CBSE Class 7 Mathematics Chapter 9: Rational Numbers NCERT Solutions

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NCERT Solutions PDF Class 7 PDF

This chapter focuses on introducing students to the concept of Rational Numbers in Mathematics for Class 7. The NCERT Solutions provide a clear understanding of what rational numbers are, how to represent them, and importantly, how to find a specified number of rational numbers between any two given rational numbers. The solutions cover various scenarios, including finding rational numbers between integers and between fractions. This detailed explanation helps students grasp the methods for comparing and ordering rational numbers, which is crucial for further mathematical studies. These solutions are designed to aid students in mastering the exercises and preparing effectively for their examinations by offering step-by-step guidance and clear reasoning.

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BoardCBSE
ClassClass 7
SubjectMathematics
Session2026
LanguageEnglish
TypeNCERT Solutions
ChapterChapter 9: Rational Numbers

Chapter summary

Chapter 9 of the Class 7 Mathematics NCERT Solutions introduces Rational Numbers. This section focuses on understanding the definition of rational numbers and practicing how to find a set of rational numbers that lie between two given rational numbers. The exercises involve finding five rational numbers between pairs of integers and pairs of fractions, requiring students to find common denominators and list numbers in sequence.

Learning outcomes

  • Understand the definition and properties of rational numbers.
  • Learn to represent rational numbers on a number line.
  • Develop the skill to find a specified number of rational numbers between two given rational numbers.
  • Practice converting fractions to equivalent fractions with common denominators.
  • Apply comparison and ordering techniques for rational numbers.

Topics covered

Paper topics

  • Definition of Rational Numbers
  • Representation of Rational Numbers
  • Finding Rational Numbers Between Two Integers
  • Finding Rational Numbers Between Two Fractions
  • Equivalent Fractions
  • Common Denominators
  • Ordering of Rational Numbers

Important topics

  • Finding rational numbers between two given numbers
  • Expressing numbers with a common denominator
  • Comparing and ordering rational numbers

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Questions and Solutions

Question 1 (ii)

List five rational numbers between -2 and -1.
Solution:

To find five rational numbers between -2 and -1, we first express these integers as fractions with a common denominator. Let's choose a denominator that allows us to find at least five numbers between them. We can use a denominator like 6.

We convert -2 and -1 into equivalent fractions with a denominator of 6:

-2 = \frac{-2 \times 6}{6} = \frac{-12}{6}

-1 = \frac{-1 \times 6}{6} = \frac{-6}{6}

Now, we need to find five rational numbers between \frac{-12}{6} and \frac{-6}{6}. We can list the numerators between -12 and -6:

-12 < -11 < -10 < -9 < -8 < -7 < -6

So, the rational numbers are:

\frac{-12}{6} < \frac{-11}{6} < \frac{-10}{6} < \frac{-9}{6} < \frac{-8}{6} < \frac{-7}{6} < \frac{-6}{6}

Simplifying the fractions where possible, we get:

-2 < \frac{-11}{6} < \frac{-5}{3} < \frac{-3}{2} < \frac{-4}{3} < \frac{-7}{6} < -1

Therefore, five rational numbers between -2 and -1 are \frac{-11}{6}, \frac{-5}{3}, \frac{-3}{2}, \frac{-4}{3}, \frac{-7}{6}.

Question 1 (i)

List five rational numbers between -1 and 0.
Solution:

To find five rational numbers between -1 and 0, we first express these integers as fractions with a common denominator. Let's choose a denominator that allows us to find at least five numbers between them. We can use a denominator like 6.

We convert -1 and 0 into equivalent fractions with a denominator of 6:

-1 = \frac{-1 \times 6}{6} = \frac{-6}{6}

0 = \frac{0 \times 6}{6} = \frac{0}{6}

Now, we need to find five rational numbers between \frac{-6}{6} and \frac{0}{6}. We can list the numerators between -6 and 0:

-6 < -5 < -4 < -3 < -2 < -1 < 0

So, the rational numbers are:

\frac{-6}{6} < \frac{-5}{6} < \frac{-4}{6} < \frac{-3}{6} < \frac{-2}{6} < \frac{-1}{6} < \frac{0}{6}

Simplifying the fractions where possible, we get:

-1 < \frac{-5}{6} < \frac{-2}{3} < \frac{-1}{2} < \frac{-1}{3} < \frac{-1}{6} < 0

Therefore, five rational numbers between -1 and 0 are \frac{-5}{6}, \frac{-2}{3}, \frac{-1}{2}, \frac{-1}{3}, \frac{-1}{6}.

Question 1 (iii)

List five rational numbers between \frac{-4}{5} and \frac{-2}{3}.
Solution:

To find five rational numbers between \frac{-4}{5} and \frac{-2}{3}, we first need to express these fractions with a common denominator. The least common multiple (LCM) of 5 and 3 is 15.

