Quarter Wit, Quarter Wisdom: Mark Up, Discount, and Profit

Quarter Wit, Quarter WisdomMark Up, Discount and Profit questions confuse a lot of people. But, actually, most of them are absolute sitters — very easy to solve — a free ride! How? We will just see. Let me begin with the previous post’s question.

Question: If a retailer marks up an article by 40% and then offers a discount of 10%, what is his percentage profit?

Let us say the retailer buys the article for $100 ($100 is his cost price of the item). He marks it up by 40% i.e. increases his cost price by 40% (100 * 140/100) and puts a tag of $140 on the article. Now, the article remains unsold and he puts it on sale – 10% off everything. So the article marked at $140, gets $14 off and sells at $126 (because 140 * 9/10 = 126). This $126 is the selling price of the article. To re-cap, we obtained this selling price in the following way:

Cost Price * (1 + Mark Up%) * (1 – Discount%) = 100 * (1 + 40/100) * (1 – 10/100) = 126 = Selling Price

The profit made on the item is $26 (obtained by subtracting 100, the retaile’s cost price, from 126, the retailer’s selling price).

He got a profit % of (26/100) * 100 = 26% (Profit/Cost Price x 100)

Or we can say that Cost Price * (1 + Profit%) = 100 * (1 + 26/100) = 126 = Selling Price

The italicized parts above show the two ways in which you can reach the selling price: using mark-up and discount or using profit. The same thing is depicted in the diagram below:



Therefore, Cost Price * (1 + Mark Up%) * (1 – Discount%)= Cost Price * (1 + Profit%)

Or

(1 + Mark Up%) * (1 – Discount%)= (1 + Profit%)

Look at the numbers here: Mark Up: 40%, Discount: 10%, Profit: 26% (Not 30% that we might expect because 40% – 10% = 30%)

Why? Because the discount offered was on $140, not on $100. So a bigger amount of $14 was reduced from the price. Hence the profit decreased. This leads us to an extremely important question in percentages – What is the base? 100 was increased by 40% but the new number 140 was decreased by 10%. So in the two cases, the bases were different. Hence, you cannot simple subtract 10 from 40 and hope to get the Profit %. Also, mind you, almost certainly, 30% will be one of the answer choices, albeit incorrect. (The GMAT doesn’t forego even the smallest opportunity of tricking you into making a mistake!)

Let’s see this concept in action on a tricky third party question:

A dealer offers a cash discount of 20%. Further, a customer bargains and receives 20 articles for the  price of 15 articles. The dealer still makes a profit of 20%. How much percent above the cost price were his articles marked?

a) 100%
b) 80%
c) 75%
d) 66+2/3%
e) 50%

This question involves two discounts:

  1. the straight 20% off
  2. discount offered by selling 20 articles for the price of 15 – a discount of cost price of 5 articles on 20 articles i.e. a discount of 5/20 = 25%

Using the formula given above:

(1 + m/100)(1 – 20/100)(1 – 25/100) = (1 + 20/100)
m = 100

Therefore, the mark up was 100%.

Answer (A)

Note: The two discounts are successive percentage discounts.

Another application of successive percentage changes is the concept of compounding. But more on that, in the next post.

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 in Detroit, Michigan, and regularly participates in content development projects such as this blog!

5 Responses

  1. TP says:

    Question… I approached the problem a different way. I’m not sure if it’s accurate or what, but here’s how I did it:

    P1=Original Price of the Items
    P2=Final Price of the Items

    P1(.80)(.75) = P2(1.20)
    P1(.60)=P2(1.20)
    Notice that P1 on the left side needs to be 2 and the P1 on the right needs to be 1, so it’s in a ratio of 2:1

    Thus, the original price has to be twice the amount as the final price. If we say that the price of the item sold was $100, then the seller had to have sold it for $200 in order to receive a 20% profit. $200 is 100% above the original cost price of $100. Correct?

  2. Karishma says:

    Yes, your approach is absolutely fine as long as you understand the following:
    P1 is the marked price or tag price or as you call it, the original price of the item before discount. Thereafter, you have discounted it by 20% and 25% to get the selling price i.e. the price at which it was actually sold.
    P2 is the cost price of the item. You equate it to the left side of the equation by multiplying with 1.2, the effect of profit earned. P2*1.2 gives selling price hence making the equation valid.
    Since P1 to P2 is 2 to 1, it means the mark up was 100% since marked price is twice of cost price.

  3. Student says:

    Hi, I’m quite confused with the question. The customer receives 20 articles for the price of 15. Doesn’t it mean that: (1+m%)*(1-25%)? Why we need to multiply (1-20%) here? Thanks!

    • Karishma says:

      Because the first line of the question says that the dealer offers a cash discount of 20%. This is in addition to the 25% discount obtained by giving away 20 articles for the price of 15. So there are 2 discounts being offered here.

      • Student says:

        Thanks! I though the situation was:
        Boss: I can offer you 20% off.
        Me: Nope.
        Boss: OK! If you buy 20 articles, You can just pay the price for 15 articles in “original” price.

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