Convert among fractions, decimals, and percents

Ontario Grade 7 Mathematics

Number

Three ways to name the same amount

Imagine a strip split into five equal parts. If three parts are shaded, the shaded amount can be named as three fifths, 0.6, or 60%. These names look different, but they describe the same part of the strip. In this lesson, you will learn how to change from one name to another. Keep the same whole in mind each time. A fraction compares equal parts with the whole. A decimal uses place value. A percent tells how many parts there would be out of 100 equal parts.

What you will learn

Explain how fractions, decimals, and percents can name the same amount.
Convert fractions to decimals and percents.
Convert decimals and percents to other forms.
Check that different forms name the same amount.

Connect the three forms

A whole is one complete item or amount. A fraction shows some equal parts of a whole. In

\frac{3}{5}

, the bottom number, 5, is the denominator. It tells how many equal parts make the whole. The top number, 3, is the numerator. It tells how many parts are being described.

A decimal is a number written with a decimal point. The first place to the right of the decimal point is tenths. The next place is hundredths. For example, 0.4 means four tenths, which can also be written as

\frac{4}{10}

.

Percent means “out of 100.” The symbol for percent is %. So 40% means 40 out of 100 equal parts. The three forms name the same kind of amount; they just show it in different ways.

Picture a strip divided into 100 equal squares, with 40 shaded. The shaded amount is

\frac{40}{100}

, or 40%. As a decimal, it is 0.40, which has the same value as 0.4. The strip is still one whole; only the way we name the shaded part changes.

1=\frac{100}{100}=1.00=100\%

The denominator counts all equal parts in a whole.
The numerator counts the parts being described.
Percent means out of 100.

Convert fractions, decimals, and percents

To change a fraction to a decimal, divide the numerator by the denominator. The division finds the amount represented by the fraction. For example,

\frac{3}{4}

means 3 divided by 4, which is 0.75.

To change a decimal to a percent, name the decimal as hundredths. For example, 0.75 is 75 hundredths, so it is 75%. Another way to see this is to multiply the decimal by 100 and write the percent sign. The digits move two places to the right because the amount is being described in hundredths.

To change a percent to a decimal, divide its number by 100. For example, 18% means 18 out of 100. As a decimal, that is 0.18.

To change a percent to a fraction, write its number over 100. Then simplify if possible. Simplifying means making an equal fraction with smaller numbers. Divide the numerator and denominator by the same number. For example,

\frac{25}{100}

becomes

\frac{1}{4}

when both numbers are divided by 25.

A fraction can become a percent by first changing it to a decimal, then changing the decimal to a percent. Some fractions do not have decimals that end. You can still use the fraction-to-decimal division method, but the examples here use decimals that end.

\frac{a}{b}=a\div b

Fraction to decimal: divide the numerator by the denominator.
Decimal to percent: express the amount in hundredths.
Percent to decimal: divide the percent number by 100.
Percent to fraction: write the number over 100 and simplify if possible.

Use place value to check

Hundredths help connect decimals and percents. In 0.36, the 3 is in the tenths place and the 6 is in the hundredths place. Together, these digits show 36 hundredths. That is why 0.36 is 36%.

A zero at the end of a decimal does not change its value. For example, 0.5 and 0.50 are equal. The first says five tenths. The second says fifty hundredths. Both name one half of a whole.

Use familiar amounts to check whether an answer makes sense. One half is

\frac{1}{2}

, 0.5, or 50%. If an amount is less than one half, its decimal should be less than 0.5 and its percent should be less than 50%.

When simplifying, divide the top and bottom by the same number. This keeps the fraction's value the same. For example, dividing both parts of

\frac{30}{100}

by 10 gives

\frac{3}{10}

. Both fractions name the same amount.

0.36=\frac{36}{100}=36\%

Hundredths connect decimal notation to percents.
A zero at the end of a decimal does not change its value.
Compare with familiar amounts to check a conversion.

Practise a conversion route

Start with the form you are given. If it is a fraction, divide to get a decimal, then express that decimal as a percent. If it is a decimal, use its place value to find the percent, or write it as a fraction using tenths or hundredths. If it is a percent, write it over 100 for a fraction or divide by 100 for a decimal.

Try these conversions: change

\frac{7}{10}

to a decimal and a percent; change 0.08 to a percent and a fraction; and change 45% to a decimal and a fraction. In each case, check that the forms describe the same part of a whole.

Show the conversion steps clearly. Keep the whole unchanged, and use place value carefully. A conversion changes the name of the amount, not the amount itself.

Choose a route based on the starting form.
Keep the same whole and the same amount.
Check the result using place value or a familiar amount.

