Division of Rational Numbers — Rules, Steps, Real Life Examples and Practice Questions

Published: 2 Nov 2026

Division of rational numbers helps us understand how one rational number fits into another. It may seem tricky at first, but with simple methods like reciprocal, KFC, and butterfly, dividing rational numbers becomes easy and fun to learn.

Basic Rules

Student Learning Outcomes

By the end of this guide on division of rational numbers, you will be able to:

Divide any rational numbers, including fractions, integers, mixed numbers, and decimals.
Use the reciprocal rule and apply the KFC and Butterfly methods correctly.
Apply sign rules while dividing positive and negative rational numbers.
Simplify answers and convert results between fraction, mixed, and decimal forms.
Solve real-life and multi-step division problems with accuracy.

Understanding the Reciprocal

Reciprocal of a Rational Number

The reciprocal of a rational number is found by interchanging the numerator and denominator with each other.
General Form:
$\text{Reciprocal of } \dfrac{a}{b} \text{ is } \frac{b}{a}$
Examples:
Reciprocal of $\dfrac{3}{4}$ is $\dfrac{4}{3}$
Reciprocal of $-\dfrac{5}{7}$ is $-\dfrac{7}{5}$
Zero has no reciprocal.

Dividing Rational Numbers

Dividing rational numbers may look tricky at first, but it’s actually very simple! Instead of dividing directly, we use a neat trick: flip the second fraction (take its reciprocal) and then multiply. After that, just simplify to get the answer in standard form.

Steps to Divide Rational Numbers

Steps

To divide $\dfrac{a}{b} \div \dfrac{c}{d}$, follow these rules:

Find the reciprocal of the second fraction $\dfrac{c}{d} \to \dfrac{d}{c}$
Change the division to multiplication.
Simplify the rational number to its lowest form, if necessary
Multiply numerators together and denominators together.
Simplify to Standard Form.

General Formula

$\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{d}{c}$

Example

$\begin{aligned} \dfrac{5}{7} \div \dfrac{2}{3} & = \dfrac{5}{7} \times \dfrac{3}{2} \\ & = \dfrac{15}{14} \end{aligned}$

Did You Know?

Division of Rational Numbers can also be done by KFC and Butterfly Method.

KFC and Butterfly Methods

The KFC Method for Division

The easiest way to divide rational numbers is the KFC method. It stands for:

KFC = Keep, Flip, Change

K – Keep the first fraction.
F – Flip the second fraction (take the reciprocal).
C – Change the division sign to multiplication.
Example:
$\dfrac{3}{4} \div \dfrac{2}{5}$

Step-by-step using KFC Method:

Worked Example

Keep: $\dfrac{3}{4}$
Flip: $\dfrac{2}{5} \rightarrow \dfrac{5}{2}$
Change: $\div \rightarrow \times$
Now multiply:
$\dfrac{3}{4} \times \dfrac{5}{2} = \dfrac{15}{8}$

Butterfly Method of Division 🦋

Butterfly Method

The butterfly method of division is a visual technique for dividing fractions by using crisscross multiplication of rational Numbers.
It looks like the wings of a butterfly.

Different Cases of Division

There are different ways to divide rational numbers. Sometimes the numbers are fractions, while other times they are mixed numbers, whole numbers, or decimals. The main idea remains the same as we use the reciprocal, but the steps can change slightly in each case. Let us go through these situations with simple examples to understand them better.

1. Division of Negative Rational Numbers

Rule

To divide negative rational numbers, follow integers sign rules:
Negative ÷ Negative = Positive
Positive ÷ Negative = Negative

Examples

Both are Negative Rational Numbers
Example:
$\begin{aligned} -\dfrac{5}{7} \div -\dfrac{2}{3} & = \dfrac{5}{7} \div \dfrac{2}{3} \\ & = \dfrac{5}{7} \times \dfrac{3}{2} \\ & = \dfrac{5 \times 3}{7 \times 2} \\ & = \dfrac{15}{14} \end{aligned}$
One Negative, One Positive Rational Number
Example:
$\begin{aligned} -\dfrac{5}{7} \div \dfrac{2}{3} & = -\dfrac{5}{7} \times \dfrac{3}{2} \\ & = \dfrac{-5 \times 3}{7 \times 2} \\ & = \dfrac{-15}{14} \\ & = -\dfrac{15}{14} \end{aligned}$

