles together = $Ct + Cs = \frac{S}{(1 +\frac{x}{100})} + \frac{S}{(1 -\frac{x}{100})}$

$Ct + Cs = S[\frac{1}{(1 + \frac{x}{100})} + \frac{1}{(1 - \frac{x}{100})}]$

$Ct + Cs = \frac{2S}{(1 - (\frac{x}{100})^2)}$

Total Selling Price of both articles together = $2S$

Overall Profit/Loss = $2S - (Ct + Cs)$

Overall Profit/Loss % = $\frac{[2S - (Ct + Cs)]}{[Ct + Cs]} \times 100$

= $[\frac{2S}{(Ct + Cs)} - 1] \times 100$

= $[\frac{2S}{[\frac{2S}{(1 - (\frac{x}{100})^2)}]} - 1] \times 100$

= $(\frac{x}{100})^2 \times 100$

= $\frac{x^2}{100}$

Overall there is a loss of $\frac{x^2}{100}$ %

Let’ see how this formula works on a GMAT Prep question.

Question: John bought 2 shares and sold them for €96 each. If he had a profit of 20% on the sale of one of the shares but a loss of 20% on the sale of the other share, then on the sale of both shares John had

(A) a profit of €10

(B) a profit of €8

(D) a loss of €10

(E) neither a profit nor a loss

Solution:

Note that the question would have been straight forward had the COST price been the same, say €100. A 20% profit would mean a gain of €20 and a 20% loss would mean a loss of €20. Overall, there would have been no profit no loss.

Here the two shares are sold at the same SALE price. One at a profit of 20% on cost price which must be lower than the sale price (to get a profit) and the other at a loss of 20% on cost price which must be higher than the sale price (to get a loss). 20% of a lower amount will be less in dollar terms and hence overall, there will be a loss.

The loss % = $\frac{(20)^2}{100}$ % = 4%.

But we need the amount of loss, not the percentage of loss.

Total Sale price of the two shares = 2 $\times$ 96 = €192

Since there is a loss of 4%, the 96% of the total cost price must be the total sale price

$(\frac{96}{100})\times$ Cost Price = Sale Price

Cost Price = €200

Loss = €200 – €192 = €8

Answer (C)

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