Equivalent forms

Fraction	Decimal	Percent
$\frac{1}{2}$	0.5	50%
$\frac{1}{4}$	0.25	25%
$\frac{3}{4}$	0.75	75%
$\frac{1}{10}$	0.1	10%

Worked example

A fraction of a strip

Convert

\frac{3}{5}

to a decimal and a percent.

Three fifths

Three of five equal parts are shaded. The amount is three fifths, 0.6, or 60%.

Find the decimal
The fraction describes 3 of 5 equal parts. Divide the numerator by the denominator to find the decimal amount.
$3\div5=0.6$
Find the percent
The decimal 0.6 means six tenths. Six tenths is the same amount as 60 hundredths, so the percent is 60%.
$0.6=0.60=60\%$

Answer:

\frac{3}{5}=0.6=60\%

Check: Sixty percent is more than 50%, so it makes sense that three fifths is more than one half.

Worked example

A decimal written as a percent and fraction

Convert 0.32 to a percent and a fraction in simplest form.

Convert to a percent
The digits in 0.32 show 32 hundredths. A percent also names hundredths, so the percent is 32%.
$0.32=32\%$
Write a fraction
Thirty-two hundredths is written as 32 over 100. Divide both numbers by 4 to make an equal fraction with smaller numbers.
$\frac{32}{100}=\frac{8}{25}$

Answer:

0.32=32\%=\frac{8}{25}

Check: Multiplying the numerator and denominator of

\frac{8}{25}

by 4 gives

\frac{32}{100}

, or 0.32.

Worked example

A percent written as a decimal and fraction

Convert 45% to a decimal and a fraction in simplest form.

Nine twentieths

Nine of twenty equal parts are shaded. This is 9/20, or 45 hundredths, or 45%.

Convert to a decimal
The percent means 45 out of 100. Dividing 45 by 100 gives the decimal form.
$45\div100=0.45$
Convert to a fraction
Write 45 out of 100 as a fraction. Divide both numbers by 5 to simplify it.
$\frac{45}{100}=\frac{9}{20}$

Answer:

45\%=0.45=\frac{9}{20}

Check: The fraction is less than one half, and 45% is also less than 50%.

Common mistakes and how to avoid them

Writing 0.4 as 4%.

Correction: 0.4 is four tenths, which is 40 hundredths. It is 40%.

Writing 7% as

\frac{7}{10}

.

Correction: Percent means out of 100, so 7% is

\frac{7}{100}

.

Changing only the numerator when simplifying.

Correction: Divide the numerator and denominator by the same number so the fraction keeps the same value.

Thinking 0.50 is greater than 0.5 because it has an extra digit.

Correction: A zero at the end of a decimal does not change its value. Both decimals equal one half.

Lesson summary

Fractions, decimals, and percents can name the same part of a whole.
Divide the numerator by the denominator to change a fraction to a decimal.
Express a decimal in hundredths to change it to a percent.
Divide a percent number by 100 to change it to a decimal.
Write a percent number over 100 to make a fraction, then simplify if possible.

Check your understanding

Question 1

Which decimal is equal to 27%?

0.027
0.27
2.7
27.0

Show answer and explanation

0.27

Twenty-seven percent means 27 out of 100. Dividing 27 by 100 gives 0.27.

Question 2

Which percent is equal to

\frac{3}{4}

?

34%
43%
75%
300%

Show answer and explanation

75%

3\div4=0.75

. The decimal 0.75 is 75 hundredths, or 75%.

Question 3

Which fraction in simplest form is equal to 20%?

$\frac{1}{5}$
$\frac{2}{5}$
$\frac{1}{20}$
$\frac{20}{10}$

Show answer and explanation

\frac{1}{5}

Twenty percent is

\frac{20}{100}

. Dividing both numbers by 20 gives

\frac{1}{5}

.

Key terms

Fraction: A number that describes equal parts of a whole.
Numerator: The top number in a fraction. It counts the parts being described.
Denominator: The bottom number in a fraction. It tells how many equal parts make the whole.
Decimal: A number written using a decimal point and place value.
Percent: A way to describe an amount out of 100.
Simplest form: A fraction whose numerator and denominator have no common factor greater than 1.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 7 Mathematics, expectation B1.7. 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.

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B1.7 · Convert among fractions, decimals, and percents

Ontario Grade 7 Mathematics

Three ways to name the same amount

What you will learn

Connect the three forms

Convert fractions, decimals, and percents

Use place value to check

Practise a conversion route

Equivalent forms

A fraction of a strip

A decimal written as a percent and fraction

A percent written as a decimal and fraction

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Grade 7 Math

About this lesson and its review