Show fraction and decimal equivalences to thousandths

Ontario Grade 6 Mathematics

Number

Show the same amount in different forms

A fraction and a decimal can name the same amount. For example, six tenths can be written as

\frac{6}{10}

or

0.6

. A thousandth is one of 1,000 equal parts of one whole. In this lesson, you will connect fractions and decimals through thousandths.

What you will learn

Read decimals to the thousandths place.
Write decimals as fractions with denominators of 10, 100, or 1,000.
Show equivalent fractions and decimals that name the same amount.

Start with equal parts

A fraction names equal parts of a whole. The bottom number is the denominator. It tells how many equal parts make the whole. The top number is the numerator. It tells how many parts are counted. For example,

\frac{3}{4}

means that 3 of 4 equal parts are counted.

Imagine one whole cut into 10 equal strips. Each strip is one tenth. Six strips make six tenths, or

\frac{6}{10}

. A decimal is another way to show an amount using place value. Place value tells what each digit is worth.

The first place to the right of the decimal point is tenths. The next place is hundredths. The third place is thousandths. Each place shows smaller parts of one whole.

\frac{1}{10}=0.1,\quad \frac{1}{100}=0.01,\quad \frac{1}{1000}=0.001

Tenths match a denominator of 10.
Hundredths match a denominator of 100.
Thousandths match a denominator of 1,000.

Read decimal places

Read the digits in a decimal from left to right. In

0.347

, the 3 is in the tenths place, the 4 is in the hundredths place, and the 7 is in the thousandths place. So the decimal shows 3 tenths, 4 hundredths, and 7 thousandths.

The same decimal can also be read as 347 thousandths. There are three digits after the decimal point, so the fraction has 1,000 as its denominator. The digits 347 tell how many thousandths are counted.

A zero at the beginning of a decimal, as in

0.347

, shows that there are no whole ones. It does not change the parts shown after the decimal point.

0.347=\frac{347}{1000}

One digit after the decimal point shows tenths.
Two digits after the decimal point show hundredths.
Three digits after the decimal point show thousandths.

Show equivalent forms

Equivalent forms have the same value. The fractions

\frac{6}{10}

and

\frac{60}{100}

are equivalent. Both name six tenths, though one counts tenths and the other counts hundredths.

To make an equivalent fraction, multiply the numerator and denominator by the same number. This changes how many equal parts the whole is divided into, and how many of those parts are counted, by the same factor. The amount stays the same.

Decimals can also show the same value with a zero at the end. For example,

0.6

,

0.60

, and

0.600

all show six tenths. The extra zeroes show that the amount can also be named in hundredths or thousandths.

\frac{6}{10}=\frac{60}{100}=\frac{600}{1000}=0.6=0.60=0.600

When making an equivalent fraction, multiply the top and bottom by the same number.
A zero at the end of a decimal does not change its value.
Match the denominator to the decimal place you are showing.

Check the place value

Before writing an equivalence, count the digits after the decimal point. Then match the fraction denominator to that place. For example, two decimal places match hundredths, so

0.23

is 23 hundredths.

You can show hundredths as thousandths by adding a zero at the end of the decimal and multiplying the fraction's numerator and denominator by 10. The value stays the same, but the amount is now counted in thousandths.

To practise, try writing

\frac{4}{10}

as a decimal and as a fraction in thousandths. Then check that both fractions name four tenths.

\frac{23}{100}=0.23=0.230=\frac{230}{1000}

A denominator of 100 matches two decimal places.
A denominator of 1,000 matches three decimal places.
Check that the digits and denominator name the same place value.

Decimal place-value guide

Place	Fraction name	Example
Tenths	$\frac{1}{10}$	$0.1$
Hundredths	$\frac{1}{100}$	$0.01$
Thousandths	$\frac{1}{1000}$	$0.001$

Worked example

Show six tenths in several ways

Write

\frac{6}{10}

as a decimal and as equivalent fractions in hundredths and thousandths.

Six tenths of a whole

The bar represents one whole divided into 10 equal parts. Six parts are counted.

