Algebra formula problem

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rayarthun

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Situation: My customer wants to pay my $100 invoice with a credit card but the credit card company charges "Fees" equal to 3.5% + $.15 of the gross amount charged to the credit card" (let's call this Gross Charge "X"). My customer has happily agreed to pay the $100 AND the Fees. Reminder: For purposes of my question below, I want to "net" exactly $100 (let's call this "Y").

Question: What is the formula for calculating the Gross Charge?

I'm stumped: If I were to simply try to enter a Gross Charge of $103.65 I know I would get less than $100 deposited in my account (as you can see from the example below). I understand that the problem is because the Fee calculation is on the "Gross" Charge after the Fees. But I just can't figure out how to solve for the Gross Charge.

Here's an example of how the credit card company calculates it:
Note: This example does not comport to my request for a deposit of exactly $100 but is inserted here to demonstrate the formula used on a "actual" amount (that being X = $113.54 and Y = 117.81). Total Fees are $4.27. Therefore $4.12 is 3.5% of X or X = 4.12/.035 = 117.7142857142857. Then I add .15 for a total of 117.8642857142857 rounded to $117.86 (if you follow IRS rounding rules). And it looks like I'm a nickle off for some reason.

Finally: Assuming the credit card company is doing it the right way, my question is: How do I use "their" formula to calculate a Gross Charge that will "net" $100.

Thanks for any help.

Ray
 
This is what algebra is for!

Unfortunately, it's hard to be sure what you mean by X; in your example, you say X = 113.54, but then you calculate X = 117.71. I'm confused.

As I understand it, X is the amount they pay with the credit card; the company then subtracts 3.5% of that, plus $0.15, and gives you the rest. Is that what Y is, which you want to be 100? I'll suppose it is, though in your example Y is greater than X.

Let's do your calculation leaving X as a variable. You have fees of 3.5% (that is, 0.035X) plus $0.15. So the net amount you receive is X - 0.035X - 0.15. You want this to equal $100. So we solve the equation,

X - 0.035X - 0.15 = 100​

Do you recall how to solve this equation for X?

In your example, if I replace 100 with 113.54 (assuming you swapped X and Y there), then when I solve for X I get $117.81, so this looks right.
 
You are correct, I did mislabel X and Y.

No, I do not recall how to solve this equation for X.
 
Okay. Here's the equation, with Y in place of 100:

X - 0.035X - 0.15 = Y​

We first combine like terms on the left; 1X - 0.035X = 0.965X. That is, 100% minus 3.5% leaves 96.5%:

0.965X - 0.15 = Y​

Now we add the 0.15 to both sides:

0.965X = Y + 0.15​

Now to solve for X, we divide by its coefficient. much like you did (which is why I thought you might want to try solving):

X = (Y + 0.15)/0.965​

That's the formula we want.

If, as in your example, Y = $113.54, we get X = (113.54 + 0.15)/0.965 = $117.81.

If Y = $100, we get X = (100 + 0.15)/0.965 = $103.78.

To check this out, 3.5% of this is 3.63; subtracting that from 103.78 leaves 100.15; subtracting the $0.15 leaves exactly $100.
 
Thanks. This is exactly what I needed. However, it looks like I may have a rounding discrepancy with the credit card company. Because, when I looked at another "actual" example, I was a penny off. Here is the other example: Gross Charge to the credit card company of $1029.00. Net to me was 992.83. Total Fees they report of $36.17 ($0.15 + 3.5% of $1,029 (rounded up on $0.50). This makes sense yet when I insert the raw numbers into the formula, my Excel spreadsheet (attached) seems to round down to 1028.99 for some reason. Any ideas?
 
Oops. Oops. It doesn't look like I can attach my Excel spreadsheet. If you need it, let me know what extension I need (I've already tried .ods and .csv).
 
You can just attach an image of the cells you want to show. Or you could just copy and paste the cells into your post.

But I don't need that. Working forward, I get 0.035*1029-0.15 = 36.165, which rounds to 36.17 (nearest penny); then 1029 - 36.17 = 992.83 as they say. If we save rounding until the end, as a spreadsheet will do, we get 1029 - 36.165 = 992.835, which rounds to 992.84. So they are giving you a penny less than they might, by doing intermediate rounding, which I presume is standard in finance. (The difference is because the exact value was exactly between 83 and 84 cents, and subtraction flips the direction in which we round.)

Working backward with my formula, if your goal were to get 992.83, then you would calculate (992.83 + 0.15)/0.965 =1028.994819, which rounds down to 1028.99. That's just the way rounding works. If we'd used the unrounded value of 992.835, we'd get back 1029 exactly. (This is why in math or engineering, we avoid intermediate rounding; in finance, I imagine, rounding is traditionally done early, as if all work were done in a field that has no digits to the right of the penny.)

Now, if you put 1028.99 into the forward formula, you'll get 992.83, just as you did for 1029. So, in fact, you would be getting the amount you wanted, but the customer would save a penny! All through the magic of rounding ...
 
Dr.Peterson said:
You can just attach an image of the cells you want to show. Or you could just copy and paste the cells into your post.

But I don't need that. Working forward, I get 0.035*1029-0.15 = 36.165, which rounds to 36.17 (nearest penny); then 1029 - 36.17 = 992.83 as they say. If we save rounding until the end, as a spreadsheet will do, we get 1029 - 36.165 = 992.835, which rounds to 992.84. So they are giving you a penny less than they might, by doing intermediate rounding, which I presume is standard in finance. (The difference is because the exact value was exactly between 83 and 84 cents, and subtraction flips the direction in which we round.)

Working backward with my formula, if your goal were to get 992.83, then you would calculate (992.83 + 0.15)/0.965 =1028.994819, which rounds down to 1028.99. That's just the way rounding works. If we'd used the unrounded value of 992.835, we'd get back 1029 exactly. (This is why in math or engineering, we avoid intermediate rounding; in finance, I imagine, rounding is traditionally done early, as if all work were done in a field that has no digits to the right of the penny.)

Now, if you put 1028.99 into the forward formula, you'll get 992.83, just as you did for 1029. So, in fact, you would be getting the amount you wanted, but the customer would save a penny! All through the magic of rounding ...
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