Use fractions, decimals, and percents flexibly, including extreme percents

Ontario Grade 8 Mathematics

Number

Connect familiar amounts, tiny percents, and percents above 100%

A fraction, a decimal, and a percent can describe the same amount in different ways. If three of four equal parts of a strip are shaded, the shaded amount is three quarters. It can also be written as

\frac{3}{4}

,

0.75

, or 75%. A percent means “out of 100.” Some amounts are smaller than one hundredth, so their percents are below 1%. Other amounts are more than one whole, so their percents are above 100%. In this lesson, you will connect the three forms and choose the one that makes a question easier to understand.

What you will learn

Explain how fractions, decimals, and percents can name the same amount.
Convert among fractions, decimals, and percents using place value and equivalent fractions.
Interpret percents below 1% and above 100%.
Choose a useful form to compare amounts and find a percent of a quantity.

1. One amount, three forms

A fraction names part of a whole. In

\frac{3}{4}

, the bottom number tells how many equal parts make one whole. The top number tells how many parts are counted.

A decimal uses place value to show an amount. The first digit to the right of the decimal point counts tenths. The next digit counts hundredths. A percent uses the symbol % and means an amount out of 100.

Picture a strip divided into four equal parts, with three shaded. The shaded amount is

\frac{3}{4}

. Three quarters are also 75 hundredths. So these forms name the same amount:

\frac{3}{4}

,

0.75

, and 75%.

Equivalent fractions are fractions that name the same amount. To change a fraction to a percent, it can help to make an equivalent fraction with 100 on the bottom. For example, eight percent means eight hundredths:

8\%=\frac{8}{100}=0.08

.

To change a decimal to a percent, express the decimal as hundredths. For example,

0.37

is 37 hundredths, so it is 37%. Going the other way, 37% is

0.37

, because both forms show 37 hundredths.

\frac{3}{4}=\frac{75}{100}=0.75=75%

A percent is an amount out of 100.
Equivalent forms do not change the amount.
Place value connects hundredths, decimals, and percents.

2. Reading percents below 1% and above 100%

A percent does not have to be between 0% and 100%. The amount 100% is one whole. A percent greater than 100% describes more than one whole. For example, 150% is 150 hundredths, or one and a half wholes.

A percent below 1% describes less than one hundredth. For example,

0.6\%=0.006

. This is small because six tenths of one percent is six thousandths of a whole. Keep the percent sign in mind: 0.6% is not the same as

0.6

.

To compare amounts, write them in the same form or use a benchmark. A benchmark is a familiar amount used to help compare another amount. Since 100% is one whole, 125% is more than one whole. Since 0.4% is less than 1%, it is less than one hundredth.

A fraction can also be greater than one when its top number is greater than its bottom number. For example,

\frac{5}{4}

is one whole and one quarter. It is also

1.25

and 125%.

100%=1

100% represents one whole.
Percents above 100% represent more than one whole.
Percents below 1% represent less than one hundredth.
Check the percent sign before comparing a percent with a decimal.

3. Find a percent of a quantity

A percent of a quantity means a share of that quantity. First change the percent to a decimal or fraction. Then multiply by the quantity. Multiplication is useful because it takes the chosen share of the quantity.

For 25% of 48, use

25\%=0.25=\frac{1}{4}

. One quarter of 48 is 12, so 25% of 48 is 12. The fraction form is convenient because 48 can be shared into four equal groups.

For a percent above 100%, the result is larger than the starting quantity. For example, 125% of 40 is one whole 40 plus one quarter of 40. That is 50. The estimate helps check the answer: it should be greater than 40.

For a percent below 1%, the share is small. For example,

0.5\%=0.005

. Half of one percent of 200 is 1. The result is small compared with 200, as expected.

Choose the form that makes the numbers easiest to work with. A familiar fraction may make equal sharing simple. A decimal may be useful for multiplication. Before finishing, compare the answer with the starting quantity and with a nearby benchmark. percent of quantity=(

\text{percent as a decimal}

)×(

\text{quantity}

)

Change a percent to a fraction or decimal before finding its share.
A percent above 100% of a positive quantity gives a larger amount.
A percent below 1% gives a very small share of a quantity.

4. Choose a form and check its size

Before calculating, ask what the percent means. Is it near 100%, much larger than 100%, or much smaller than 1%? Then choose a useful representation and estimate the answer’s size.

For example, 200% of 18 means two wholes, so the answer must be twice 18. In contrast, 0.2% of 18 is much smaller than one whole, so the answer must be a small decimal.

Try these conversions: write

0.35

as a percent; write 175% as a decimal; and decide whether 0.8% is greater or less than

\frac{1}{100}

. Use hundredths or thousandths to explain each choice.

Place value can catch an error. If 0.6% were changed to

0.06

, the answer would be too large:

0.06

is 6%. The correct decimal for 0.6% is

0.006

.

