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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

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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.

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  • $\begingroup$ 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 $\endgroup$
    – Fin27
    Oct 30, 2021 at 20:38
  • $\begingroup$ And please break down the the equation from where you got that 1250 , what exactly have you equated $\endgroup$
    – Fin27
    Oct 30, 2021 at 20:39
  • $\begingroup$ @Fin27 The motive is to pay the minimum amount as the final transaction (where you do not receive any cashback). $\endgroup$ Oct 30, 2021 at 20:49
  • $\begingroup$ 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. $\endgroup$ Oct 30, 2021 at 20:52
  • $\begingroup$ 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 $\endgroup$
    – Fin27
    Oct 31, 2021 at 3:41

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