Eight Standard >> Rational numbers | Multiplication of rational numbers and division of rational numbers

Multiplication and division of rational numbers

Multiplication of Rational Numbers:

When you multiply two rational numbers, you're essentially finding the product of the fractions. To do this, you multiply their numerators together and multiply their denominators together.

For example:
\(\frac{a}{b}*\frac{c}{d}\)=\(frac{a*c}{b*d}\)

For example, let's consider two fractions: \(\frac{2}{3}\) and \(\frac{5}{7}\).

(\(\frac{3}{4}\)) * (\(\frac{5}{6}\)) = \(\frac{\frac{3}{5}}{\frac{4}{6}}\) = \(\frac{15}{12}\)

As you can see, we multiplied the numerators (2 and 5) to get 10, and we multiplied the denominators (3 and 7) to get 21. The result, \(\frac{10}{21}\) is a new rational number.

It's important to simplify the result if possible. In this case, \(\frac{10}{21}\) cannot be simplified further because there is no common factor other than 1 for the numerator and denominator.

Division of Rational Numbers:

When dividing rational numbers, you're essentially finding the quotient of one fraction divided by another fraction. To do this, remember the rule: "Dividing by a fraction is the same as multiplying by its reciprocal." The reciprocal of a fraction ( \(\frac{a}{b}\)) is simply swapping the numerator and denominator ( \(\frac{b}{a}\)).

So, the division of two rational numbers ( \(\frac{a}{b}\)) and ( \(\frac{c}{d}\)) can be expressed as:

( \(\frac{a}{b}\)) ÷ ( \(\frac{c}{d}\)) = ( \(\frac{a}{b}\)) * ( \(\frac{d}{c}\))

For example, let's consider the following division: (\(\frac{4}{5}\)) ÷ (\(\frac{2}{3}\)).

(\(\frac{4}{5}\)) ÷ (\(\frac{2}{3}\)) = (\(\frac{4}{5}\)) * (\(\frac{3}{2}\))

Now, perform the multiplication:

 \(\frac{(4 * 3) }{(5 * 2) }\) = \(\frac{12}{10}\)

To simplify the result, find the greatest common divisor (GCD) of the numerator and denominator, which is 2 in this case:

12 ÷ 2 = 6
10 ÷ 2 = 5

So, \(\frac{12}{10}\) simplifies to \(\frac{6}{5}\).

Always try to simplify the result after division to its simplest form if possible.

Rational numbers, which include both fractions and whole numbers, are an essential part of mathematics and are used in various everyday situations, such as measurements, calculations, and comparisons. Understanding how to perform operations like multiplication and division with rational numbers can be valuable in many practical scenarios.