Math CentralQuandaries & Queries


Question from z:

Hello math Central,
thank you for your help with my query,

i have found your answer to another to calculate the g.s.t. and or p.s.t. from the BASE amount on an invoice
In 1 calculation, one arrives at the Original cost of goods

But this requires an additional calculation & step to extract the gst and or pst FROM the pretax amount from a sale/purchase

My question to you is,
What is the math formula and method I can use, IN 1 STEP to extract ONLY both(pst/gst) & or gst TAXES from the total sale/purchase?

i.e.,, in one of your examples, you utilized $214.00 as the total invoice sale
in that example, i understand to obtain the original cost of goods, FROM the totaled invoice ($214),
making a reverse-division of 1.07 ,will provide me with the original cost of goods
FROM the totaled invoice($241.00) is $200.00

What are the formula's, in a 1-step process where i can isolate the taxes ONLY from the Totaled invoice ($214.00)
or from an amount including pst & gst , such as $220.00 ?

I would like the formula i can use which will show me on my calculator the gst amount eg. $14.00 FRom the $214.00
and the formula I can use in 1-calculation to obtain the pst and gst ,,i.e. $10.00 pst, and $10.00 gst
FROM the total invoice eg. $220.00 -where the base cost of goods is $200

thank you
i look forward to hearing from you


I'm not sure I completely understand what you want but I'll give it a try. I'm going to start with your example.

Suppose you have an item where the total invoice is $214.00. You are in Saskatchewan where the gst and pst are both 5% so you paid 10% tax on this item. Thus the cost of the item before tax was $214.00/1.10 = $194.55 and hence the total tax was $214.00 -$194.55 = $19.45. Since the two tax rates are the same (5%) the gst and pst are both $19.45/2 = $9.73.

To do this in one calculation let's suppose the total invoice is $T then the gst and pst are each

0.05 × $T/(1.10).

This calculation is only appropriate in Saskatchewan where the gst and pst are both 5% and only applies to items where the gst and pst are both charged. There is a Wikipedia page that outlines the gst, pst and hst rates across the country.

I hope this helps,


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