Multiplying Rational Numbers – Rules, Real Life Examples, and Practice Questions

Multiplying rational numbers is the process of finding the product of two or more numbers in the form $\dfrac{p}{q}$ where $q \neq 0$. These numbers can be fractions, integers, mixed numbers, or decimals.
In this guide, you will learn how to multiply rational numbers, when to simplify or cross-cancel, and see clear examples for mixed numbers, negatives, decimals, and more. You will be surprised how simple multiplying rational numbers can be once you see it step by step.

Student Learning Outcomes
By the end of this guide on multiplication of rational numbers, you will be able to:

• Multiply any rational numbers (fractions, integers, mixed numbers, decimals).
• Use cross cancellation / butterfly method to simplify before multiplying.
• Apply sign rules for negatives and convert between mixed/decimal/fraction forms.
• Solve multi-factor multiplication problems and check your answers.

How to Multiply Rational Numbers

To multiply rational numbers, follow these rules:

1. Simplify the fraction to its lowest form if possible.
2. Multiply the numerators (top numbers).
3. Multiply the denominators (bottom numbers).
4. Reduce the result to its Standard form (lowest form).

General Form:
The general form for Multiplication of Rational Numbers is:
$
\text{Product} = \dfrac{\text{Product of Numerators}}{\text{Product of Denominators}}
$

Product of Two Rational Numbers Formula

If $\dfrac{a}{b} \text{and} \dfrac{c}{d}$ are two rational numbers, then Product of Two Rational Numbers Formula is:
$
\dfrac{a}{b} \times \dfrac{c}{d}=\dfrac{a \times c}{b \times d}
$
Example
Multiply $\dfrac{3}{4} \text{by} \dfrac{5}{7}$
Solution
$
\begin{aligned}
\dfrac{3}{4} \times \dfrac{5}{7}
&= \dfrac{3 \times 5}{4 \times 7} \\
&= \dfrac{15}{28}
\end{aligned}
$

Methods of Multiplying Rational Numbers

Rational numbers can be multiplied in different ways depending on the numbers involved.

1. Simple Multiplication (No Cancellation)

When there’s no common factor between any numerator and denominator, simply multiply the numerators and denominators.

• Example # 1
  $ \dfrac{3}{4} \times \dfrac{5}{6} = \dfrac{15}{24} $
• Example # 2
  $
  \dfrac{7}{4} \times \dfrac{7}{9} = \dfrac{49}{36}
  $

2. Simplifying Rational Numbers (Fractions) Before Multiplying

1. Look at each fraction (top and bottom) separately.
2. If they have a common factor, divide both by that number.
3. Then multiply
   General Form:
   $
   \dfrac{a}{b} \times \dfrac{c}{d}
   $

• If $a$ and $b$ have a common factor, simplify.
• If $c$ and $d$ have a common factor, simplify.
  Example # 1:
  $
  \begin{aligned}
  \dfrac{36}{40} \times \dfrac{3}{5}
  & = \dfrac{\cancel{36}^{9}}{\cancel{40}^{10}} \times \dfrac{3}{5} \\
  & = \dfrac{9}{10} \times \dfrac{3}{5} \\
  & = \dfrac{27}{50}
  \end{aligned}
  $
  Example # 2:
  $
  \begin{aligned}
  \dfrac{36}{4} \times \dfrac{7}{28}
  & = \dfrac{\cancel{36}^{9}}{\cancel{4}^{1}} \times \dfrac{\cancel{7}^{1}}{\cancel{28}^{4}} \\
  & = \dfrac{9}{1} \times \dfrac{1}{4} \\
  & = \dfrac{9}{4}
  \end{aligned}
  $

3. Cross Cancellation (Rational Numbers/ Fractions)

1. Look across the fractions (numerator of one and denominator of the other).
2. If they have a common factor, divide both by that number.
3. Then multiply
   General Form:
   $
   \dfrac{a}{b} \times \dfrac{c}{d}
   $

• If $a$ and $d$ have a common factor, simplify.
• If $b$ and $c$ have a common factor, simplify.
  Example:
  $
  \begin{aligned}
  \dfrac{3}{35} \times \dfrac{25}{7}
  &= \dfrac{3}{\cancel{35}^{7}} \times \dfrac{\cancel{25}^{5}}{7} \\
  &= \dfrac{3}{7} \times \dfrac{5}{7} \\
  &= \dfrac{15}{49}
  \end{aligned}
  $

Cross Cancellation is also called Butterfly method.

