Let me explain the concept with easier values.
Let us say cost price of an item is $100. I, a merchant, mark it up by 40% and put a tag on it of $140. Now, I have a sale and I offer everything at 10% discount. So something that is marked at $140, will get $14 off and will be sold at $126. The profit I made on the item is $26 (= 126 - 100 (which was my cost price)). This profit is equal to a profit % of 26/100 = 26% (Profit/CP x 100)
Note here that my mark up % was 40%, I gave discount of 10% but my profit is only 26%, not 30%. This is because the 40% mark up was on cost price while when I gave discount, I gave 10% on the marked price (which was way more than cost price). The diagram below will make this clearer.
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Now, if I go back to your question, let us say the cost price of the item was $100. It was marked up to $133.33 (to generate a profit of 33.33%) but the actual profit received was only 20% i.e. the item was sold at $120. Then, discount % = (133.33 - 120)/133.33 x 100 = 10%
Same thing can be done using multipliers (as shrouded1 has done above)
When cost price, c, increases by 33.33%, it becomes c + 33.33c/100 = c(1 + 33.33/100) = c4/3. This is the marked price (also called tag price)
Profit = 20% so selling price must have been c + 20c/100 = c6/5
Discount = 4c/3 - 6c/5 = 2c/15
Discount is always given on the marked price. Discount % = (2c/15)/(4c/3) x 100 = 10%
We can make up a quick formula.
If m% is the mark up %, d% is the discount % and p% is the profit %, then
cost price x ( 1 + m/100) = marked price
marked price x (1 - d/100) = selling price
which means: cost price x ( 1 + m/100) x (1 - d/100) = selling price
We know, cost price x (1 + p/100) = selling price
From the 2 equations above, \(
( 1 + \frac{m}{100}) * (1 - \frac{d}{100}) = (1 + \frac{p}{100})\)
If I am to use this formula in your question,
\((1 + \frac{33.33}{100})(1 - \frac{d}{100}) = (1 + \frac{20}{100})\)
\(\frac{4}{3}x (1 - \frac{d}{100}) = \frac{6}{5}\)
d = 10
_________________
Karishma
Veritas Prep GMAT Instructor
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