Math CentralQuandaries & Queries


Question from carole, a parent:

I work at a company where we use transport companies and we often get credits
on their invoices. I need to know how to subtract the sales taxes from these credit amounts.
(5% TPS and 7.5% TVQ).
Is the equation: Amount / 1.05 = then this amount / by 1.075=?

Hi Carole,

What you suggest is correct in Quebec because of the way that the taxes are imposed. If an item has a cost of $C then the cost with the 5% federal tax included is 1.05 × $C. The 7.5% provincial tax is then calculated on this amount so the final cost $F to your customer is

$F = 1.075 × 1.05 × $C.

Thus if your customer gives you a credit of $F then

$C = $F/(1.075 × 1.05).

If you need to determine the individual taxes paid you need to be slightly more careful.

The amount on which the provincial tax was calculated is $F/1.075 so the provincial tax paid was $F - ($F/1.075) which simplifies to

provincial tax paid = (0.075/1.075) × $F.

The amount of federal tax paid was $F/1.075 - $C which simplifies to

federal tax paid = (0.05/(1.075 × 1.05) × $F.

Hence, for example if you get a credit of $200 then the provincial tax paid was

(0.075/1.075) × $200 = $13.95

and the amount of federal tax paid was

(0.05/(1.075 × 1.05) × $200 = $8.86

and your cost was then $200 - (13.95 + 8.86) = $177.19.

I hope this helps,

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