B. Number · Number: Number Sets, Powers and Rational Numbers in Action

Percentages: Taxes, Discounts, Percent Change and Interest

B3.5 A1 A2About 75 minutes

Learning goals

I can find a percent of an amount and use multipliers like 1.13 and 0.75 for increases and decreases.
I can calculate percent change and explain whether it is an increase or a decrease.
I can work backward to find an original amount, such as a price before 13% HST.
I can solve problems with successive percent changes, simple interest and percents in data.

Words to know

percent
A ratio out of 100. 13% = 13/100 = 0.13.

multiplier
The decimal you multiply by to apply a percent change: +13% → × 1.13; −25% → × 0.75.

percent change
(new − original) ÷ original × 100%. Positive means increase; negative means decrease.

HST
Harmonized Sales Tax. In Ontario it is 13% on most purchases.

simple interest
Interest paid only on the original amount (principal): I = Prt, where P is the principal, r is the annual rate as a decimal, and t is time in years.

principal
The amount of money first saved, invested or borrowed.

Remember from Grade 8

In Grade 8 you used percents over 100% and under 1%, found discounts and 13% HST, and solved percent problems with proportions. Grade 9 makes this faster and more powerful with multipliers, and adds percent change, working backward and repeated changes.

The multiplier idea

To find 13% of $80: 0.13 × 80 = $10.40.
To add 13%: the total is 100% + 13% = 113% of the price, so multiply by 1.13: 1.13 × 80 = $90.40.
To take off 25%: you pay 100% − 25% = 75%, so multiply by 0.75.

One multiplication does the whole job.

Adding 13% HST to an $80 price. The total is 113% of the original price, so multiply by 1.13. (Bar lengths are to scale.)

Worked example

Example 1: Discount, then tax

Skates cost $64.00. They are 25% off, then 13% HST is added. What is the total?

Sale price: you pay 75%, so 0.75 × 64 = $48.00.
Add HST: 1.13 × 48 = $54.24.
One step: 64 × 0.75 × 1.13 = $54.24.

$54.24

Worked example

Example 2: Work backward

A receipt shows a total of $135.60, including 13% HST. What was the price before tax?

The total is 113% of the price: 1.13 × p = 135.60.
Divide by the multiplier: p = 135.60 ÷ 1.13 = 120.00.
Check: 1.13 × 120 = 135.60 ✓. Note: taking 13% of $135.60 and subtracting gives $117.97, which is wrong, because the 13% was calculated on the smaller amount.

$120.00

Worked example

Example 3: Percent change

a) A town grows from 12 500 to 13 750 people. b) The price of a litre of gas drops from $1.60 to $1.40. Find each percent change.

a) Change = 13 750 − 12 500 = 1250. 1250 ÷ 12 500 = 0.10 = +10%.
b) Change = 1.40 − 1.60 = −0.20. −0.20 ÷ 1.60 = −0.125 = −12.5%.
Always divide by the original value.

a) +10% b) −12.5%

Repeated percent changes multiply

When one percent change is followed by another, the second is a percent of the new amount. So the multipliers multiply; the percents do not simply add.

20% off, then another 10% off: 0.80 × 0.90 = 0.72. You pay 72%, so the total discount is 28%, not 30%.
Up 50%, then down 50%: 1.5 × 0.5 = 0.75. You end 25% lower than you started.

A $100 item at 20% off, then 10% off the sale price. The second discount is only 10% of $80 ($8), so the final price is $72: 28% off in total.

Worked example

Example 4: Simple interest

Maya puts $1500 in a savings account that pays 3.5% simple interest per year. How much interest does she earn in 2 years? What is the total?

I = Prt = 1500 × 0.035 × 2.
1500 × 0.035 = 52.50 per year, × 2 = $105.
Total = 1500 + 105 = $1605.

$105 interest; $1605 total. (In the Financial Literacy unit you will compare this with compound interest, where interest also earns interest.)

