B. Number · Operations: Exponents, Factors, Fractions, Decimals, Integers, Percents and Proportions

Percents, Fractions and Decimals to Know by Heart

B2.2 A1About 40 minutes

Learning goals

I can recall common fraction, decimal and percent equivalents, like 1/8 = 0.125 = 12.5%.
I can explain why each equivalent is true.
I can choose the easiest form (fraction, decimal or percent) to calculate in my head.

Words to know

percent (%)
A ratio that compares a number to 100. 35% means 35 out of 100.

equivalent
Having the same value. 3/4, 0.75 and 75% are equivalent.

terminating decimal
A decimal that ends, like 0.125.

repeating decimal
A decimal with a digit or group of digits that repeats forever, like 0.333... for 1/3.

Remember from Grade 6

In Grade 6 you found 50%, 25%, 10%, 5%, 15% and 1% of numbers in your head, and you changed scores like 18 out of 25 into a percent (72%). In Grade 7 you will learn a set of equivalents by heart, so that you can switch between fractions, decimals and percents instantly.

Let's start with a shortcut

What is 12.5% of 64?

That looks hard. But if you know that 12.5% = 1/8, it is easy: 64 ÷ 8 = 8.

Knowing equivalents by heart lets you pick the easiest form for the job. Sometimes a fraction is easiest, sometimes a decimal, sometimes a percent.

Because 12.5% = 1/8, finding 12.5% of 64 is just cutting 64 into 8 equal parts.

Fraction	Decimal	Percent	How to remember it
1/2	0.5	50%	One half of 100 is 50.
1/4, 3/4	0.25, 0.75	25%, 75%	Quarters, like 25¢ coins.
1/5, 2/5, 3/5, 4/5	0.2, 0.4, 0.6, 0.8	20%, 40%, 60%, 80%	Each fifth is 20%.
1/10, 3/10, 7/10	0.1, 0.3, 0.7	10%, 30%, 70%	Tenths: the digit is the number of tens of percent.
1/8, 3/8, 5/8, 7/8	0.125, 0.375, 0.625, 0.875	12.5%, 37.5%, 62.5%, 87.5%	1/8 is half of 1/4: half of 25% is 12.5%.
1/3, 2/3	0.333..., 0.666...	33 1/3%, 66 2/3%	Thirds repeat forever.
1/20	0.05	5%	Half of 1/10.
1/25	0.04	4%	25 × 4 = 100, so 1/25 = 4/100.
1/50	0.02	2%	Half of 4%.
1/100	0.01	1%	One hundredth.

Commonly used equivalents. The "..." means the digit repeats forever.

Why are they true?

A percent is "out of 100", so make an equivalent fraction with denominator 100 when you can:

3/5 = 60/100 = 0.60 = 60%
1/25 = 4/100 = 0.04 = 4%

For eighths, halve the quarters: 1/4 = 25%, so 1/8 = 12.5%. Then 3/8 = 3 × 12.5% = 37.5%.

For thirds, divide: 1 ÷ 3 = 0.333..., which never ends. 1/3 is 33 1/3% (a little more than 33%). Sometimes it is written as 0.3 with a bar over the 3 to show that it repeats.

Fraction strips lined up with a percent ruler. One eighth is half of a quarter (12.5%), one fifth is 20%, and one third is a little more than 33%.

Worked example

Example 1: Pick the friendly form

Find 75% of 36, 0.2 × 45 and 33 1/3% of 45.

75% = 3/4. 36 ÷ 4 = 9, and 9 × 3 = 27.
0.2 = 1/5. 45 ÷ 5 = 9.
33 1/3% = 1/3. 45 ÷ 3 = 15.

27, 9 and 15

Worked example

Example 2: Compare different forms

Order from least to greatest: 0.3, 1/3, 30.5%.

Change all to decimals: 0.3 = 0.300, 1/3 = 0.333..., 30.5% = 0.305.
0.300 < 0.305 < 0.333...

0.3, 30.5%, 1/3

Worked example

Example 3: A test score

Ayaan got 21 out of 24 on a quiz. What percent is that?

21/24 in lowest terms: the GCF is 3, so 21/24 = 7/8.
7/8 = 87.5% (from the equivalents).

87.5%

Think like a mathematician

Build the hard ones from the easy ones. If you forget 5/8, think: 5/8 = 1/2 + 1/8 = 50% + 12.5% = 62.5%. If you forget 2/3, think: 2 × 33 1/3% = 66 2/3%. Explaining how you rebuilt it is a great way to remember it next time.

Hands-on activity

Equivalents memory game

You need: 24 small paper squares (cut from scrap paper with a grown-up's help, or use sticky notes) and a pencil.

Make 8 sets of 3 cards: one with a fraction, one with the matching decimal, one with the matching percent. Use: 1/2, 1/4, 3/4, 1/5, 1/8, 3/8, 1/3, 1/20. For example: 1/8, 0.125, 12.5%.
Mix them up and place them face down in rows.
On your turn, flip 3 cards. If all three match, keep them and go again. If not, flip them back.
The player with the most sets at the end wins.
Play again with new sets: 2/5, 5/8, 7/8, 2/3, 1/25, 1/50, 7/10, 1/100.

Math talk: "Which equivalents are easiest for you to remember? Which ones do you rebuild from others?"

Just for fun

Eating 12.5% of a pizza sounds very scientific. It just means you ate one slice of a pizza cut into 8.

Your turn

Write 3/4 as a percent.

Write 0.2 as a fraction in lowest terms.

Write 12.5% as a fraction.

Write 2/3 as a decimal and as a percent.

Find 40% of 35 in your head.

Find 33 1/3% of 27.

Find 0.125 × 48.

More practice: warm-up

Write 50% as a fraction.

Write 0.1 as a percent.

Write 1/4 as a decimal.

Write 3/10 as a percent.

Write 1/5 as a percent.

More practice: on your own

Write 37.5% as a fraction.

Write 0.6 as a fraction in lowest terms and as a percent.

Find 66 2/3% of 90.

Find 87.5% of 40.

Order from least to greatest: 0.62, 5/8, 60%.

Challenge

A hockey team won 15 of its 24 games. What percent did it win?

Use 1/20 = 5% to find the percent for 1/40. Then find 1/40 of 200.

Why can't 1/3 be written as a decimal that ends? Use division to explain.

Check your understanding

Quiz

Which is equal to 1/8?

What is 3/5 as a percent?

What is 33 1/3% of 60?

Which fraction is equal to 0.75?

Which is the greatest?

What is 5% as a fraction in lowest terms?

For parents and teachers

Big idea: The Ontario curriculum (B2.2) asks Grade 7 students to understand and recall commonly used percents, fractions and decimal equivalents. Recall is the goal, but understanding comes first: students should be able to rebuild an equivalent they forget (for example, 1/8 is half of 1/4).

Common difficulties:

Reading digits literally: thinking 1/8 = 0.8 or 1/5 = 15%.
Rounding thirds to 33% and forgetting that 1/3 is a repeating decimal.
Comparing decimals by length (thinking 0.62 > 0.625).

How to help: Play the memory game, and use equivalents in daily life: "25% off is the same as a quarter off". Quiz each other in the car. This lesson builds A1 skills: making connections between representations and choosing efficient strategies.