What is the Butterfly Method? 🦋

• The Butterfly Method is a simple, visual way to perform cross cancellation when multiplying fractions (rational numbers).
• You cancel common factors diagonally — the crossing lines look like the wings of a butterfly, that is why it is called Butterfly Method.
  $
  \dfrac{a}{b} \times \dfrac{c}{d}
  $
  Example:
  $
  \begin{aligned}
  \dfrac{6}{16} \times \dfrac{4}{9}
  &= \dfrac{\cancel{6}^{2}}{\cancel{16}^{4}} \times \dfrac{\cancel{4}^{1}}{\cancel{9}^{3}} \\
  &= \dfrac{2}{4} \times \dfrac{1}{3} \\
  &= \dfrac{\cancel{2}^1}{\cancel{4}^2} \times \dfrac{1}{3} \\
  &= \dfrac{1}{2} \times \dfrac{1}{3} \\
  &= \dfrac{1}{6}
  \end{aligned}
  $

4. Multiplying Negative Rational Numbers

1. To multiply negative rational numbers, follow integers sign rules:
   Negative × Negative = Positive
   Positive × Negative = Negative
2. Then multiply
   a) Both are Negative Rational Numbers
   Example:
   $ \dfrac{-7}{8} \times \dfrac{-3}{5} = \dfrac{21}{40} $
   b) One Negative, One Positive Rational Number
   Example:
   $
   \dfrac{-7}{8} \times \dfrac{3}{5} = \dfrac{-21}{40}
   $

5. Multiplying Rational and Whole (integers) Number

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

1. Multiply Whole number with Fraction
   Example:
   $
   \begin{aligned}
   8 \times \dfrac{5}{36}
   &= \dfrac{\cancel{8}^{2}}{1} \times \dfrac{5}{\cancel{36}^{9}} \\
   &= \dfrac{2}{1} \times \dfrac{5}{9} \\
   &= \dfrac{10}{9} \\
   &= 1\dfrac{1}{9}
   \end{aligned}
   $
2. Multiply Fraction with Whole Number
   Example:
   $
   \begin{aligned}
   \dfrac{5}{36} \times 5
   &= \dfrac{5}{36} \times \dfrac{5}{1} \\
   &= \dfrac{25}{36}
   \end{aligned}
   $

6. Multiplying Rational and Mixed Numbers

1. Convert the mixed number to an improper fraction first.
2. Simplify if possible.
3. Then multiply

• Multiply Mixed Number by Fraction
  Example:
  $
  \begin{aligned}
  1\dfrac{1}{2} \times \dfrac{4}{5}
  &= \dfrac{3}{2} \times \dfrac{4}{5} \\
  &= \dfrac{12}{10} \\
  &= \dfrac{6}{5} \\
  &= 1\dfrac{1}{5}
  \end{aligned}
  $
• Multiply Fraction by Mixed Number
  Example:
  $
  \begin{aligned}
  \dfrac{1}{5} \times 3\dfrac{1}{5}
  &= \dfrac{1}{5} \times \dfrac{16}{5} \\
  &= \dfrac{16}{25}
  \end{aligned}
  $

7. Multiplying Rational Number by Zero

• Multiplication of rational numbers with zero is zero.
  Examples:
  $ \dfrac{7}{9} \times 0 = 0 $
  $ 0 \times \dfrac{99999}{45574} = 0 $

8. Multiplying Two Reciprocals

• The product of a rational number and its reciprocal always gives 1 (except when numerator is 0).
  General Form:
  $
  \dfrac{a}{b} \times \dfrac{b}{a} = 1
  $
  Example:
  $
  \dfrac{5}{6} \times \dfrac{6}{5} = 1
  \quad \text{(reciprocal of } \dfrac{5}{6} \text{ is } \dfrac{6}{5})
  $
  $
  \Rightarrow \dfrac{\cancel{5}}{\cancel{6}} \times \dfrac{\cancel{6}}{\cancel{5}} = 1
  $

9. Multiplying Rational Number and Decimal

1. Change the decimal into a fraction.
2. Simplify if possible.
3. Then multiply.
   Example:
   $
   \begin{aligned}
   \dfrac{3}{5} \times 0.3
   &= \dfrac{3}{5} \times \dfrac{3}{10} \\
   &= \dfrac{9}{50}
   \end{aligned}
   $

10. Multiplication of Same Rational Numbers

The multiplication of rational numbers with same numerators and same denominators means multiplying a rational number by itself. This is called finding the square of that rational number.

• Formula:
  If $\dfrac{a}{b}$ is a rational number, then
  $
  \dfrac{a}{b} \times \dfrac{a}{b} = \dfrac{a \times a}{b \times b} = \dfrac{a^2}{b^2}
  $
  where $b \neq 0$.
  Example:
  Multiply $\dfrac{5}{8}$ by $\dfrac{5}{8}$:
  $
  \begin{aligned}
  \dfrac{5}{8} \times \dfrac{5}{8} &= \dfrac{5 \times 5}{8 \times 8} \\
  &= \dfrac{25}{64}
  \end{aligned}
  $

11. Multiplication of More Than Two Rational Numbers

1. When multiplying more than two rational numbers, multiply all the numerators together and all the denominators together, then simplify the result.
2. You can also simplify before multiplying to make the calculation easier.