Read the denominator
The denominator is 10, so the fraction counts tenths. Six tenths is written with a 6 in the tenths place.
$\frac{6}{10}=0.6$
Show hundredths
Multiply the numerator and denominator by 10. The fraction still names the same amount, now counted in hundredths.
$\frac{6}{10}=\frac{60}{100}$
Show thousandths
Multiply the numerator and denominator of $\frac{60}{100}$ by 10. The equivalent fraction now counts thousandths.
$\frac{60}{100}=\frac{600}{1000}$

Answer:

\frac{6}{10}=\frac{60}{100}=\frac{600}{1000}=0.6

Check: Each form names six tenths of one whole. The decimal can also be written as

0.60

or

0.600

.

Worked example

Write a decimal as a fraction

Write

0.284

as a fraction in thousandths.

Count the decimal places
There are three digits after the decimal point. The decimal shows thousandths, so use 1,000 as the denominator.
$0.284=\frac{\square}{1000}$
Use the digits
The digits after the decimal point are 284. They show that 284 thousandths are counted.
$0.284=\frac{284}{1000}$

Answer:

0.284=\frac{284}{1000}

Check: The denominator is 1,000 because the decimal has three places after the decimal point.

Worked example

Connect hundredths and thousandths

Write

\frac{23}{100}

as a decimal and show the same value in thousandths.

Read the denominator
The denominator is 100, so the fraction counts hundredths. Write two digits after the decimal point.
$\frac{23}{100}=0.23$
Show thousandths
Add a zero at the end of the decimal to show thousandths. Multiply the numerator and denominator by 10 to make the equivalent fraction.
$0.23=0.230,\quad \frac{23}{100}=\frac{230}{1000}$

Answer:

\frac{23}{100}=0.23=0.230=\frac{230}{1000}

Check: The fraction in thousandths has 10 times as many counted parts and 10 times as many equal parts as the fraction in hundredths. The value stays the same.

Common mistakes and how to avoid them

Writing

0.6

as

\frac{6}{100}

because the decimal contains a 6.

Correction: The 6 is in the tenths place, so

0.6=\frac{6}{10}

. Hundredths need two places after the decimal point.

Changing only the numerator when making an equivalent fraction.

Correction: Multiply the numerator and denominator by the same number. For example,

\frac{6}{10}=\frac{60}{100}

.

Thinking that

0.23

and

0.230

have different values.

Correction: A zero at the end does not change the value. Both decimals show 23 hundredths.

Lesson summary

Tenths, hundredths, and thousandths match denominators of 10, 100, and 1,000.
Use the number of decimal places to choose the denominator when writing a decimal as a fraction.
Multiply the numerator and denominator by the same number to make an equivalent fraction.
A zero at the end of a decimal can show another place value without changing the amount.

Check your understanding

Question 1

Which fraction is equivalent to

0.7

and has a denominator of 1,000?

$\frac{7}{1000}$
$\frac{70}{1000}$
$\frac{700}{1000}$
$\frac{7000}{1000}$

Show answer and explanation

\frac{700}{1000}

0.7

is seven tenths. Seven tenths is equivalent to 700 thousandths, so the fraction is

\frac{700}{1000}

.

Question 2

What decimal is equivalent to

\frac{45}{100}

?

$0.045$
$0.45$
$4.5$
$0.4500$

Show answer and explanation

0.45

The denominator 100 means hundredths. Forty-five hundredths is

0.45

.

Question 3

Which statement is true?

$0.308=\frac{308}{100}$
$0.308=\frac{38}{1000}$
$0.308=\frac{308}{1000}$
$0.308=\frac{30}{1000}$

Show answer and explanation

0.308=\frac{308}{1000}

There are three digits after the decimal point, so the decimal shows 308 thousandths:

\frac{308}{1000}

.

Key terms

Fraction: A number that shows equal parts of a whole.
Numerator: The top number in a fraction. It tells how many parts are counted.
Denominator: The bottom number in a fraction. It tells how many equal parts make the whole.
Equivalent: Having the same value, even when written in different ways.
Thousandth: One of 1,000 equal parts of a whole.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 6 Mathematics, expectation B1.6. 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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B1.6 · Show fraction and decimal equivalences to thousandths

Ontario Grade 6 Mathematics

Show the same amount in different forms

What you will learn

Start with equal parts

Read decimal places

Show equivalent forms

Check the place value

Decimal place-value guide

Show six tenths in several ways

Write a decimal as a fraction

Connect hundredths and thousandths

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Grade 6 Math

About this lesson and its review