Operations with Rational Numbers Examples in Math

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

Extending addition, subtraction, multiplication, and division to the full set of rational numbers—including fractions, decimals, mixed numbers, and their negative counterparts.

Once you can handle integers and fractions separately, combine the skills: apply the sign rules you know from integers to fractions and decimals. $-\frac{2}{3} + \frac{1}{4}$ uses common denominators AND sign rules at the same time.

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: Operations with rational numbers apply integer sign rules to fractions, decimals, and mixed numbers, so you manage common denominators and signs in the same step.

Common stuck point: The procedure for operations with rational numbers is the easy part; the trap is dropping the sign while finding common denominators. Asking "Are the signed numbers also fractions or decimals, needing fraction rules with sign tracking?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Are the signed numbers also fractions or decimals, needing fraction rules with sign tracking?

Worked Examples

Example 1

easy

Calculate

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

Answer

\dfrac{11}{12}

First step

Find the LCD of 3 and 4: LCD = 12.

Full solution

2
$\frac{2}{3} = \frac{8}{12}$ and $\frac{1}{4} = \frac{3}{12}$ .
3
Add: $\frac{8}{12} + \frac{3}{12} = \frac{11}{12}$ .
4
$\frac{11}{12}$ is already in lowest terms.

To add fractions, convert to a common denominator (LCD=12), add numerators, and simplify.

Example 2

medium

Calculate

\frac{5}{6} \times \frac{3}{10}

and simplify.

Example 3

medium

Compute

3\dfrac{1}{4} - 1\dfrac{2}{3}

Example 4

medium

Compute

\dfrac{3}{4} + \dfrac{1}{6} - \dfrac{5}{12}

Example 5

hard

Compute

\left(-\dfrac{3}{4}\right) + \dfrac{5}{6} \times \dfrac{3}{10}

Example 6

hard

Compute

\dfrac{5}{6} - \left( \dfrac{1}{2} - \dfrac{2}{3} \right) \times 3

Example 7

challenge

Compute

\dfrac{ \frac{1}{2} + \frac{1}{3} }{ \frac{1}{2} - \frac{1}{3} }

Practice Problems

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

Example 1

easy

Calculate

\frac{3}{4} - \frac{1}{6}

Example 2

medium

Calculate

\frac{7}{8} \div \frac{3}{4}

Example 3

easy

Compute

-3 + 7

Example 4

easy

Compute

-5 - 3

Example 5

easy

Compute

-4 \times 3

Example 6

easy

Compute

-6 \div (-2)

Example 7

easy

Compute

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

Example 8

easy

Compute

\frac{2}{3} \times \frac{3}{5}

Example 9

easy

Compute

0.5 + 0.25

Example 10

easy

Compute

-\frac{3}{4} + \frac{3}{4}

Example 11

medium

Compute

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

Example 12

medium

Compute

-\frac{2}{3} \div \left(-\frac{1}{2}\right)

Example 13

medium

Compute

2\frac{1}{2} \times \frac{2}{5}

Example 14

medium

Compute

-3 - (-7)

Example 15

medium

Compute

\frac{3}{4} - \frac{5}{6}

Example 16

medium

Compute

(-2)^3

Example 17

medium

Compute

\frac{2}{3} \div 4

Example 18

medium

Compute

-0.6 \times 0.5

Example 19

challenge

Compute

-\frac{1}{2} + \frac{2}{3} - \frac{1}{6}

Example 20

challenge

Evaluate

\frac{-3}{4} \times \frac{8}{9} \div \left(-\frac{2}{3}\right)

Example 21

challenge

A diver descends

\frac{3}{4}

m every second for

8

seconds, starting at the surface (

0

). Find the final depth as a signed number.

Example 22

medium

Compute

1\frac{1}{2} \div 3

Example 23

easy

Compute

-8 + 5

Example 24

easy

Compute

-9 \times (-4)

Example 25

easy

Compute

\dfrac{3}{5} + \dfrac{1}{5}

Example 26

easy

Compute

\dfrac{5}{8} - \dfrac{1}{4}

Example 27

easy

Compute

-0.2 + 0.7

Example 28

easy

Compute

\dfrac{2}{7} \times 0

Example 29

medium

Compute

\dfrac{5}{6} + \dfrac{2}{9}

Example 30

medium

Compute

-\dfrac{4}{5} \times \dfrac{15}{16}

Example 31

medium

Compute

\dfrac{-3}{7} \div \dfrac{6}{14}

Example 32

medium

Compute

-\dfrac{5}{8} - \left(-\dfrac{1}{4}\right)

Example 33

medium

Compute

0.25 \times \dfrac{8}{3}

Example 34

medium

Compute

\left(-\dfrac{2}{5}\right)^2

Example 35

medium

Compute

-1.2 \div 0.3

Example 36

hard

Compute

\dfrac{ -\frac{2}{3} + \frac{1}{6} }{ \frac{1}{2} }

Example 37

hard

A bank account starts at

0

, deposits

\$45.75

, and then has a withdrawal of

\$60.20

. What is the balance?

Example 38

hard

Compute

\left(\dfrac{-1}{2}\right)^3 + \left(\dfrac{1}{2}\right)^2

Example 39

hard

A recipe calls for

2\dfrac{1}{3}

cups of flour. How much flour is in

\dfrac{3}{4}

of the recipe?

Example 40

challenge

Compute

\dfrac{1}{1 \cdot 2} + \dfrac{1}{2 \cdot 3} + \dfrac{1}{3 \cdot 4} + \dfrac{1}{4 \cdot 5}

Example 41

challenge

x = -\dfrac{2}{3}

and

y = \dfrac{3}{4}

, compute

x^2 - 2xy + y^2

← Back to Operations with Rational Numbers Formula Explained Practice Mode

Related Concepts

Integer Operations Adding Fractions with Unlike Denominators Multiplying Fractions Dividing Fractions Decimals Expressions

Background Knowledge

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

integer operationsadding fractions unlike denominatorsmultiplying fractionsdividing fractionsdecimals