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D-bay, an online clothes and fashion store, is giving a 20% cashback on purchases worth €1000 or above made on its website. (i.e. when you purchase something worth, say, €3000, your account will be credited back with an amount of €600). If Alex has €10,000 with him, what is the maximum value of the products which he can purchase from the website ?

My approach:-

First spending = €10000, Cashback received = €2000

Second spending = €2000, Cashback received = €400

Third spending = €400, Now no cashback would be received as spending is < €1000

which gives me a total of €10000+€2000+€400=€12400

but the answer is given as €12450

Fin27
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1 Answers1

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Your approach looks fine. However, give a closer look for Third Spending since, the amount is less than €$1000$ you are not getting any cashback. In an ideal scenario you want non-cashback eligible amount (which is €400 in your example) to be as small as possible. Consider the following example: You have €1000 in hand, instead of spending it in one go you spend €500 two times. Is the total spending same? No, when you spend €1000 in one go, total spending is €1000 + €200, in the second case its just €500+500=€1000. This is because you did not receive any cashback on both of the €500. Hence, the < €1000 transaction should be minimum.

Let's try to optimize that, How about we spend €$x$ in second spending? We want the amount for third transaction to be €1000 so that our non-cashback eligible amount becomes €200 which is the minimum possible. This translates to:

$$ \left(\frac{20}{100}\times x \right)+2000-x = 1000 $$ where $\left(\frac{20}{100}\times x \right)$ is the cashback received from second transaction and $2000-x$ is the balance amount. You equate it to $€1000$ so that the next immediate transaction is of €1000 and the final terminal transaction (<€1000 and hence, non-cashback eligible) is of €200 which is our goal.

Hence, $x=€1250$. Your transactions look like:

First spending = €10000, Cashback received = €2000

Second spending = €1250, Cashback received = €250, Balance = €2000 - €1250 = €750

Third spending = €250 + €750 = €1000, Cashback received = €200

Fourth spending = €200, Now no Cashback would be received as spending is < €1000

Total Spending = €(10000+1250+1000+200) = €12450

We cannot do better than this because the minimum cashback possible is 20% of €1000 = €200 and our final spending = €200. Hence we are not losing out on any cashback possible.

Shiv Tavker
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  • What is the logic behind minimising the last cashback to 1000 , if our motive is to rather maximise our cashbacks, why do I need to restrict my cashback to 1000 – Fin27 Oct 30 '21 at 20:38
  • And please break down the the equation from where you got that 1250 , what exactly have you equated – Fin27 Oct 30 '21 at 20:39
  • @Fin27 The motive is to pay the minimum amount as the final transaction (where you do not receive any cashback). – Shiv Tavker Oct 30 '21 at 20:49
  • Since the cashback from second last transaction becomes ur final transaction. Its best to keep that cashback as low as possible. Hence instead of spending 2000 directly and getting 400 cashback you spend it in parts to have final transaction (non-cashback transaction) = 200. – Shiv Tavker Oct 30 '21 at 20:52
  • But ultimately all the amount is being spent only , be it the one received from cashback or the amount on which cashback can't be achieved , then why are we focusing to make the non-cashback transaction to be minimum – Fin27 Oct 31 '21 at 03:41
  • My question is If we really want to maximize the amounts received from cashback, why would I not then maximise my last cashback too – Fin27 Oct 31 '21 at 03:43
  • And also why did you choose second transaction specifically only ? – Fin27 Oct 31 '21 at 03:47
  • Say if you make a spending of €x, and x>€1000. Ultimately you are going to spend €(x + x/5). Implying your total spending increases. Consider the following example: You have €1000 in hand, instead of spending it in one go you spend €500 two times. Is the total spending same? No, when you spend €1000 in one go, total spending is €1000 + €200, in the second case its just €500+500=€1000. This is because you did not receive any cashback on both of the €500. Hence, the < €1000 transaction should be minimum. – Shiv Tavker Oct 31 '21 at 07:29
  • That's an interesting question now, why just the second transaction? Let's think backwards your last transaction is always going to be <€1000 because other wise you will get a cashback and spend that resulting in another transaction.. so technically it cannot be called the last transaction. So now we know we want the last transaction = €200. Second last should be €1000 in that case. This brings us to optimize third last transaction which is of €2000. Why not alter the first transaction? the cashback received is already greater than €1000 and hence not a terminal transaction. – Shiv Tavker Oct 31 '21 at 07:39