# Multiplying rational numbers

In this post we will learn methods to multiply two or more rational numbers with solved examples.

The process is similar to multiplication of fraction number which we have studied in Grade 5 Math.

## How to multiply rational numbers ?

We know that rational number can occur in both fraction and decimal form.

So there are three possible cases for multiplication of rational numbers;

(a) Fraction x Fraction

(b) Fraction x Decimal

(c) Decimal x Decimal

We will learn all the three cases in detail.

### Fraction & Fraction Multiplication

In this case both the rational number are present in fraction form.

Here the method of multiplication is similar to fraction multiplication.

For the product of rational numbers follow the below step;

(a) Multiply the numerator and denominator separately.

(b) If possible, reduce the fraction to its lowest terms.

I hope you understood the process. Let us see some examples for further understanding.

Example 01
Multiply the below rational numbers.

\mathtt{\frac{2}{3} \times \frac{5}{7}}

Solution
Note that both the rational numbers are in fraction form.

Multiply the numerator and denominator separately and find the solution.

\mathtt{\Longrightarrow \frac{2}{3} \times \frac{5}{7}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2\ \times 5}{3\times 7}}\\\ \\ \mathtt{\Longrightarrow \ \frac{10}{21}}

Hence, 10 / 21 is the solution.

Example 02
Multiply the rational numbers.

\mathtt{\frac{8}{11} \times \frac{22}{9}}

Solution
Both the rational numbers are present in fraction form. So we will multiply the numerator and denominator separately.

\mathtt{\Longrightarrow \frac{8}{11} \times \frac{22}{9}}\\\ \\ \mathtt{\Longrightarrow \ \frac{8\ \times 22}{11\times 9}}

Note that denominator 11 & numerator 22 are divisible to each other. So we can simplify the expression as;

\mathtt{\Longrightarrow \ \frac{8\ \times \ \cancel{22} \ \mathbf{2} \ }{\cancel{11} \ \times \ 9}}\\\ \\ \mathtt{\Longrightarrow \ \frac{16}{9}}

Hence, 16 / 9 is the solution.

Example 03
Multiply the rational numbers.

\mathtt{\frac{12}{5} \times \frac{1}{3} \times \frac{15}{18}}

Solution
Multiply the numerators and denominator separately.

\mathtt{\Longrightarrow \frac{12}{5} \times \frac{1}{3} \times \frac{15}{18}}\\\ \\ \mathtt{\Longrightarrow \ \frac{12\ \times 1\times 15\ }{5\ \times \ 3\times 18}}

The above multiplication can be simplified by;

⟹ dividing numerator 12 by denominator 3

⟹ dividing numerator 15 by denominator 5

\mathtt{\Longrightarrow \ \frac{\cancel{12} \ \mathbf{4} \ \times 1\times \cancel{15} \ \mathbf{3} \ }{\cancel{5\ } \times \ \cancel{3} \times 18}}\\\ \\ \mathtt{\Longrightarrow \ \frac{12}{18}}

The fraction can be further simplified by dividing numerator and denominator by 6.

\mathtt{\Longrightarrow \ \frac{12\div 6}{18\div 6}}\\\ \\ \mathtt{\Longrightarrow \frac{2}{3}}

Hence, 2/3 is the solution of given fraction.

### Fraction and Decimal multiplication

When the rational numbers are given in the form of fraction and decimal then you can do multiplication by following below steps;

(a) Convert decimal into fraction

We will try to make all numbers in fraction format so that multiplication gets easier.

(b) Multiply numerator and denominator of fractions separately

(c) If possible, reduce the fraction to its lowest terms

I hope you understood the above steps. Let us solve some problems for better clarity.

Example 01
Multiply the below numbers

\mathtt{2.7\ \times \ \frac{2}{5}}

Solution
Note that the rational numbers are present both in decimals and fraction form.

To multiply the numbers, first convert decimal into fraction.

\mathtt{\Longrightarrow 2.7\ \times \ \frac{2}{5}}\\\ \\ \mathtt{\Longrightarrow \frac{27}{10} \times \frac{2}{5}}

Now all numbers are in form of fraction. Multiply the numerators and denominators separately.

