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Arithmetic - Profit on One, Loss on One Formula

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

= x^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

 

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 % = (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

 

Answer (C)

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