B. Number · Number Sense: Scientific Notation, Real Numbers, Square Roots and Percents

Solving Problems with Percents Over 100% and Under 1%

B1.4About 40 minutes

Learning goals

I can find a percent of a number when the percent is more than 100% or less than 1%.
I can find what percent one number is of another, even when the answer is more than 100% or less than 1%.
I can find a total including 13% HST in Ontario.
I can find the whole when I know a part and its percent.

Words to know

HST
Harmonized Sales Tax. In Ontario it is 13% and is added to the price of many items. A total with HST is 113% of the price.

interest
Money a bank pays you for keeping your savings there, often given as a percent per year.

percent of
Multiplying by the percent as a decimal or fraction. 135% of 120 = 1.35 × 120.

the whole
The amount that counts as 100%.

A snowy winter

In one town, the normal snowfall for a winter is 120 cm. This winter, the weather report says snowfall was 135% of normal. How much snow fell?

135% means more than the whole amount: all of the normal 120 cm (100%), plus 35% more.

135% of 120 = 1.35 × 120 = 162 cm.

A small-town street after a heavy snowfall, with deep snowbanks and cars buried in snow. — A winter like this one could easily bring more than the normal amount of snow. Snowfall above normal is described with a percent greater than 100%. AI-generated image

The normal snowfall, 120 cm, is 100%. This winter's bar goes past the 100% mark: 162 cm is 135% of normal.

Three kinds of percent problems

1. Find a percent of a number. Change the percent to a decimal (or a fraction) and multiply.

135% of 120 = 1.35 × 120 = 162
0.5% of 2 000 = 0.005 × 2 000 = 10

2. Find what percent one number is of another. Divide the part by the whole, then multiply by 100.

45 is what percent of 30? 45 ÷ 30 = 1.5 = 150%
3 is what percent of 600? 3 ÷ 600 = 0.005 = 0.5%

3. Find the whole. If you know a part and its percent, divide the part by the percent (as a decimal).

12 is 0.5% of what number? 12 ÷ 0.005 = 2 400. Check: 0.005 × 2 400 = 12. ✓

Mental math tip: 1% is the whole ÷ 100, and 0.1% is the whole ÷ 1 000. So 0.5% of 3 000 = half of 1% of 3 000 = half of 30 = 15.

Worked example

Example 1: Sales tax in Ontario

A pair of winter boots costs $80 before tax. Ontario's HST is 13%. What is the total?

The total is the price (100%) plus the tax (13%), so it is 113% of the price.
113% = 1.13. 1.13 × 80 = 90.40.
Check: 13% of 80 = 0.13 × 80 = 10.40, and 80 + 10.40 = 90.40. ✓

The total is $90.40.

The price is 100%. The 13% HST sits on top of it, so the total is 113% of the price: 1.13 × $80 = $90.40.

Worked example

Example 2: A tiny percent

A savings account pays 0.8% interest per year. How much interest does $1 500 earn in one year?

0.8% = 0.008.
0.008 × 1 500 = 12.
Check with 1%: 1% of 1 500 = 15, so 0.8% should be a bit less than 15. ✓

$12 in interest

Worked example

Example 3: Maple syrup

Maple syrup makers often say it takes about 40 L of sap to make 1 L of syrup. What percent of the sap's volume is the syrup? What percent of the syrup's volume is the sap?

Syrup as a percent of sap: 1 ÷ 40 = 0.025 = 2.5%.
Sap as a percent of syrup: 40 ÷ 1 = 40 = 4 000%.
The same comparison gives a small percent one way and a huge percent the other way. It depends on which amount is the whole.

The syrup is about 2.5% of the sap. The sap is about 4 000% of the syrup.

Worked example

Example 4: Find the whole

A store gives 0.5% of its sales to the local food bank. Last month it gave $45. What were its sales?

$45 is 0.5% of the sales. 0.5% = 0.005.
Sales = 45 ÷ 0.005 = 9 000.
Check: 0.005 × 9 000 = 45. ✓

Sales were $9 000.

Is my answer reasonable?

More than 100% of a number → the answer is bigger than the number.
Less than 1% of a number → the answer is less than one hundredth of the number.
100% of a number → the number itself.

If your answer breaks one of these rules, check your decimal point.

Just for fun: 110% effort

When a coach asks you to give 110%, they are asking for 1.1 times everything you have. Mathematically, that's all of your effort plus a bonus tenth you have to find somewhere. Maybe check under the bench?

Hands-on activity

Receipt and growth check

You need: a store receipt from your home (or a flyer), a calculator, and if possible a record of your height from when you were younger (a growth chart, a doorframe mark or a health record a grown-up can share).

Receipt: find the subtotal of the taxed items and the HST. Is the HST about 13% of the taxed subtotal? Some items, like many basic groceries, have no HST, so check which items were taxed. Find the total as a percent of the taxed subtotal.
Growth: measure your height now. What percent of your younger height is it? (It will be more than 100%!)

What to look for: on a receipt with a taxed subtotal of $40.00, the HST should be $5.20, and the total $45.20 is 113% of $40. For growth: if you were 110 cm and now you are 154 cm, 154 ÷ 110 = 1.4 = 140%.

Math talk: "Why can your height now be more than 100% of your old height, but a test score usually can't be more than 100%?"

Your turn

Find 150% of 60.

Find 0.5% of 800.

Find 250% of 12.

Find 0.2% of 5 000.

A jacket costs $45 before tax. What is the total with 13% HST?

18 is what percent of 12?

3 is what percent of 600?

More practice: warm-up

Find 100% of 37.

Find 200% of 15.

Find 1% of 400.

Find 0.1% of 1 000.

Find 110% of 50.

More practice: on your own

A puppy had a mass of 4 kg. Now its mass is 325% of that. What is its mass now?

A town got 90 cm of snow last winter. This winter it got 160% of that. How much snow fell this winter?

A savings account pays 0.75% interest per year. How much interest does $2 400 earn in a year?

A school had 400 students. Now it has 460. What percent of the old number is the new number?

In a survey of 2 500 fish in a lake, 0.4% had tags. How many fish had tags?

Challenge

Using the rule of thumb that 40 L of sap makes about 1 L of syrup, how much syrup can a family make from 250 L of sap?

The HST on an item is $3.90. Ontario's HST is 13%. What was the price before tax, and what is the total?

A price rises to 125% of the original. Later, the new price is cut to 80% of itself. How does the final price compare with the original?

Check your understanding

Quiz

What is 120% of 50?

What is 0.5% of 400?

An item costs $20 before tax. What is the total with 13% HST?

30 is what percent of 20?

Which number is less than 1% of 1 000?

This year's snowfall is 100% of normal. That means:

For parents and teachers

Big idea: B1.4 includes solving problems with percents of more than 100% or less than 1%. Students find a percent of a number, find what percent one number is of another, and find the whole. Ontario's 13% HST gives a real everyday use: a total with tax is 113% of the price.

Common difficulties:

Treating 0.5% as 0.5 or 5%. Use the 1% benchmark: 1% of 800 is 8, so 0.5% must be smaller.
Not believing percents can be more than 100%. Use growth examples: a puppy, your child's height, a savings jar.
Choosing the wrong whole. "What percent of the sap is the syrup?" and "What percent of the syrup is the sap?" give very different answers.

How to help: Check receipts together and talk about which items have HST. Estimate first ("about 10% of $80 is $8, so 13% is a bit more"), then calculate.