Math CentralQuandaries & Queries


Question from Cynthia, a parent:

I paid $15.98 for an item. Both GST (.96) & PST ($1.12) were added for a total of $18.06. I know that .96 is GST (6%) and $1.12 is PST (7%).
Other than looking at the receipt, if all I have is the total amount paid, how do I calculate how much of the total are taxes?
This is what I've tried so far:
I multiply $18.06 by 0.11504 (13%) and come up with the correct total tax $2.08 (13%).
When I multiply $18.06 by 0.05660 (6%) I get an incorrect amount: $1.02 but the actual is only $.96.
When I multiply $18.06 by 0.06542 (7%) I get an incorrect amount: $1.26 but the actual is $1.12.
What am I doing wrong?

We have two responses for you

Hi Cynthia,

You need to recover the amount of the bill before the taxes are added since the GST is 6% of this pre-tax amount and the PST is 7% of the pre-tax amount. You correctly found the total taxes to be $2.08 so the pre-tax amount is

$18.06 - $2.08 = $15.98

Hence the GST is

0.06 × $15.98 = $0.96

and the PST is

0.07 × $15.98 = $1.12




In all, you payed $18.06 which is 113% of the marked price $15.98, of which 13/113 = 0.11504 is the tax. The part of GST is 6/113 = 0.05310, rather than 6/106 = 0.05660 as you computed. So the GST you payed is 0.05310 x $18.06 = $0.96. The part of PST is 7/113 = 0.06195 rather than 7/107 = 0.06542 as you computed. So the PST you payed is 0.06195 x $18.06 = $1.12. Also, the marked price is
(100/113) x $18.06 = (0.88496) x $18.06 = $15.98.

For the cash register, it is as easy to compute 0.05310 x $18.06 as it is to compute 0.06 x 15.98. So, for the stores, it would be easy to adopt a policy of always putting the "honest'' price (like $18.06) including all taxes rather than prices without taxes. It is done in most other countries. After all, for the customer the important information is the price they will really have to pay, not not which part of it will remain to the store after the taxes are paid.

In fact, the store owners already know that, but they use it to lie: If you go to the sale where the store announces "We pay the GST and the PST'' what it really means is that "all taxes are included in the price''. So if you buy an item marked $18.06, the store will pay $0.96 in GST, rather than 0.06 x $18.06 = $1.08.

Here it is customary for stores to not include taxes in prices, and also to have prices where a few pennies are dropped off a simple round number: $15.98 instead of $16.00. When the taxes were 15%, I would make a game out of rounding up, then adding 10% ($1.60) and half of that ($0.80), and then subtract two pennies to get the correct $18.38 price with taxes before the cashier could punch it up. But I believe that most people adopt the opposite attitude: All the numbers in real life are complicated lies, so unconsciously they stop thinking instead of starting to count. And they have to go through elementary math, high school algebra and calculus with that involuntary reflex. No wonder they find math hard.


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