Worked example

Example 5: Percents in data

A class of 28 students voted: 7 for skating, 12 for snowshoeing, 9 for tobogganing. What percent chose each? If the whole school of 560 voted the same way, how many would choose snowshoeing?

Skating: 7/28 = 0.25 = 25%. Snowshoeing: 12/28 ≈ 0.4286 ≈ 42.9%. Tobogganing: 9/28 ≈ 0.3214 ≈ 32.1%.
Check: 25 + 42.9 + 32.1 = 100% ✓.
School: 12/28 × 560 = 240 students.

25%, about 42.9%, about 32.1%; about 240 students

Percent or percentage points?

If an interest rate rises from 2% to 3%, it went up 1 percentage point, but that is a 50% increase in the rate (1 ÷ 2 = 0.5). News reports sometimes mix these up, so read carefully.

Hands-on activity

Receipt and flyer detective

You need: a store flyer and a receipt (or an online shopping page), a calculator.

Pick 3 sale items. For each, find the percent off: (sale − regular) ÷ regular × 100%.
Find each final price with 13% HST using one multiplication (for example 0.70 × 1.13 = 0.791).
Take a receipt total that includes HST. Work backward to find the pre-tax total. Does it match the subtotal printed on the receipt? (Some items, like basic groceries, are not taxed, so it may not match exactly. Can you explain why?)
Find a "buy one, get one 50% off" deal. What percent do you save overall on two items? (Model answer: you pay 150% for 200% worth, so you save 50/200 = 25%.)

Just for fun

Riddle: Why did the 13% HST refuse to be removed by taking 13% off the total?

Because it knew it was 13% of a different number. (That's why you divide by 1.13!)

Your turn

Find the total cost of a $45 hoodie with 13% HST.

A $250 bike is 30% off. What is the sale price before tax? After tax?

A total including 13% HST is $56.50. Find the pre-tax price.

A hockey team's ticket price rises from $80 to $92. Find the percent change.

Snow depth drops from 48 cm to 30 cm. Find the percent change.

A price goes up 10%, then down 10%. Is it back to the original? Explain.

Find the simple interest on $800 at 4% per year for 3 years.

More practice: warm-up

Find 20% of 65.

What multiplier increases an amount by 13%?

What multiplier decreases an amount by 40%?

Write 0.085 as a percent.

What percent is 9 out of 12?

More practice: on your own

A video game costs $69.99. It is 15% off, then 13% HST is added. Find the total.

A town's population fell 8% to 27 600. What was it before?

Find the overall percent change for +25% followed by −20%.

A $1200 loan has simple interest of $189 after 3 years. What is the annual interest rate?

In a survey, 18% of 650 people said winter is their favourite season. How many people is that?

Challenge

A store says "40% off, then an extra 25% off the sale price." A friend says that's 65% off. What is the real discount?

An investment grows 10% each year for 3 years, with each increase based on the new amount. By what total percent did it grow? Compare with simple interest at 10% for 3 years.

Pose and solve: write a percent problem involving a price, a discount and 13% HST where the final total is exactly $113.00.

Check your understanding

Quiz

What is the total for a $60 item with 13% HST?

Which multiplier gives a 35% discount?

A price goes from $50 to $40. What is the percent change?

A total with 13% HST is $90.40. What was the pre-tax price?

An item is 50% off, then another 20% off the sale price. What is the overall discount?

What is the simple interest on $2000 at 2.5% per year for 4 years?

For parents and teachers

Big idea: This lesson completes expectation B3.5 with percentages in financial literacy (taxes, discounts, interest), data and measurement. Multipliers (1.13, 0.75) make multi-step problems quick and are the foundation for exponential growth and compound interest later in high school.

Common difficulties:

Removing tax by subtracting 13% of the total. Divide by 1.13 instead.
Adding successive percents (20% + 10% = 30%). Multiply the multipliers.
Dividing by the new value in percent change. Always divide by the original.
Confusing interest with the total amount.

How to help: Let your teen calculate the final price at the till before you pay, or check a receipt's HST. Talk about loans and savings accounts in your family's own experience if you are comfortable doing so.