Real-Life Example

A company lost $\dfrac{5}{6}$ tons of material in total. Each smaller loss was $\dfrac{1}{3}$ ton. How many such losses were there?
$\begin{aligned} -\dfrac{5}{6} \div -\dfrac{1}{3} &= \dfrac{5}{6} \div \dfrac{1}{3} \\ &= \dfrac{5}{6} \times \dfrac{3}{1} \\ &= \dfrac{5 \times 3}{6 \times 1} \\ &= \dfrac{15}{6} \\ &= \dfrac{5}{2} \end{aligned}$
There were $\dfrac{5}{2}$ or $2\dfrac{1}{2}$ equal losses.

2. Division of Rational and Whole (Integers) Numbers

Rule

To divide rational numbers and whole numbers, convert the whole number into a fraction(denominator = 1). $ a = \dfrac{a}{1} $

Examples

Divide Fraction by Whole Number
Example:
$\begin{aligned} \dfrac{4}{5} \div 6 & = \dfrac{4}{5} \div \dfrac{6}{1} \\ & = \dfrac{4}{5} \times \dfrac{1}{6} \\ & = \dfrac{\cancel{4}^2}{5} \times \dfrac{1}{\cancel{6}^3} \\ & = \dfrac{2}{5} \times \dfrac{1}{3} \\ & = \dfrac{2}{15} \end{aligned}$
Divide Whole Number by Fraction
Example:
$\begin{aligned} 6 \div \dfrac{2}{3} & = \dfrac{6}{1} \div \dfrac{2}{3} \\ & = \dfrac{6}{1} \times \dfrac{3}{2} \\ & = \dfrac{\cancel{6}^3}{1} \times \dfrac{3}{\cancel{2}^1} \\ & = \dfrac{3}{1} \times \dfrac{3}{1} \\ & = \dfrac{9}{1} \\ & = 9 \end{aligned}$

Real-Life Example

Example:
Sarah has $\dfrac{7}{8}$ kg of chocolate. She wants to divide it equally among 4 friends. How much does each friend get?
Solution
$\begin{aligned} \dfrac{7}{8} \div 4 &= \dfrac{7}{8} \div \dfrac{4}{1} \\ &= \dfrac{7}{8} \times \dfrac{1}{4} \\ &= \dfrac{7 \times 1}{8 \times 4} \\ &= \dfrac{7}{32} \end{aligned}$
Each friend gets $\dfrac{7}{32}$ kg of chocolate.

3. Division of Rational and Mixed Numbers

Rule

To divide rational numbers and Mixed numbers, convert the Mixed number into an improper fraction.

Examples

Divide Fraction by Mixed Number
Example:
$\begin{aligned} \dfrac{2}{5} \div 1\dfrac{1}{2} & = \dfrac{2}{5} \div \dfrac{3}{2} \\ & = \dfrac{2}{5} \times \dfrac{2}{3} \\ & = \dfrac{4}{15} \end{aligned}$
Divide Mixed Number by Fraction
Example:
$\begin{aligned} 2\dfrac{1}{3} \div \dfrac{1}{2} & = \dfrac{7}{3} \div \dfrac{1}{2} \\ & = \dfrac{7}{3} \times \dfrac{2}{1} \\ & = \dfrac{14}{3} \end{aligned}$

Real-Life Example

Sarah baked $3\dfrac{1}{2}$ kg of cake. She wants to pack it in portions of $\dfrac{3}{4}$ kg. How many portions can she make?
Solution
$\begin{aligned} 3\dfrac{1}{2} \div \dfrac{3}{4} &= \dfrac{7}{2} \div \dfrac{3}{4} \\ &= \dfrac{7}{2} \times \dfrac{4}{3} \\ &= \dfrac{7 \times 4}{2 \times 3} \\ &= \dfrac{28}{6} \\ &= \dfrac{14}{3} = 4\dfrac{2}{3} \end{aligned}$
She can make $4\dfrac{2}{3}$ portions.

