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% = (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 + x/100) = S
Ct = S/(1 + x/100)
Cost Price of the article on which loss was made = Cs
Cs (1 – x/100) = S
Cs = S/(1 – x/100)
Total Cost Price of both articles together = Ct + Cs = S/(1 + x/100) + S/(1 – x/100)
Ct + Cs = S[1/(1 + x/100) + 1/(1 – x/100)]
Ct + Cs = 2S/(1 – (x/100)^2)
Total Selling Price of both articles together = 2S
Overall Profit/Loss = 2S – (Ct + Cs)
Overall Profit/Loss % = [2S – (Ct + Cs)]/[Ct + Cs] * 100
= [2S/(Ct + Cs) – 1] * 100
= [2S/[2S/(1 – (x/100)^2)] – 1] * 100
= (x/100)^2 * 100
Overall there is a loss of (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
(C) a loss of $8
(D) a loss of $10
(E) neither a profit nor a loss
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 % = (20)^2/100 % = 4%.
But we need the amount of loss, not the percentage of loss.
Total Sale price of the two shares = 2*96 = $192
Since there is a loss of 4%, the 96% of the total cost price must be the total sale price
(96/100)*Cost Price = Sale Price
Cost Price = $200
Loss = $200 – $192 = $8
Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!