Convert the fractions to equivalent fractions with a denominator of 15:

\frac{-4}{5} = \frac{-4 \times 3}{5 \times 3} = \frac{-12}{15}

\frac{-2}{3} = \frac{-2 \times 5}{3 \times 5} = \frac{-10}{15}

We need to find five rational numbers between \frac{-12}{15} and \frac{-10}{15}. Since there are no integers between -12 and -10, we need to use a larger common denominator. Let's use 45 (which is 15 * 3).

Convert the fractions to equivalent fractions with a denominator of 45:

\frac{-12}{15} = \frac{-12 \times 3}{15 \times 3} = \frac{-36}{45}

\frac{-10}{15} = \frac{-10 \times 3}{15 \times 3} = \frac{-30}{45}

Now we need to find five rational numbers between \frac{-36}{45} and \frac{-30}{45}. We can list the numerators between -36 and -30:

-36 < -35 < -34 < -33 < -32 < -31 < -30

So, the rational numbers are:

\frac{-36}{45} < \frac{-35}{45} < \frac{-34}{45} < \frac{-33}{45} < \frac{-32}{45} < \frac{-31}{45} < \frac{-30}{45}

Simplifying the fractions where possible:

\frac{-35}{45} = \frac{-7}{9}

\frac{-33}{45} = \frac{-11}{15}

Therefore, five rational numbers between \frac{-4}{5} and \frac{-2}{3} are \frac{-7}{9}, \frac{-34}{45}, \frac{-11}{15}, \frac{-32}{45}, \frac{-31}{45}.

Question 1 (iv)

List five rational numbers between \frac{-1}{2} and \frac{2}{3}.
Solution:

To find five rational numbers between \frac{-1}{2} and \frac{2}{3}, we first need to express these fractions with a common denominator. The least common multiple (LCM) of 2 and 3 is 6.

Convert the fractions to equivalent fractions with a denominator of 6:

\frac{-1}{2} = \frac{-1 \times 3}{2 \times 3} = \frac{-3}{6}

\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}

Now we need to find five rational numbers between \frac{-3}{6} and \frac{4}{6}. We can list the numerators between -3 and 4:

-3 < -2 < -1 < 0 < 1 < 2 < 3 < 4

So, the rational numbers are:

\frac{-3}{6} < \frac{-2}{6} < \frac{-1}{6} < \frac{0}{6} < \frac{1}{6} < \frac{2}{6} < \frac{3}{6} < \frac{4}{6}

Simplifying the fractions where possible:

\frac{-2}{6} = \frac{-1}{3}

\frac{0}{6} = 0

\frac{2}{6} = \frac{1}{3}

\frac{3}{6} = \frac{1}{2}

Therefore, five rational numbers between \frac{-1}{2} and \frac{2}{3} are \frac{-1}{3}, \frac{-1}{6}, 0, \frac{1}{6}, \frac{1}{3}.

Common mistakes

  • Errors in finding a common denominator for fractions.
  • Incorrectly listing numbers between the calculated equivalent fractions.
  • Misinterpreting negative signs when comparing rational numbers.
  • Forgetting to simplify the final list of rational numbers if required.

Revision tips

  • Focus on the method of finding a common denominator before listing numbers.
  • Practice converting integers into fractions with a suitable denominator.
  • Double-check the signs of the rational numbers when ordering them.
  • Review the examples to understand how to find more than five rational numbers if needed.

Practice MCQs

Q1. How many rational numbers can be found between any two given rational numbers?

Q2. To find rational numbers between two given rational numbers, what is the first step?

Q3. Which of the following is a rational number between -1 and 0?

Q4. What is the equivalent fraction of -4/5 with a denominator of 45?

Q5. Which pair of numbers has more rational numbers between them?

Frequently asked questions

What are rational numbers?

Rational numbers are numbers that can be expressed as a fraction p/q, where p and q are integers and q is not equal to zero. Examples include 1/2, -3/4, 5, and 0.

How do I find rational numbers between -1 and 0?

To find rational numbers between -1 and 0, you can express them with a common denominator, for example, -6/6 and 0/6. Then, you can list the integers between -6 and 0 in the numerator, such as -5/6, -4/6, -3/6, -2/6, -1/6.

What is the method to find rational numbers between two fractions like -4/5 and -2/3?

First, find a common denominator for -4/5 and -2/3. The least common multiple of 5 and 3 is 15. Convert the fractions: -4/5 = -12/15 and -2/3 = -10/15. To find more numbers, you can use a larger common denominator, like 45: -4/5 = -36/45 and -2/3 = -30/45. Then list the numbers between them, such as -35/45, -34/45, -33/45, -32/45, -31/45.

Are the solutions provided for Class 7 Maths Chapter 9 accurate?

Yes, these solutions are based on the standard NCERT curriculum for Class 7 Mathematics and follow the correct mathematical procedures to solve the problems.

How can these NCERT solutions help in exam preparation?

These solutions offer clear, step-by-step explanations for each problem, helping students understand the methods and reasoning. Practicing with these solutions can build confidence and improve problem-solving skills for exams.

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