Rational Numbers Examples in Math

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

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

Concept Recap

Numbers that can be expressed as a ratio of two integers ( $\frac{a}{b}$ where $b \neq 0$ ).

Any number you can write as a fraction, including decimals that end or repeat.

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: A rational number is a ratio of two integers, which is exactly the numbers whose decimals end or repeat.

Common stuck point: The procedure for rational numbers is the easy part; the trap is calling every decimal rational. Asking "Can this number be written as one integer divided by another (with the decimal ending or repeating)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Can this number be written as one integer divided by another (with the decimal ending or repeating)?

Worked Examples

Example 1

easy

Place the following numbers in order from least to greatest:

\frac{3}{4}

0.6

\frac{7}{10}

Answer

0.6 < \frac{7}{10} < \frac{3}{4}

First step

Convert all to decimals:

\frac{3}{4} = 0.75

0.6 = 0.6

\frac{7}{10} = 0.7

Full solution

2
Order the decimals: $0.6 < 0.7 < 0.75$ .
3
In original form: $0.6 < \frac{7}{10} < \frac{3}{4}$ .

To compare rational numbers in different forms, convert them all to the same representation—usually decimals—then order them. Every rational number can be expressed as a terminating or repeating decimal.

Example 2

medium

Express

0.\overline{36}

as a fraction in simplest form.

Example 3

easy

Plot

\frac{1}{2}

-\frac{1}{4}

, and

\frac{3}{4}

on a number line. List them from least to greatest.

Example 4

medium

Show that

\sqrt{16}

is rational but

\sqrt{20}

is not.

Example 5

hard

Show

\frac{a}{b} \div \frac{c}{d}

is rational when

b, c, d \ne 0

. Use

a=3, b=4, c=2, d=5

Example 6

hard

Find a fraction equivalent to

\frac{15}{25}

in lowest terms, and identify its decimal form.

Example 7

challenge

Prove the sum of a rational and an irrational is irrational.

Practice Problems

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

Example 1

easy

Which is greater:

\frac{5}{8}

\frac{3}{5}

Example 2

medium

Order these numbers from least to greatest:

-\frac{1}{2}

-0.45

-\frac{3}{5}

Example 3

easy

\frac{3}{4}

a rational number?

Example 4

easy

Is the integer

7

a rational number?

Example 5

easy

Is the decimal

0.5

rational?

Example 6

easy

Is the repeating decimal

0.\overline{3}

rational?

Example 7

easy

-\frac{2}{5}

rational?

Example 8

easy

Write the integer

-4

as a ratio of two integers.

Example 9

easy

Convert

0.25

to a fraction in lowest terms.

Example 10

easy

Which is rational:

\sqrt{9}

\sqrt{2}

Example 11

medium

Convert the repeating decimal

0.\overline{6}

to a fraction.

Example 12

medium

\frac{0}{5}

rational? Is

\frac{5}{0}

rational?

Example 13

medium

Convert

0.\overline{12}

to a fraction in lowest terms.

Example 14

medium

Add

\frac{1}{3}+\frac{1}{6}

and state whether the sum is rational.

Example 15

medium

Multiply

\frac{2}{3}\cdot\frac{3}{4}

and state whether the product is rational.

Example 16

medium

Order from least to greatest:

\frac{1}{2},\ 0.4,\ \frac{3}{5}

Example 17

medium

A decimal terminates exactly when its reduced denominator has only which prime factors?

Example 18

medium

\frac{\sqrt{16}}{2}

rational?

Example 19

challenge

Show that

0.\overline{9}=1

, and explain why this means

0.\overline{9}

is rational.

Example 20

challenge

Prove that the sum of a rational number and an irrational number is irrational.

Example 21

challenge

Find a rational number strictly between

\frac{1}{3}

and

\frac{1}{2}

Example 22

medium

Express

\frac{7}{8}

as a decimal and confirm it terminates.

Example 23

easy

\frac{11}{4}

a rational number? Justify in one sentence.

Example 24

easy

Convert

0.4

to a fraction in lowest terms.

Example 25

easy

Convert

\frac{3}{5}

to a decimal.

Example 26

medium

Express

0.\overline{27}

as a fraction in simplest form.

Example 27

medium

Which is larger:

\frac{7}{9}

\frac{8}{11}

Example 28

medium

Compute

\frac{2}{3} + \frac{1}{4}

and express as a fraction in lowest terms.

Example 29

medium

Compute

\frac{5}{6} - \frac{1}{4}

Example 30

medium

Find a rational number strictly between

\frac{1}{3}

and

\frac{1}{2}

Example 31

medium

Convert

1.\overline{6}

to a fraction.

Example 32

medium

Is the sum of two rational numbers always rational?

Example 33

hard

Express

0.1\overline{6}

as a fraction in simplest form.

Example 34

hard

Compute

\frac{2}{3} \times \frac{9}{8}

in lowest terms.

Example 35

hard

Find the additive inverse and multiplicative inverse of

-\frac{3}{7}

Example 36

hard

Is the difference of any two rationals rational? Prove it.

Example 37

hard

\pi

rational? Briefly justify.

Example 38

challenge

Find a rational number whose decimal expansion has a repeating block of length exactly 6.

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Related Concepts

Fractions Decimals Integers Irrational Numbers Real Numbers

Background Knowledge

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

fractionsdecimalsintegers