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: Rational number operations unify integer arithmetic and fraction arithmetic: find common denominators for addition/subtraction, multiply across for multiplication, flip-and-multiply for division, and track signs throughout.

Common stuck point: Mixing up the procedures: students sometimes try to find common denominators when multiplying fractions, or multiply across when adding. Each operation has its own rule.

Sense of Study hint: Ask yourself which operation you are doing, then apply the matching rule: common denominators for +/-, multiply across for x, flip-and-multiply for /.

Worked Examples

Example 1

easy
Calculate \(\frac{2}{3} + \frac{1}{4}\).

Solution

  1. 1
    Find the LCD of 3 and 4: LCD = 12.
  2. 2
    \(\frac{2}{3} = \frac{8}{12}\) and \(\frac{1}{4} = \frac{3}{12}\).
  3. 3
    Add: \(\frac{8}{12} + \frac{3}{12} = \frac{11}{12}\).
  4. 4
    \(\frac{11}{12}\) is already in lowest terms.

Answer

\(\dfrac{11}{12}\)
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.

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 \(\frac{7}{8} \div \frac{3}{4}\).
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Background Knowledge

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

integer operationsadding fractions unlike denominatorsmultiplying fractionsdividing fractionsdecimals