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I was reading a text book and came across the following interesting problem, mostly faced by shopkeepers:

If $x\%$ discount on an article is given on cash payment, find the percentage that should be marked above the cost price so as to make a profit of $y\%$.

Can someone please explain me how to solve this problem?

Thanks in advance!

1 Answers 1

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Note well. See Ross Millikan's comment below; it seems I'm using an incorrect notion of "profit" here.


For an example: suppose the item costs you $\$ $1, and you wanted a $5\%$ profit. If you offer no discounts, you could simply price it at $\$ $1.05, and when a customer buys it you get the $5\%$ profit.

But say you offer a $5\%$ discount for paying cash. If you price the item at $\$ $1.05, then the $5\%$ discount means that if I pay cash, I only need to pay you $1.05 - .05 = 1.00$ (rounding down), so instead of the 5 percent profit, you make no money.

If you want to make 5 percent profit even after you give me the discount, that means that you want the advertised price to be such that once you give me the 5% discount, then the discounted price will be $\$ $1.05.

So what we want is that the price, after taking 5\% off, will be $\$ $1.05. That is, $$\left(1 - \frac{5}{100}\right)P = 1.05.$$ Solving for $P$ we get $$P = \frac{1.05}{.95} \approx 1.1052$$ so you want to price it at either $\$ $1.10 or $\$ $1.11.

Now, let's deal with the problem in abstract and in generality.

So, what is it we want? We want to take the cost $C$, and add a certain percentage, $(1+\frac{p}{100})C$ so that, if we take $x\%$ off this price, the result will be $(1+\frac{y}{100})C$ ($\frac{y}{100}$ because it is $y\%$ profit).

Now, $x$ and $y$ are given and fixed, as is $C$. So what we want is to find the value of $p$ such that: $$\Biggl(\left(1 + \frac{p}{100}\right)C\Biggr)\left(1-\frac{x}{100}\right) = \left(1 + \frac{y}{100}\right)C.$$ You can cancel $C$, and you end up with the equation $$\left(1+\frac{p}{100}\right)\left(1 - \frac{x}{100}\right) = \left(1+\frac{y}{100}\right).$$ Now simply express $p$ in terms of $x$ and $y$, and that is the "percentage that should be marked above the cost price so as to make a profit of y% [after the cash discount is taken off]".

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    Unfortunately, the usual definition of profit margin is profit/revenue, not profit/cost. So, if the object costs \$1 and you want 10% profit, you need to price at \$10/9 or about \$1.11. The good news is that if you normally work at a 5% profit margin and then give a 5% discount, you break even. The difference gets larger when the margin goes up. For a 40% profit margin, you need to mark up 66 2/3%2011-06-19
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    @Ross: Hmmm... the problem said "profit", not "profit margin". If they mean the same, how is "profit margin" defined in terms of "profit" (either profit/revenue or profit/cost). But I don't do a lot of economics applications, so no doubt you are correct.2011-06-19
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    When profit is in dollars it is well-defined. When it is in percent, it depends upon the denominator. So (as I understand it) you should have $$\left(1+\frac{p}{100}\right)\left(1 - \frac{x}{100}\right) = \left(1+\frac{y}{100+y}\right).$$ When $y$ is small, it doesn't matter much.2011-06-19
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    @Ross: Okay; I'll add a note at the top. Thanks for the correction.2011-06-19