Estimate first by comparing with 1% and 100%.
Choose the form that makes the calculation clear.
Check the size of the result to catch place-value errors.

Equivalent forms to remember

Fraction	Decimal	Percent	Meaning
$\frac{1}{100}$	$0.01$	1%	One hundredth
$\frac{1}{2}$	$0.5$	50%	One half
$1$	$1.0$	100%	One whole
$\frac{3}{2}$	$1.5$	150%	One and a half

Worked example

Convert a fraction to a decimal and a percent

Write

\frac{7}{20}

as a decimal and as a percent.

Seven twentieths

Seven of 20 equal parts are shaded. The same amount is 35 of 100 equal parts.

Make hundredths
The denominator is 20. Multiply the top and bottom by 5 to make 100 on the bottom. The fraction keeps the same value because both numbers are multiplied by the same number.
$\frac{7}{20}=\frac{35}{100}$
Write both forms
Thirty-five hundredths is the decimal $0.35$ . As a percent, 35 hundredths is 35%.
$\frac{7}{20}=0.35=35%$

Answer:

\frac{7}{20}=0.35=35\%

.

Check: 35% is less than 50%, and

\frac{7}{20}

is less than one half, so the result is reasonable.

Worked example

Interpret a percent greater than 100%

Write 165% as a decimal and as a mixed number.

Use hundredths
The percent means 165 hundredths. As a decimal, 165 hundredths is $1.65$ . The 1 shows one whole, and the 65 shows 65 hundredths.
$165%=\frac{165}{100}=1.65$
Write a mixed number
There are 100 hundredths in one whole. After taking one whole from 165 hundredths, 65 hundredths remain. Divide the top and bottom of that fraction by 5 to simplify it.
$\frac{165}{100}=1\frac{65}{100}=1\frac{13}{20}$

Answer:

165\%=1.65=1\frac{13}{20}

.

Check: Since 165% is greater than 100%, both the decimal and the mixed number are greater than one whole.

Worked example

Find a percent below 1% of a quantity

Find 0.4% of 250.

Change the percent to a decimal
To change a percent to a decimal, divide by 100. So 0.4% is $0.004$ , which is less than one hundredth.
$0.4%=\frac{0.4}{100}=0.004$
Find the share
Multiply the quantity by the decimal share. The result is 1, which is much smaller than 250, as expected for a percent below 1%.
$0.004\times250=1$

Answer: 0.4% of 250 is 1.

Check: 1% of 250 is 2.5, so 0.4% must be less than 2.5. The answer 1 fits.

Common mistakes and how to avoid them

Writing 0.6% as

0.6

.

Correction: Divide by 100 to change a percent to a decimal:

0.6\%=0.006

.

Thinking a percent must be between 0% and 100%.

Correction: 100% is one whole. A greater percent describes more than one whole.

Changing only the denominator when making equivalent fractions.

Correction: Multiply or divide the numerator and denominator by the same number so the fraction keeps the same value.

Assuming a tiny percent of a quantity must be a whole number.

Correction: A share can be a decimal. Estimate its size before deciding whether an answer makes sense.

Lesson summary

Fractions, decimals, and percents are different ways to name amounts.
A percent means an amount out of 100.
100% is one whole; percents above it are more than one whole.
Percents below 1% are less than one hundredth.
To find a percent of a quantity, convert the percent to a decimal or fraction and multiply.

Check your understanding

Question 1

Which decimal is equal to 0.7%?

$0.7$
$0.07$
$0.007$
$7$

Show answer and explanation

0.007

Divide by 100 to change a percent to a decimal:

0.7\%=0.007

.

Question 2

Which statement describes 140%?

It is 14 hundredths of a whole.
It is one whole and 40 hundredths more.
It is less than one whole.
It is 140 wholes.

Show answer and explanation

It is one whole and 40 hundredths more.

140\%=\frac{140}{100}=1.4

, so it is one whole and 40 hundredths.

Question 3

What is 20% of 35?

0.7
7
17.5
70

Show answer and explanation

7

20\%=0.2

, and

0.2\times35=7

.

Key terms

Percent: A way to name an amount out of 100.
Equivalent fractions: Fractions that name the same amount, even though their top and bottom numbers differ.
Hundredth: One of 100 equal parts of a whole; written as $\frac{1}{100}$ or $0.01$ .
Benchmark: A familiar amount used to help compare or estimate another amount.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 8 Mathematics, expectation B1.4. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

Official curriculum reference

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B1.4 · Use fractions, decimals, and percents flexibly, including extreme percents

Ontario Grade 8 Mathematics

Connect familiar amounts, tiny percents, and percents above 100%

What you will learn

1. One amount, three forms

2. Reading percents below 1% and above 100%

3. Find a percent of a quantity

4. Choose a form and check its size

Equivalent forms to remember

Convert a fraction to a decimal and a percent

Interpret a percent greater than 100%

Find a percent below 1% of a quantity

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Grade 8 Math

About this lesson and its review