• Straight multiplication
  Example
  $
  \begin{aligned}
  \dfrac{2}{3} \times \dfrac{5}{4} \times \dfrac{7}{2}
  &= \dfrac{2 \times 5 \times 7}{3 \times 4 \times 2} \\
  &= \dfrac{70}{24} \\
  &= \dfrac{\cancel{70}^{35}}{\cancel{24}^{12}} \\
  &= \dfrac{35}{12} \\
  &=2 \dfrac{11}{12}
  \end{aligned}
  $
• Simplifying before multiplying
  Example
  $
  \begin{aligned}
  \dfrac{2}{15} \times \dfrac{5}{4} \times \dfrac{7}{2}
  &= \dfrac{\cancel{2}^{1}}{\cancel{15}^3} \times \dfrac{\cancel{5}^1}{4} \times \dfrac{7}{\cancel{2}^{1}} \\
  &= \dfrac{1}{3} \times \dfrac{1}{4} \times \dfrac{7}{1} \\
  &= \dfrac{1 \times 7}{3 \times 4} \\
  &= \dfrac{7}{12}
  \end{aligned}
  $

Quick Tips for Multiplication of Rational Numbers

• Always simplify before and after multiplication.
• Multiply: numerator × numerator and denominator × denominator.
• Follow sign rules for negative numbers.
  • Negative × Negative = Positive
  • Positive × Negative = Negative
• Convert mixed numbers and decimals to fractions first.
• A whole number is written as $\dfrac{a}{1}$ for multiplication.
• Use cross cancellation to make calculations easier.
• Use brackets for negative numbers to avoid sign mistakes.

Practice Questions

Practice Question # 1
Simplify $5 \times -\dfrac{12}{25} \times 2\dfrac{1}{2} \times \left(-\dfrac{10}{3}\right)$
Solution:

1. Convert mixed numbers to fractions:
   $2\dfrac{1}{2} = \dfrac{5}{2}$
2. Write all terms as fractions:
   $\dfrac{5}{1} \times \dfrac{-12}{25} \times \dfrac{5}{2} \times \dfrac{-10}{3}$
3. Cancel common factors step by step:
   Cancel the first 5 (numerator) with 25 (denominator):
   $\dfrac{1}{1} \times \dfrac{-12}{5} \times \dfrac{5}{2} \times \dfrac{-10}{3}$
   Cancel the 5 in the denominator with the 5 in the numerator (from the third fraction):
   $\dfrac{1}{1} \times \dfrac{-12}{1} \times \dfrac{1}{2} \times \dfrac{-10}{3}$
   Cancel $-12 \ \& \ 3$ and also $2 \ \& -10$:
   $\dfrac{1}{1} \times \dfrac{-4}{1} \times \dfrac{1}{1} \times \dfrac{-5}{1}$
4. Multiply numerators and denominators:
   $= \dfrac{1 \times -4 \times 1 \times -5}{1 \times 1 \times 1 \times 1}$
   Negative × Negative = Positive
   $= \dfrac{20}{1}$
   $= 20$

Practice Question # 2
Simplify $-3\dfrac{1}{4} \times \dfrac{-8}{5} \times \dfrac{15}{16}$
Solution:

1. Convert the mixed number to an improper fraction:
   $-3\dfrac{1}{4} = \dfrac{-13}{4}$
2. Write all numbers as fractions:
   $\dfrac{-13}{4} \times \dfrac{-8}{5} \times \dfrac{15}{16}$
3. Cancel common factors step by step:
   Cancel (8) in numerator with (16) in denominator.
   Cancel (5) in denominator with (15) in numerator.
   $\dfrac{-13}{4} \times \dfrac{-\cancel{8}^{-1}}{\cancel{5}^{1}} \times \dfrac{\cancel{15}^{3}}{\cancel{16}^{2}}$
4. Solve the remaining terms:
   $\dfrac{-13}{4} \times \dfrac{-1}{1} \times \dfrac{3}{2}$
5. Multiply numerators and denominators:
   $\dfrac{-13 \times -1 \times 3}{4 \times 1 \times 2}$
   Negative × Negative = Positive
   $= \dfrac{39}{8}$
6. Convert to a mixed number:
   $= 4\dfrac{7}{8}$

Practice Question # 3
Simplify $-2 \times \dfrac{3}{5} \times 4\dfrac{1}{2} \times \left(-\dfrac{10}{9}\right)$
Solution:

1. Convert the mixed number to an improper fraction:
   $4\dfrac{1}{2} = \dfrac{9}{2}$
2. Write all numbers as fractions:
   $\dfrac{-2}{1} \times \dfrac{3}{5} \times \dfrac{9}{2} \times \dfrac{-10}{9}$
3. Cancel common factors step by step:
   Cancel (2) in numerator with (2) in denominator.
   Cancel (9) in numerator with (9) in denominator.
   Cancel (5) in denominator with (10) in numerator.
   $\dfrac{\cancel{-2}^{-1}}{1} \times
   \dfrac{3}{\cancel{5}^{1}} \times
   \dfrac{\cancel{9}^{1}}{\cancel{2}^{1}} \times
   \dfrac{\cancel{-10}^{-2}}{\cancel{9}^{1}}$
4. Solve the remaining terms:
   $\dfrac{-1}{1} \times \dfrac{3}{1} \times \dfrac{1}{1} \times \dfrac{-2}{1}$
5. Multiply numerators and denominators:
   $\dfrac{-1 \times 3 \times 1 \times -2}{1 \times 1 \times 1 \times 1}$
   Negative × Negative = Positive
   $= \dfrac{6}{1}$
   $= 6$

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