\mathtt{\Longrightarrow \frac{27}{10} \times \frac{2}{5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{27\times 2}{10\times 5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{54}{50}}

Hence, 54/50 is the solution.

The solution can be further simplified by dividing numerator and denominator by 2.

\mathtt{\Longrightarrow \frac{54}{50}}\\\ \\ \mathtt{\Longrightarrow \ \frac{54\div 2}{50\div 2}}\\\ \\ \mathtt{\Longrightarrow \frac{27}{25}}

Hence, 27 / 25 is the final solution.

Example 02
Solve the below multiplication.

\mathtt{\frac{6}{13} \times \frac{7}{2} \ \times 0.15}

Solution
First convert the decimal into fraction.

\mathtt{\Longrightarrow \ \frac{6}{13} \times \frac{7}{2} \ \times 0.15}\\\ \\ \mathtt{\Longrightarrow \frac{6}{13} \times \frac{7}{2} \times \frac{15}{100}}

Now multiply the numerator and denominator separately.

\mathtt{\Longrightarrow \ \frac{6\times 7\times 15}{13\times 2\times 100}}

The above multiplication can be simplified by;

(a) Dividing 6 on numerator with denominator 2.

(b) Dividing numerator 15 & denominator 100 by number 3.

\mathtt{\Longrightarrow \frac{2\times 7\times 3}{13\times 20}}

Now divide numerator 2 by denominator 20.

\mathtt{\Longrightarrow \frac{\cancel{2} \times 7\times 3}{13\times \cancel{20} \ \mathbf{10}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{21}{130}}

Hence, 21/130 is the final solution.

Example 03
Multiply the below rational numbers

\mathtt{0.2\times 0.01\ \times \frac{9}{16}}

Solution
First convert the decimals into fraction.

\mathtt{\Longrightarrow \ 0.2\times 0.01\ \times \frac{9}{16}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2}{10} \times \frac{1}{100} \times \frac{9}{16}}

Now multiply the numerator and denominator separately.

\mathtt{\Longrightarrow \frac{2\times 9}{1000\times 16}}

For simplification, divide numerator 2 by denominator 16.

\mathtt{\Longrightarrow \frac{\cancel{2} \times 9}{1000\times \cancel{16} \ \mathbf{8}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{9}{8000}}

Hence, 9/8000 is the solution.

### Decimal and decimal multiplication

In this case, all the given rational numbers are in the form of decimals.

To multiply the numbers, first convert all decimals into fraction and then do the multiplication as explained above.

Let us solve some examples for further understanding.

Example 01
Multiply the below rational numbers.

\mathtt{\ 0.25\times 0.13}

Solution
Convert decimals into fraction.

\mathtt{\Longrightarrow \ \frac{25}{100} \times \frac{13}{100}}

Multiply numerator and denominator separately.

\mathtt{\Longrightarrow \frac{25\times 13}{100\times 100}}

Divide numerator 25 by denominator 100.

\mathtt{\Longrightarrow \frac{\cancel{25} \times 13}{\cancel{100\ }\mathbf{4} \times 100}}\\\ \\ \mathtt{\Longrightarrow \ \frac{13}{400}}\

Hence, 13/400 is the solution.

Example 02
Multiply the rational numbers.

\mathtt{\ 0.1\times 0.45\ }

Solution
Convert the decimals into fraction.

\mathtt{\Longrightarrow \ 0.1\times 0.45\ }\\\ \\ \mathtt{\Longrightarrow \ \frac{1}{10} \times \frac{45}{100}}

Divide numerator 45 and denominator 100 by 5.

\mathtt{\Longrightarrow \frac{\cancel{45} \ \mathbf{9}}{10\times \cancel{100} \ \mathbf{20}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{9}{200}}

Hence, 9/200 is the solution.

Alternate Method

\mathtt{\Longrightarrow \ 0.1\times 0.45\ }\\\ \\ \mathtt{\Longrightarrow \ \frac{1}{10} \times \frac{45}{100}}\\\ \\ \mathtt{\Longrightarrow \frac{45}{1000}}

Converting back to decimal we get;

\mathtt{\Longrightarrow \ 0.045}

Hence, 0.045 is the solution