Decimal Representation Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Decimal Representation.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Writing fractions as digits to the right of a decimal point, using place values of tenths, hundredths, thousandths, etc.

Just like $234 = 200 + 30 + 4$ , we have $2.34 = 2 + 0.3 + 0.04$ .

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Decimal representation writes the fractional part of a number as tenths, hundredths, thousandths to the right of the point.

Common stuck point: The procedure for decimal representation is the easy part; the trap is comparing decimals by digit count so 0.45 beats 0.5. Asking "Are the digits after a point standing for tenths, hundredths, thousandths of a whole?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Are the digits after a point standing for tenths, hundredths, thousandths of a whole?

Worked Examples

Example 1

easy

Convert

\dfrac{5}{8}

to a decimal by long division, and determine whether it terminates or repeats.

Answer

\dfrac{5}{8} = 0.625

(terminating decimal)

First step

Divide

5 \div 8

8

goes into

50

six times (

48

), remainder

2

. So far:

0.6

Full solution

2
$8$ goes into $20$ twice ( $16$ ), remainder $4$ . Decimal: $0.62$ .
3
$8$ goes into $40$ five times ( $40$ ), remainder $0$ . Decimal: $0.625$ .
4
Remainder is $0$ , so the decimal terminates: $\dfrac{5}{8} = 0.625$ .

A fraction in lowest terms terminates as a decimal if and only if its denominator has no prime factors other than

2

and

5

. Here

8 = 2^3

, so the decimal terminates.

Example 2

medium

Convert

0.\overline{142857}

to a fraction in simplest form.

Example 3

medium

Write

3.246

in expanded form using place-value fractions.

Example 4

medium

Write

0.4 + 0.05 + 0.006

as a single decimal.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Write each fraction as a decimal and classify as terminating or repeating: (a)

\dfrac{3}{4}

, (b)

\dfrac{2}{9}

Example 2

medium

Convert

0.3\overline{6}

to a fraction in simplest form. (Note: only the

6

repeats.)

Example 3

easy

Write

\frac{7}{10}

as a decimal.

Example 4

easy

What is the value of the digit

3

0.34

Example 5

easy

Write

\frac{1}{4}

as a decimal.

Example 6

easy

Which is larger:

0.5

0.125

Example 7

easy

Expand

2.34

as a sum of place values.

Example 8

easy

0.4

equal to

0.40

Example 9

easy

Write

\frac{3}{100}

as a decimal.

Example 10

easy

Round

3.7

to the nearest whole number.

Example 11

medium

Convert

\frac{5}{8}

to a decimal.

Example 12

medium

Order least to greatest:

0.09, 0.1, 0.099

Example 13

medium

Write

0.\overline{6}

as a fraction.

Example 14

medium

What decimal is exactly halfway between

0.2

and

0.3

Example 15

medium

Convert

0.375

to a fraction in lowest terms.

Example 16

medium

A length is

2.5

cm and another is

\frac{12}{5}

cm. Which is longer?

Example 17

medium

Round

3.14159

to the nearest hundredth.

Example 18

medium

Express

0.04

as a percent and as a fraction.

Example 19

challenge

The decimal

0.\overline{142857}

equals

\frac{1}{7}

. Use this to write

0.\overline{285714}

as a fraction.

Example 20

challenge

Without dividing, explain why

\frac{1}{3}

cannot be written as a terminating decimal.

Example 21

challenge

A number rounds to

4.6

when rounded to the nearest tenth. What is the range of possible original values?

Example 22

medium

Add

0.6 + 0.45

and write the answer as a decimal.

Example 23

easy

What is the place value of the digit

7

0.073

Example 24

easy

Which is larger,

0.42

0.4

Example 25

easy

Write

\frac{8}{10}

as a decimal.

Example 26

easy

Write

0.6

in expanded form using fractions.

Example 27

easy

How many tenths are in

0.5

Example 28

medium

Convert

\frac{9}{25}

to a decimal.

Example 29

medium

Order from least to greatest:

0.305

0.35

0.3

Example 30

medium

Convert

0.\overline{1}

to a fraction in simplest form.

Example 31

medium

Convert

\frac{13}{20}

to a decimal.

Example 32

medium

Round

7.4583

to the nearest hundredth.

Example 33

medium

Write

0.5

as a percent and as a fraction in simplest form.

Example 34

medium

Order from greatest to least:

0.207

0.27

0.2

Example 35

medium

Subtract:

0.5 - 0.27

Example 36

hard

Convert

\frac{4}{11}

to a decimal (mark the repeating digits).

Example 37

hard

A value rounds to

2.4

at the nearest tenth. Give the smallest possible value.

Example 38

hard

Find a decimal strictly between

0.3

and

0.31

Example 39

hard

Write

0.\overline{142857} \times 3

as a fraction in simplest form, using the fact that

0.\overline{142857}=\frac{1}{7}

Example 40

hard

Which fraction has a terminating decimal:

\frac{5}{16}

\frac{5}{18}

Example 41

challenge

Find the

50

th decimal digit of

\frac{1}{7}=0.\overline{142857}

Example 42

challenge

Express

0.1\overline{6}

(only the

6

repeats) as a fraction in simplest form.

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Related Concepts

Place Value Fractions Operations with Rational Numbers Percent as Ratio

Background Knowledge

These ideas may be useful before you work through the harder examples.

place valuefractions