4. Division of Decimals and Rational Numbers

Rule

To divide rational numbers and Decimals numbers, convert the decimals into fractions, then divide normally.

Examples

Divide Fraction by Decimal
Example:
$\begin{aligned} \dfrac{3}{4} \div 0.5 & = \dfrac{3}{4} \div \dfrac{5}{10} \\ & = \dfrac{3}{4} \times \dfrac{10}{5} \\ & = \dfrac{3}{4} \times \dfrac{\cancel{10}^2}{\cancel{5}^1} \\ & = \dfrac{3}{4} \times \dfrac{2}{1} \\ & = \dfrac{\cancel{6}^3}{\cancel{4}^2} \\ & = \dfrac{3}{2} \end{aligned}$
Divide Decimal by Fraction
Example:
$\begin{aligned} 0.6 \div \dfrac{7}{11} & = \dfrac{6}{10} \div \dfrac{7}{11} \\ & = \dfrac{6}{10} \times \dfrac{11}{7} \\ & = \dfrac{6 \times 11}{10 \times 7} \\ & = \dfrac{\cancel{66}^{33}}{\cancel{70}^{35}} \\ & = \dfrac{33}{35} \end{aligned}$

Real-Life Example

$0.8$ kg of sugar is packed in packets of $\dfrac{2}{5}$ kg. How many packets are needed?
Solution
$\begin{aligned} 0.8 \div \dfrac{2}{5} &= \dfrac{8}{10} \div \dfrac{2}{5} \\ &= \dfrac{4}{5} \div \dfrac{2}{5} \\ &= \dfrac{4}{5} \times \dfrac{5}{2} \\ &= \dfrac{4 \times 5}{5 \times 2} \\ &= \dfrac{4}{2} = 2 \end{aligned}$
There are 2 packets.

5. Division of Two Mixed Numbers

Rule and Example

To divide Two Mixed numbers, convert the both mixed numbers into improper fractions.
Example:
$\begin{aligned} 1\dfrac{1}{4} \div 2\dfrac{1}{2} & = \dfrac{5}{4} \div \dfrac{5}{2} \\ & = \dfrac{5}{4} \times \dfrac{2}{5} \\ & = \dfrac{10}{20} \\ & = \dfrac{1}{2} \end{aligned}$

Real-Life Example

A rope of $5\dfrac{3}{4}$ meters is divided into parts of $2\dfrac{1}{2}$ meters. How many parts?
Solution
$\begin{aligned} 5\dfrac{3}{4} \div 2\dfrac{1}{2} &= \dfrac{23}{4} \div \dfrac{5}{2} \\ &= \dfrac{23}{4} \times \dfrac{2}{5} \\ &= \dfrac{23 \times 2}{4 \times 5} \\ &= \dfrac{46}{20} \\ &= \dfrac{23}{10} = 2\dfrac{3}{10} \end{aligned}$
There are $2\dfrac{3}{10}$ parts.

6. Division of Mixed and Whole (Integers) Numbers

Rule and Example

To divide Whole and Mixed numbers, change the mixed number and whole number into fractions before solving.
Example:
$\begin{aligned} 3\dfrac{1}{2} \div 2 & = \dfrac{7}{2} \div \dfrac{2}{1} \\ & = \dfrac{7}{2} \times \dfrac{1}{2} \\ & = \dfrac{7}{4} \end{aligned}$

Real-Life Example

A worker completes $7\dfrac{1}{2}$ hours of work in 3 days equally. How many hours did he work each day?
Solution
$\begin{aligned} 7\dfrac{1}{2} \div 3 &= \dfrac{15}{2} \div 3 \\ &= \dfrac{15}{2} \times \dfrac{1}{3} \\ &= \dfrac{15 \times 1}{2 \times 3} \\ &= \dfrac{15}{6} \\ &= \dfrac{5}{2} = 2\dfrac{1}{2} \end{aligned}$
He worked $2\dfrac{1}{2}$ hours each day.

