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Apologies in advance if I have not formatted this problem correctly.

Context: I need to find a way to calculate a number that will equal a service fee applied to a product, taking into account the service fee will also be applied to this number.

The system is sort of a middle man between a client and supplier. Currently the the supplier takes the cost of the service charge. We need to be able to allow the client to take this charge. And to do that it it needs to be added as a line item on the invoice, the problem is the service charge will be applied to the line item also as it is calculated using the total transaction value. So in the case above a £2 line item could be added to offset the original service charge, but the 2% would also be applied leaving that (£0.04) unaccounted for.

Example: The cost of a product is £100, the an service fee would be (2%) £2.

In this case the number couldn't be £2 because a 2% fee would also be applied to the £2 leaving £0.04.

When I first looked at this problem I originally thought the value could be:

((Cost of Product) * 0.02) + ((Cost of Product) * 0.02) * 0.02)

But this is wrong also as there is still a small amount remaining.

Is there an easy way to calculate what the value should be?

Liam Russell
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    Just for context, you want to apply service fee on the service fee? – Dhanvi Sreenivasan Aug 03 '21 at 12:04
  • I see two options here: Either you are really confused by something, or you are applying a service fee to a service fee. The latter is of course possible, but I don't know how that would occur in real life. Any comments? If you have a £100 product then the service fee is £2 and the total is £102. End of story. What's there more to calculate? – Matti P. Aug 03 '21 at 12:16
  • If I understand correctly, you are just after the sum of the infinite series $.02+.02^2+.02^3+\cdots$. That is a standard Geometric Series and the usual formulas apply. – lulu Aug 03 '21 at 12:32
  • Sorry for the confusion, the system is sort of a middle man between a client and supplier. Currently the the supplier takes the cost of the service charge. We need to be able to allow the client to take this charge. And to do that it it needs to be added as a line item on the invoice, the problem is the service charge will be applied to the line item also as it is calculated using the total transaction value. So in the case above a £2 line item could be added to offset the original service charge, but the 2% would also be applied leaving that (£0.04) unaccounted for. – Liam Russell Aug 03 '21 at 12:35
  • @lulu Thank you I will look into this – Liam Russell Aug 03 '21 at 12:36
  • Do you want to apply first a service fee of $2%$ on product, say $SF_1$, i.e. $SF_1=2%$ of Cost of Product, then again $2%$ on this service fee $SF_1$, say $SF_2$ , i.e $SF_2$= $2%$ of $SF_1$, and then again a service fee of $2%$ on $SF_2$, say $SF_3$ and so on.. $SF_3$= $2%$ of $SF_2=2%$ of ($2%$ of $SF_1$). This way your total cost would be [Cost of Product + $SF_1 +SF_2+SF_3+....$] . This makes it a question of Infinite G.P. And so, the required sum will be $\frac{P}{1-0.02}=\frac{P}{0.98}$ where $P$ is the cost of product. – Aman Kushwaha Aug 03 '21 at 12:38
  • Does this answer your question? A is greater than B by 25% then by what percentage B is less than A? – Ethan Bolker Aug 03 '21 at 14:55

2 Answers2

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You want to make the client pay some extra price as line item to account for the service charge which the supplier had to pay. But your problem is that, when you add line item having price say $T$ such that $T= \frac{2}{100} P$ , the supplier still had to pay the service charge on line item (equal to $2\%$ of $T$) which is unaccounted. $\tag{*} \label{*}$ So what you decide to do is that, you again add that unaccounted amount in the line item so that your new price of line item is $T'$ such that $$T'= T+ \frac{2}{100} T= \frac{2}{100} P+\frac{2}{100}\frac{2}{100}P$$ Yet again, you face the same problem but the unaccounted amount is now lesser. So you do this infinite times to reduce the unaccounted amount to zero. So your final price of line item should be $T_{final}= \frac{2}{100} P+\frac{2}{100}\frac{2}{100}P +....= 0.02 P+(0.02)^2 P +(0.02)^3 P+...=\frac{0.02 P}{1-0.02}=\frac{P}{49}$.

This $T_{final}$ is the price of line item, i.e the number you were referring to in the question. The total price in the invoice will be $P+ T_{final} =P+\frac{P}{49}=+\frac{P}{0.98}$

$\ref{*}$ From here you can simply add the price of line item in that service charge with the following equation as done by @TonyK in his answer, instead of doing that infinite sum.

$T=\frac{2}{100}(P+T)$

Solving for $T$ you'll directly get $T =\frac{P}{49}$

Aman Kushwaha
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  • It looks like you have misunderstood the question. (Which is strange, because I see you got it right in your comment to the OP!) – TonyK Aug 03 '21 at 13:07
  • @TonyK Maybe you are right but what I understood in the comment is also strange because the OP didn't mention anything like the word infinite or the service charge being added infinitely many times. After reading the comment (where words like offset is used) by OP 30 mins ago, I think now that this is what he actually asked, however I'm not completely sure. – Aman Kushwaha Aug 03 '21 at 13:09
  • @TonyK Yes, you are actually right. Now that I realised that that the infinite GP sum and simply including the line item in the $2%$ service charge are serving the same purpose, I edited my answer. Thank you – Aman Kushwaha Aug 03 '21 at 14:07
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Given the supplier's base price $P$, you want to know the total price $T$ so that after the middle man takes 2% of $T$, what's left is $P$. (Right?)

So we have $T-0.02T=P$, which you can rearrange as $T=P/0.98$.

And if $P=£200$, then $T=£204.08$ to the nearest penny. The middle man takes 2% of this, which is $£4.08$ to the nearest penny. This leaves $P=£200$ for the supplier, as desired.

TonyK
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