Grade by Grade
B. Number · Operations: Divisibility, Decimals, Fractions, Ratios and Percents

Solving Problems with Percents

B2.12 B2.3 A1About 45 minutes
Learning goals
  • I can explain that a percent is a ratio that compares a number to 100.
  • I can write a part of a whole as a percent.
  • I can find a percent of a number, including sale prices and sales tax.
  • I can compare amounts using percents.

Words to know

percent (%)
A ratio that compares a number to 100. 72% means 72 out of 100, or 72:100.
discount
The amount of money taken off a price during a sale.
sale price
The price after the discount is taken off.
sales tax
Money added to the price of many things you buy. In Ontario, the HST (harmonized sales tax) is 13%.

Remember from earlier

In Grade 5 you matched fractions, decimals and percents, like 3/4 = 0.75 = 75%. In this unit you found 50%, 25%, 10%, 5%, 15% and 1% of numbers in your head. Now you'll use percents to solve problems: test scores, sales and taxes.

Let's start with a quiz

On a spelling quiz, Noah got 18 out of 25. Is that better than Elsa's 80%?

A percent is a ratio out of 100. So change 18:25 into an equivalent ratio out of 100. 25 × 4 = 100, so 18 × 4 = 72.

18 out of 25 = 72 out of 100 = 72%. Elsa's 80% is higher.

18 out of 25 is the same as 72 out of 10018 / 25× 4each squaresplits into 472 / 100 = 72%
Each square of the 25-grid becomes 4 squares of the 100-grid. So 18 out of 25 shades 72 out of 100: Noah scored 72%.

A percent is a ratio

Part to whole as a percent: make an equivalent ratio with 100 as the second term.

  • 13 out of 20 → × 5 → 65 out of 100 = 65%
  • 45 out of 60 → simplify to 3 out of 4 → × 25 → 75 out of 100 = 75%

A percent of a number: build it from benchmark percents.

  • 35% of 60 = 30% + 5% = 18 + 3 = 21
  • 12% of 250 = 10% + 2% = 25 + 5 = 30

Discount and tax: find the percent, then subtract (sale) or add (tax).

0%$025%$2050%$4075%$60100%$80percentdollars
A double number line for 25% of $80. 100% is $80, so each 25% jump is $20. The shaded part shows 25% = $20.
Worked example

Example 1: Sale price

A jacket costs $80. It is 25% off. What is the sale price?

  1. 25% of $80: 25% is 1/4, and $80 ÷ 4 = $20. That's the discount.
  2. Sale price: $80 − $20 = $60.
  3. Another way: you pay 100% − 25% = 75% of the price. 75% of $80 = $60.

The sale price is $60.

Worked example

Example 2: Adding HST

In Ontario, many items have 13% HST added at the till. A board game costs $40 before tax. What is the total?

  1. 13% = 10% + 1% + 1% + 1%.
  2. 10% of $40 = $4.00. 1% of $40 = $0.40, so 3% = $1.20.
  3. Tax: $4.00 + $1.20 = $5.20.
  4. Total: $40.00 + $5.20 = $45.20.

The total is $45.20.

Worked example

Example 3: A percent that isn't a benchmark

35% of 60 = ?

  1. 10% of 60 = 6, so 30% = 3 × 6 = 18.
  2. 5% is half of 10%: 3.
  3. 35% = 30% + 5% = 18 + 3 = 21.

35% of 60 = 21

What is the question asking?

Read carefully: is the question asking for the discount (how much you save) or the sale price (how much you pay)? With tax, is it asking for the tax or the total?

Just for fun

A sign says "100% off!" If that were true, you'd walk out with everything for free. Always read the fine print, and do the math!

Hands-on activity

Flyer shopping trip

You need: a store flyer (paper or online), paper, a pencil and a calculator for checking.

  1. Find 3 items on sale with a percent off (or make up a percent like 10%, 20% or 25% off).
  2. For each item, find the discount and the sale price.
  3. Add 13% HST to one item. Use 10% + 3%.
  4. Check one answer with a calculator: price × 0.13 gives the tax.
  5. Which item has the biggest discount in dollars? Is it the same item with the biggest percent off?

Math talk: "Why can a smaller percent off sometimes save you more money?"

Your turn

1

30% of 50 = ?

2

75% of 40 = ?

3

Write 9 out of 10 as a percent.

4

Write 13 out of 20 as a percent.

5

A video game costs $45. It is 20% off. What is the sale price?

6

Find the 13% HST on a $20 item.

7

Write 7 out of 25 as a percent.

More practice: warm-up

1

Write 50 out of 100 as a percent.

2

Write 1/4 as a percent.

3

Write 3 out of 10 as a percent.

4

100% of 37 = ?

5

10% of 90 = ?

More practice: on your own

1

40% of 85 = ?

2

Write 45 out of 60 as a percent.

3

A $120 pair of skates is 15% off. What is the sale price?

4

What is the total cost of a $30 item with 13% HST?

5

12% of 250 = ?

Challenge

1

A $70 toboggan is 30% off. Then 13% HST is added to the sale price. What is the total?

2

10% of a number is 8.5. What is the number?

3

Which is the better score: 21 out of 25 or 40 out of 50? Explain with percents.

Check your understanding

Quiz

1.

What is 17 out of 20 as a percent?

2.

A $36 shirt is 25% off. What is the sale price?

3.

What is 13% of $10?

4.

What is 60% of 45?

5.

A percent compares a number to…

6.

Which is the greatest?

For parents and teachers

Big idea: B2.12 includes percents as a kind of ratio: a percent compares a number to 100. Grade 6 students find a part as a percent (18 out of 25 = 72%), find a percent of a number, and solve problems about discounts and taxes. The mental math percents from B2.3 (10%, 5%, 1%, etc.) are the building blocks, so B2.3 is tagged too.

Common difficulties:

  • Answering with the discount instead of the sale price (or the tax instead of the total).
  • Thinking 18 out of 25 is 18%. It is only a percent when the whole is 100.
  • Applying the percent to the wrong amount: in the toboggan problem, tax is on the sale price.

How to help: Shopping is the best practice. Before paying, ask your child to estimate the HST and the total. Compare a "$10 off" coupon with a "15% off" coupon on a real price and talk about which saves more and why (A1: reasoning and making connections).