In a Shop, The Marked Price of an Article is Worked Out in Such a Way That it Generates a Profit of 33 1 by 3 GMAT Problem Solving

Question: In a shop, the marked price of an article is worked out in such a way that it generates a profit of 33\(\frac{1}{3}\). What should be the discount percent allowed on the marked price during a sale, so that the final profit made is 20%?

1. 33\(\frac{1}{3}\)
2. 12\(\frac{1}{2}\)
3. 10%
4. 8\(\frac{1}{2}\)
5. 6\(\frac{1}{2}\)

“In a shop, the marked price of an article is worked out in such a way that it generates a profit of 33\(\frac{1}{3}\).”-  is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide 2021”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:

Approach Solution 1:

There is only one approach to solve the problem statement

For better understanding of the candidates, the concept is explained with easier values.
Let us say the cost price of an item is $100. A merchant marks it up by 40% and puts a tag on it of $140.
Now, the merchant has a sale and offers everything at a 10% discount.
So something that is marked at $140, will get $14 off and will be sold at $126. The profit made on the item is $26 (= 126 - 100 (which was the cost price)).
This profit is equal to a profit % of 26/100 = 26% (Profit/CP x 100)
Note here that the markup % was 40%, the merchant gave a discount of 10% but the profit was only 26%, not 30%.
This is because the 40% mark up was on cost price when the discount was given. The merchant gave 10% on the marked price (which was way more than cost price). The diagram below will make this clearer.

Now, going back to the 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.
Therefore, the 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.

Hence, the Discount % = (2c/15)/(4c/3) x 100 = 10%
We can make up a quick formula.
If m% is the markup %, 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,\( (( 1 + \frac{m}{100}) * (1 - \frac{d}{100}) = (1 + \frac{p}{100}))\)
Now, if we use the formula in the question, we get:

=>\((1+\frac{33.33}{100})(1- \frac{d}{100}) = (1+\frac{20}{100})\)

=>\(\frac{4}{3}(1-\frac{d}{100}) = 6/5\)

=>x=10

Correct Answer: C

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