Maharashtra Board Class 7 Maths part 1 Chapter 5 Operations on Rational Numbers PDF Download

Operations on Rational Numbers

Rational Numbers

In previous standards, we have learnt that the counting numbers 1, 2, 3, 4, ..... are called natural numbers. We know that natural numbers, zero, and the opposite numbers of natural numbers together form the group of integers. We are also familiar with fractions like \(\frac{7}{11}\), \(\frac{2}{5}\), \(\frac{1}{7}\).

Is there then, a group that includes both integers and fractions? Let us see.

\(4 = \frac{12}{3}\); \(7 = \frac{7}{1}\); \(-3 = \frac{-3}{1}\); \(0 = \frac{0}{2}\)

Thus, we also know that all integers can be written in the form \(\frac{m}{n}\). If \(m\) is any integer and \(n\) is any non-zero integer, then the number \(\frac{m}{n}\) is called a rational number.

This group of rational numbers includes all types of numbers mentioned before.

Complete the table given below.

Teacher's Note

A rational number is any number that can be written as a fraction. Like how we write -3 as \(\frac{-3}{1}\), we can write any integer as a fraction.

Exam Trick

Remember: Any integer can be written as a rational number by putting 1 in the denominator. For example, 5 = \(\frac{5}{1}\).

Points to Remember

Natural numbers are 1, 2, 3, 4, ..... and so on.
Integers include natural numbers, zero, and negative numbers.
A rational number is \(\frac{m}{n}\) where m and n are integers and n is not zero.
All integers are rational numbers.
All fractions are rational numbers.

Operations on Rational Numbers

Rational numbers are written like fractions using a numerator and a denominator. That is why, operations on rational numbers are carried out as on fractions.

(1) \(\frac{5}{7} + \frac{9}{11} = \frac{55 + 63}{77} = \frac{118}{77}\)

(2) \(\frac{1}{7} - \frac{3}{4} = \frac{4 - 21}{28} = \frac{-17}{28}\)

(3) \(2\frac{1}{7} + 3\frac{8}{14} = \frac{15}{7} + \frac{50}{14} = \frac{30}{14} + \frac{50}{14} = \frac{80}{14} = \frac{40}{7}\)

(4) \(\frac{9}{13} \times \frac{4}{7} = \frac{9 \times 4}{13 \times 7} = \frac{36}{91}\)

(5) \(\frac{3}{5} \times \frac{(-4)}{5} = \frac{3 \times (-4)}{5 \times 5} = \frac{-12}{25}\)

(6) \(\frac{9}{13} \times \frac{26}{3} = \frac{3 \times 2}{1} = \frac{6}{1}\)

Teacher's Note

When we add fractions, we need to make the denominators the same. Just like we count money in rupees and paise, we need to count fractions in the same parts.

Exam Trick

For addition and subtraction, make denominators same. For multiplication, multiply numerators together and denominators together. For division, flip the second fraction and multiply.

Points to Remember

To add or subtract rational numbers, make the denominators equal first.
To multiply rational numbers, multiply the numerators and denominators.
To divide one rational number by another, multiply by the opposite (reciprocal).
The opposite of \(\frac{a}{b}\) is \(\frac{b}{a}\).
We cannot divide by zero ever.

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