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Sales tax

Ratio, rates and money · level 4, lesson 4 of 6

A question to start

Kim has saved exactly 60 dollars for a video game. The price tag in the shop in Toronto says 60.00. She takes the game to the till, hands over her money, and the cashier says: "That's 67.80, please."

Kim is 7.80 dollars short. Nobody made a mistake. The extra money is sales tax, and in Canada it is usually added at the till, not shown on the price tag.

Where does the 7.80 come from? And how can Kim work out, before she gets to the till, what she will really pay? By the end of this lesson you will be able to do it in your head.

The idea

What sales tax is

A tax is money that people and businesses must pay to the government. Governments use taxes to pay for schools, hospitals, roads, parks and much more. A sales tax is a tax added on top of the price when you buy something. The shop collects it from you and passes it on to the government.

Sales tax is a percent of the price. That is why it is in this level: every sales tax question is a "percent of an amount" question from lesson 2.

Sales tax in Canada

In Canada there is a federal tax called the GST (Goods and Services Tax) of 5%. Most provinces add a tax of their own.

  • In Ontario, the federal and provincial taxes are combined into one HST (Harmonized Sales Tax) of 13%: 5% federal plus 8% provincial. Ontario has used 13% since July 2010.
  • In Alberta there is no provincial sales tax, so shoppers pay only the 5% GST.
  • Other provinces have their own rates and rules.

Tax rates are set by governments and can change, so always check the current rate where you live. In this lesson we use Ontario's 13% for most examples.

Not everything is taxed. In Canada, basic groceries such as milk, bread, fruit and vegetables have no GST or HST, while snacks, soft drinks and restaurant meals do. In Ontario, some items (for example children's clothing and printed books) pay only the 5% federal part. The rules have many details; a receipt shows which items were taxed.

Working out the tax

The tax is 13% of the price. Build it from easy pieces:

\[13\% = 10\% + 3\%.\]

For the 60 dollar game:

  • 10% of 60 is 6.00 dollars.
  • 1% of 60 is 0.60 dollars, so 3% is 1.80 dollars.
  • Tax: \(6.00 + 1.80 = 7.80\) dollars.
  • Total: \(60.00 + 7.80 = 67.80\) dollars.

That is exactly what the cashier said.

One step: multiply by 1.13

There is a quicker way to get the total. You pay the whole price (100%) plus the tax (13%), which is 113% of the price:

price: 100% 13% total: 113% of the price

And 113% as a decimal is 1.13. So

\[\text{total} = \text{price} \times 1.13.\]

For the game: \(60 \times 1.13 = 67.80\) dollars. Same answer, one step. Where the tax is 5%, multiply by 1.05.

Why is it allowed to combine the two steps? Because price + 13% of price is "1 lot of the price plus 0.13 lots of the price", which is 1.13 lots of the price. This number, 1.13, is called a multiplier. You will use multipliers for every kind of percent change in later levels.

A bill with several items

When you buy several taxable things, the tax is worked out on the subtotal (the total before tax), then rounded to the nearest cent:

Item Price
Notebook 3.49
Pens 5.99
Backpack 32.99
Subtotal 42.47
HST 13% 5.52
Total 47.99

The tax is \(42.47 \times 0.13 = 5.5211\) dollars, which rounds to 5.52 dollars.

Paying cash with no pennies

Canada stopped making pennies in 2012 and stopped handing them out to shops in 2013. When you pay cash, the total is rounded to the nearest 5 cents. When you pay by card, you pay the exact amount.

Total ends in Cash rounds to
1 or 2 cents down to 0
3 or 4 cents up to 5
6 or 7 cents down to 5
8 or 9 cents up to the next 0

So the 47.99 bill above costs 48.00 dollars in cash, and 47.99 dollars by card.

Estimating tax quickly

13% is very close to \(\tfrac18\), which is 12.5%. So a quick estimate of the tax is "divide by 8", and of the total, "add an eighth". For a 40 dollar item: \(40 \div 8 = 5\), so the total is about 45 dollars (exactly \(40 \times 1.13 = 45.20\)). Since 13% is a little more than 12.5%, the real tax is a little more than this estimate.

Tax-included prices

Some prices already include the tax, such as the price shown on a gas pump. If a price includes 13% tax, it is 113% of the price before tax. To find the price before tax, divide by 1.13. A tax-included price of 113 dollars was 100 dollars before tax, with 13 dollars of tax.

