   SEARCH HOME Math Central Quandaries & 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,
Penny     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.