7. Dividing Fractions (Rational Numbers) with Same Numerators

Rule and Example

If the numerators are the same, divide the second denominator by the first denominator.
General Form:
$\dfrac{a}{b} \div \dfrac{a}{c} = \dfrac{c}{b}$
Example:
$\dfrac{5}{7} \div \dfrac{5}{9} = \dfrac{9}{7}$
Explanation
$\begin{aligned} \dfrac{5}{7} \div \dfrac{5}{9} & = \dfrac{5}{7} \times \dfrac{9}{5} \\ & = \dfrac{\cancel{5}^1}{7} \times \dfrac{9}{\cancel{5}^1} \\ & = \dfrac{1}{7} \times \dfrac{9}{1} \\ & = \dfrac{9}{7} \end{aligned}$

Real-Life Example

A student studies $\dfrac{6}{8}$ hours one day and $\dfrac{6}{4}$ hours the next day. Compare the times using division.
Solution
$\begin{aligned} \dfrac{6}{8} \div \dfrac{6}{4} &= \dfrac{6}{8} \times \dfrac{4}{6} \\ &= \dfrac{6 \times 4}{8 \times 6} \\ &= \dfrac{4}{8} \\ &= \dfrac{1}{2} \end{aligned}$
The first day’s study time is half of the second day’s time.

8. Dividing Fractions (Rational Numbers) with Same Denominators

Rule and Example

If the denominators are the same, divide only the numerators (Divide first numerator by second).
General Form:
$\dfrac{a}{c} \div \dfrac{b}{c} = \dfrac{a}{b}$
Example:
$\dfrac{4}{9} \div \dfrac{5}{9} = \dfrac{4}{5}$
Explanation
$\begin{aligned} \dfrac{4}{9} \div \dfrac{5}{9} & = \dfrac{4}{9} \times \dfrac{9}{5} \\ & = \dfrac{4}{\cancel{9}^1} \times \dfrac{\cancel{9}^1}{5} \\ & = \dfrac{4}{5} \end{aligned}$

Real-Life Example

A student reads $\dfrac{20}{27}$ of a book, while another reads $\dfrac{5}{27}$. How many times more did the first student read?
Solution
$\begin{aligned} \dfrac{20}{27} \div \dfrac{5}{27} &= \dfrac{20}{27} \times \dfrac{27}{5} \\ &= \dfrac{20 \times 27}{27 \times 5} \\ &= \dfrac{20}{5} \\ &= 4 \end{aligned}$
The first student read 4 times more than the second.

9. Division of Same Rational Numbers

Rule and Example

Any number divided by itself is always equal to 1.
Example:
$\dfrac{7}{8} \div \dfrac{7}{8} = 1$
Explanation
$\begin{aligned} \dfrac{7}{8} \div \dfrac{7}{8} & = \dfrac{7}{8} \times \dfrac{8}{7} \\ & = \dfrac{\cancel{7}^1}{\cancel{8}^1} \times \dfrac{\cancel{8}^1}{\cancel{7}^1} \\ & = 1 \end{aligned}$

Real-Life Example

A boy ate $\dfrac{3}{5}$ of a pizza today and the same $\dfrac{3}{5}$ yesterday. What is $\dfrac{3}{5} \div \dfrac{3}{5}$?
Solution
$\begin{aligned} \dfrac{3}{5} \div \dfrac{3}{5} &= \dfrac{3}{5} \times \dfrac{5}{3} \\ &= \dfrac{3 \times 5}{5 \times 3} \\ &= 1 \end{aligned}$
The result is 1 — both are equal amounts.

10. Division of Zero and Rational Numbers

Rule

Zero divided by any number is 0, but dividing by 0 is undefined.

Important: Zero Rules

Zero Divided by Rational Number
$0 \div \text{any number} = 0$
Example:
$0 \div \dfrac{5}{6} = 0$
Rational Number Divided by Zero
$\text{Any number} \div 0$ is undefined
Example:
$\dfrac{3}{4} \div 0 = \text{undefined}$

Real-Life Example

A teacher has $\dfrac{4}{7}$ of a book and tries to divide it among 0 students. What is the result?
Solution:
Division by zero is undefined — we cannot divide any number by zero.