Notice that taking 13% off 113 dollars does not get you back to 100: 13% of 113 is about 14.69, and \(113 - 14.69 = 98.31\). The 13% was a percent of the price before tax, not of the total. You will meet this kind of "reverse percent" again in level 6.

Worked examples

1. The video game. 60 dollars with 13% HST.

  • Tax: \(10\% + 3\% = 6.00 + 1.80 = 7.80\) dollars.
  • Total: \(60 \times 1.13 = 67.80\) dollars.

2. Some taxed, some not. At an Ontario grocery store you buy milk (5.49), bread (3.29) and a bag of chips (4.99). Only the chips are taxed.

  • Tax on the chips: \(4.99 \times 0.13 = 0.6487\), which rounds to 0.65 dollars.
  • Total: \(5.49 + 3.29 + 4.99 + 0.65 = 14.42\) dollars.
  • In cash: 14.42 ends in 2, so it rounds down to 14.40 dollars.

3. Ontario and Alberta. A bike costs 250 dollars before tax.

  • In Ontario (13%): \(250 \times 1.13 = 282.50\) dollars.
  • In Alberta (5%): \(250 \times 1.05 = 262.50\) dollars.
  • Difference: 20 dollars, which is the 8% provincial part: 8% of 250 is 20. ✓

4. Can I afford it? Two shirts cost 22 dollars each in Ontario. Is 50 dollars enough?

  • Subtotal: \(2 \times 22 = 44\) dollars.
  • Total: \(44 \times 1.13 = 49.72\) dollars. Yes, just.
  • Change from 50 dollars: by card, 0.28 dollars; in cash, the total rounds to 49.70, so the change is 0.30 dollars.

5. A tax-included price. A tax-included price is 56.50 dollars with 13% HST. What was the price before tax?

  • \(56.50 \div 1.13 = 50\) dollars before tax.
  • The tax was \(56.50 - 50 = 6.50\) dollars. Check: 13% of 50 is 6.50. ✓

Try it

  1. Find the 13% HST on a 20 dollar item, and the total.
  2. Find the total for a 45 dollar item with 13% HST.
  3. In Alberta, how much does an 80 dollar item cost with 5% GST?
  4. A bill has items at 12.99, 7.50 and 3.25 dollars, all taxed at 13%. Find the subtotal, the tax and the total. What would you pay in cash?
  5. Use the multiplier to find the total for a 149.99 dollar item with 13% HST.
  6. Estimate the 13% tax on a 72 dollar item by dividing by 8. Then find the exact tax. Is the estimate a bit high or a bit low? Why?
  7. A price including 13% HST is 226 dollars. What was the price before tax?
  8. Explain in your own words why "price times 1.13" gives the total with 13% tax.
Answers
  1. 10% is 2.00 and 3% is 0.60, so the tax is 2.60 dollars. Total 22.60 dollars.
  2. \(45 \times 1.13 = 50.85\) dollars. (Tax: \(4.50 + 1.35 = 5.85\).)
  3. 5% of 80 is 4, so \(80 + 4 = 84\) dollars. (Or \(80 \times 1.05 = 84\).)
  4. Subtotal: \(12.99 + 7.50 + 3.25 = 23.74\) dollars. Tax: \(23.74 \times 0.13 = 3.0862\), about 3.09 dollars. Total: 26.83 dollars. In cash, 83 cents ends in 3, so it rounds up to 26.85 dollars.
  5. \(149.99 \times 1.13 = 169.4887\), which rounds to 169.49 dollars.
  6. Estimate: \(72 \div 8 = 9\) dollars. Exact: \(72 \times 0.13 = 9.36\) dollars. The estimate is a bit low, because dividing by 8 finds 12.5%, and 13% is a little more than that.
  7. \(226 \div 1.13 = 200\) dollars. (Check: \(200 \times 1.13 = 226\).)
  8. For example: "You pay all of the price, which is 100% of it, and also 13% of it as tax. Together that is 113% of the price, and 113% written as a decimal is 1.13."

Common mistake

Forgetting the tax. In Canada most price tags do not include sales tax. When you plan what you can afford, add the tax first. A quick "add an eighth" estimate is better than nothing.

Removing tax by subtracting 13%. To go back from a tax-included price, divide by 1.13. Subtracting 13% of the total takes off too much, because the total is bigger than the price the tax was based on.

Where it leads