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

Moving Flexibly Between Fractions, Decimals and Percents

B1.4About 40 minutes

Learning goals

I can change any fraction, mixed number, decimal or percent into the other two forms.
I can work with percents greater than 100% and less than 1%.
I can write repeating decimals as percents, like 2/3 = 66 2/3%.
I can choose the form that makes a problem easiest.

Words to know

percent
A number out of 100. 35% = 35/100 = 0.35.

equivalent
Equal in value. 3/4, 0.75 and 75% are equivalent.

mixed number
A whole number and a fraction together, like 2 3/8.

benchmark
A well-known value used to check or estimate: 1/2 = 50%, 1 = 100%, 1/100 = 1%.

Remember from Grade 7

In Grade 7 you converted between fractions, decimals and percents, including percents over 100% (like 125% = 1.25) and under 1% (like 0.5% = 0.005). You also learned common equivalents by heart. This year the goal is to use all three forms interchangeably and flexibly to solve problems: choosing whichever form makes the work easiest.

Three ways to say the same thing

The school's food drive collected 2 3/8 times last year's amount. The principal says it is 237.5% of last year. A spreadsheet shows 2.375. All three are the same number!

The conversion toolbox

Fraction → decimal: divide the numerator by the denominator. 3/8 = 3 ÷ 8 = 0.375.
Decimal → percent: multiply by 100 (move the decimal point 2 places right). 2.375 → 237.5%.
Percent → decimal: divide by 100 (move 2 places left). 0.4% → 0.004.
Decimal or percent → fraction: write it over a power of 10 and simplify. 0.004 = 4/1 000 = 1/250.
Repeating decimals: keep the fraction part. 2/3 = 0.666… = 66 2/3% (about 66.7%).

Benchmarks to check: less than 1/100 means less than 1%. More than 1 whole means more than 100%.

One number, three forms. Follow an arrow to change forms: divide to go from a fraction to a decimal, × 100 and ÷ 100 to move between decimals and percents, and write a percent over 100 to get a fraction.

One number line, three labels. Each tick shows a fraction (top), a decimal (middle) and a percent (bottom) that are equal. The first section shows 0 to 1 whole (100%); everything after it is more than 100%.

Fraction	Decimal	Percent
1/1 000	0.001	0.1%
1/200	0.005	0.5%
1/8	0.125	12.5%
1/6	0.166 6…	16 2/3% (≈ 16.7%)
1/3	0.333…	33 1/3% (≈ 33.3%)
5/8	0.625	62.5%
2/3	0.666…	66 2/3% (≈ 66.7%)
1 1/4	1.25	125%
3/2 = 1 1/2	1.5	150%
2 3/8	2.375	237.5%

Some useful equivalents, including percents under 1% and over 100%.

Worked example

Example 1: A mixed number

Write 2 3/8 as a decimal and as a percent.

The whole number stays: 2.
3/8 = 3 ÷ 8 = 0.375, so 2 3/8 = 2.375.
Multiply by 100: 2.375 × 100 = 237.5.

2 3/8 = 2.375 = 237.5%

Worked example

Example 2: A tiny percent

Write 0.4% as a decimal and as a fraction in simplest form.

Divide by 100: 0.4 ÷ 100 = 0.004.
0.004 is 4 thousandths: 4/1 000.
Simplify by dividing by 4: 1/250.

0.4% = 0.004 = 1/250

Worked example

Example 3: A repeating decimal

Write 5/6 as a percent.

5 ÷ 6 = 0.833 3… (the 3 repeats).
Multiply by 100: 83.333…%.
The repeating 0.333… is 1/3, so this is 83 1/3%. Rounded, it is about 83.3%.

5/6 = 83 1/3% ≈ 83.3%

Worked example

Example 4: Choose the easiest form

Find 25% of 48, 0.5% of 3 000 and 150% of 22.

25% = 1/4, and 1/4 of 48 = 48 ÷ 4 = 12. The fraction is easiest.
0.5% = 0.005, and 0.005 × 3 000 = 15. (Or: 1% of 3 000 is 30, so 0.5% is half: 15.)
150% = 1 1/2, so take all of 22 plus half of 22: 22 + 11 = 33.

12, 15 and 33

Watch out!

0.5% is not 0.5. 0.5% = 0.005. Always divide the percent by 100.
1.5 is not 1.5%. 1.5 = 150%.
0.06 = 6%, not 60%.
Check with benchmarks: a number bigger than 1 must be more than 100%.

Just for fun

Three friends share a pizza. Ava eats 1/4, Jordan eats 0.25 and Sam eats 25%. They spend the rest of lunch arguing about who ate the most.

(It's a tie. It was always a tie.)

Hands-on activity

Label detective

You need: grocery flyers, food packages and a calculator.

Find 8 numbers written as percents, decimals or fractions. Look for "% Daily Value" on Canadian nutrition labels, sale percents in flyers, fractions in recipes, and prices per 100 g.
For each number, write the other two forms. For example, 15% Daily Value = 0.15 = 3/20.
Look for (or make up) one over 100% and one under 1%. A food can have more than 100% of the daily value of a nutrient, and some ingredients are listed in tiny amounts.
Sort your 8 numbers from least to greatest.

Model answers: 20% off = 0.2 = 1/5. 3/4 cup = 0.75 = 75%. 120% = 1.2 = 1 1/5. 0.5% = 0.005 = 1/200.

Math talk: "Which form did the label or flyer use? Why do you think the writer chose that form?"

Your turn

Write 1 1/4 as a decimal and a percent.

Write 0.6% as a decimal and a fraction in simplest form.

Write 2/3 as a decimal and a percent.

Write 3.5 as a percent and a mixed number.

Write 0.125 as a fraction and a percent.

Write 7/200 as a decimal and a percent.

Write 180% as a decimal and a mixed number.

More practice: warm-up

Write 50% as a fraction.

Write 0.75 as a percent.

Write 1/10 as a decimal and a percent.

Write 200% as a whole number.

Write 1% as a decimal.

More practice: on your own

Write 0.25% as a decimal and a fraction.

Write 1/6 as a percent.

Write 4 3/5 as a decimal and a percent.

Write 0.08% as a decimal and a fraction.

Which is greatest: 1.3, 135% or 1 1/3?

Challenge

Write 1/7 as a percent, rounded to the nearest tenth of a percent.

Write 0.004 5 as a percent and as a fraction in simplest form.

What is 250% of 1/5? Give your answer as a fraction and a decimal.

Check your understanding

Quiz

Which is equal to 1 3/4?

What is 0.3% as a decimal?

Which is equal to 2/3?

What is 2.4 as a percent?

Which is equal to 3/8?

Which is greatest?

For parents and teachers

Big idea: B1.4 asks Grade 8 students to use fractions, decimal numbers and percents, including percents of more than 100% or less than 1%, interchangeably and flexibly to solve a variety of problems. This lesson strengthens the conversions from Grade 7, including mixed numbers and repeating decimals, with an emphasis on choosing the easiest form.

Common difficulties:

Small percents: writing 0.5% as 0.5 or 0.05. Remind your child: percent → decimal always means divide by 100.
Large percents: thinking a percent can't be more than 100. Talk about "double" = 200%.
Repeating decimals: rounding too early. 2/3 is 66 2/3%, not 66%.

How to help: Read flyers and nutrition labels together and ask "What is that as a fraction?" Ask your child which form they would use for a quick mental calculation and why.