11. Division by the Reciprocal of a Rational Number

Rule and Example

To divide a rational number by its reciprocal, gives the square of that numbers.
Example:
$\dfrac{2}{3} \div \dfrac{3}{2} = \dfrac{4}{9}$
Explanation:
$\begin{aligned} \dfrac{2}{3} \div \dfrac{3}{2} & = \dfrac{2}{3} \times \dfrac{2}{3} \\ & = \dfrac{2 \times 2}{3 \times 3} \\ & = \dfrac{4}{9} \end{aligned}$

Real-Life Example

A farmer uses $\dfrac{5}{9}$ of his land. If he divides this by its reciprocal, what fraction does it equal?
Solution
$\begin{aligned} \dfrac{5}{9} \div \dfrac{9}{5} &= \dfrac{5}{9} \times \dfrac{5}{9} \\ &= \dfrac{5 \times 5}{9 \times 9} \\ &= \dfrac{25}{81} \end{aligned}$
The result is $\dfrac{25}{81}$ — the square of the number.

Quick Tips

Always change division into multiplication by using the reciprocal of the second number.
Divide: $\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{d}{c}$
Simplify before and after dividing to keep numbers small.
Follow the same sign rules:
Negative ÷ Negative = Positive
Positive ÷ Negative = Negative
Convert mixed numbers and decimals into fractions before dividing.
Write whole numbers as $\dfrac{a}{1}$ when dividing.
Use cross-cancellation after changing to multiplication.
Use brackets around negative numbers to avoid sign confusion.

Practice Questions

Practice Question 1

$-2 \div \dfrac{3}{5} \div 4\dfrac{1}{2} \div \left(-\dfrac{10}{9}\right)$
Solution:

Convert the mixed number to an improper fraction
$4\dfrac{1}{2} = \dfrac{9}{2}$
Rewrite the expression using fractions
$= \dfrac{-2}{1} \div \dfrac{3}{5} \div \dfrac{9}{2} \div \left(-\dfrac{10}{9}\right)$
Convert divisions to multiplications by reciprocals
$= \dfrac{-2}{1} \times \dfrac{5}{3} \times \dfrac{2}{9} \times \left(-\dfrac{9}{10}\right)$
$= \dfrac{-2}{1} \times \dfrac{5}{3} \times \dfrac{2}{9} \times \dfrac{-9}{10}$
Cancel common factors step by step
Cancel (9) in numerator with (9) in denominator.
Cancel (5) in numerator with (10) in denominator.
$= \dfrac{-2}{1} \times \dfrac{1}{3} \times \dfrac{2}{1} \times \dfrac{-1}{2}$
Now again cancel (2) in numerator with (2) in denominator.
$= \dfrac{-2}{1} \times \dfrac{1}{3} \times \dfrac{1}{1} \times \dfrac{-1}{1}$
Multiply numerators and denominators
$= \dfrac{-2 \times 1 \times 1 \times (-1)}{1 \times 3 \times 1 \times 1}$
$= \dfrac{2}{3}$

Practice Question 2

$\dfrac{-3}{4} \div 2\dfrac{1}{2} \div \dfrac{5}{6} \div 0.8$
Solution:

Convert mixed and decimal numbers to fractions
$2\dfrac{1}{2} = \dfrac{5}{2}, \quad 0.8 = \dfrac{8}{10}$
Rewrite the expression using fractions
$\dfrac{-3}{4} \div \dfrac{5}{2} \div \dfrac{5}{6} \div \dfrac{8}{10}$
Change all divisions to multiplications by reciprocals
$\dfrac{-3}{4} \times \dfrac{2}{5} \times \dfrac{6}{5} \times \dfrac{10}{8}$
Cancel common factors step by step
Cancel (4) in denominator with (2) in numerator.
Cancel (6) in numerator with (8) in denominator.
Cancel (5) in denominator with (10) in numerator.
$\dfrac{-3}{2} \times \dfrac{1}{5} \times \dfrac{3}{1} \times \dfrac{2}{4}$
Now cancel (2) in numerator with (4) in denominator.
$\dfrac{-3}{2} \times \dfrac{1}{5} \times \dfrac{3}{1} \times \dfrac{1}{2}$