Find rate when changing discount and vice-versa while keeping a fix final price

Question

I am a web developer and building an accounting system for a client however I am almost good at programming but very bad at accounting and maths; and my client is not able to explain/communicate properly and present the equation of the calculations. So, here is the matter:

This is about creating bill of sale of products and calculating rate, discount, tax and total. Screenshot of a related portion of the software

Suppose,

Original price (op) = 100
Discount (d) = 5
So, selling price (sp) (including tax) = 95
Tax rate (tr) = 15%

There are textboxes for showing product rate (pr), tax amount (ta) and taxable value (tv) which need to be calculated from about data. I am able to calculate these initially as following:

tv = sp / (1 + (tr / 100))
   = 95 / (1 + (15 / 100))
   = 95 / 1.15
   = 82.61

ta = tv * (tr / 100)
   = 82.61 * (15 / 100)
   = 82.61 * 0.15
   = 12.39

pr = tv + d
   = 82.61 + 5
   = 87.61

But I don't know and the client is not able to explain properly how to find other value (discount or product rate) if I change any of these two. If I change the discount then product rate and tax amount need to be recalculated, and if I change the product rate then discount and tax amount need to be recalculated. The tax rate (tr) 15 and selling price (sp) 95 are constants/fixed/readonly/non-editable.

According to this description, please give equations to find discount and product rate if any of these two is changed.

Answer 1

Assume you have a new pair of product rate ($pr$) and discount ($d$). Then $ tv = pr - d $

Tax amount ($ ta $) can then be calculated in the exact same way as before from the new taxable value ($tv$). Selling price ($sp$) equals

$ sp = \dfrac{100+tr}{100} \cdot tv $.

Let's see what happens if we assume a constant selling price ($sp$) and a constant tax rate ($tr$). Then

$tv = \dfrac{sp}{1 + \dfrac{tr}{100}}$

This means that taxable value ($tv$) is also constant. Now this implies that tax amount is also constant. In this case all that can change if we change the discount is
$pr = tv + d$
and if we change the product rate all that changes is discount
$d = pr - tv$.

Answer 2

Given that sp and tr are constant and that tv is a calculated value from sp and tr, by definition tv is also constant.

And considering that ta also is calculated based off of constant values, ta is also constant in both below scenarios.

ta = tv * tr / 100

Scenario 1, for discount changes,

pr = tv + d

Scenario 2, for product rate changes,

d = pr - tv

Is there anything i am missing?