Rational Numbers - Class 7 Mathematics (CBSE)
Based on the 2025-26 NCERT syllabus for Class 7 Mathematics. This chapter extends the number system to rational numbers, covering operations, comparison, and representation on the number line.
1. Why this chapter matters
Rational numbers bridge the gap between integers and real numbers. Understanding rational numbers is essential for algebra, coordinate geometry, and calculus in higher classes. In CBSE exams, this chapter contributes 6-8 marks and is a direct precursor to Class 8 Rational Numbers.
2. Why do we need rational numbers?
Integers cannot represent quantities between two integers. For example, the length of a pencil may be 5.5 cm, which is between 5 and 6. Such quantities require rational numbers.
A rational number is any number that can be expressed as p/q where p and q are integers and q is not zero.
3. Positive and negative rational numbers
Positive rational numbers
Both numerator and denominator have the same sign. Examples: 3/4, (-5)/(-7) = 5/7.
Negative rational numbers
One of numerator or denominator (but not both) is negative. Examples: -3/5, 7/(-6) = -7/6.
Zero as a rational number
0 = 0/q for any non-zero q. Zero is neither positive nor negative.
4. Standard form of a rational number
A rational number is in standard form when:
- The denominator is positive.
- The numerator and denominator have no common factor other than 1.
- The only common factor between numerator and denominator is 1.
Example: 8/12 in standard form = 2/3 (divide numerator and denominator by 4).
5. Comparison of rational numbers
Rules for comparison
- Positive rational numbers are greater than negative rational numbers.
- For positive rationals with same denominator, compare numerators.
- For positive rationals with different denominators, convert to like denominators (LCM) then compare.
- For negative rationals, the number with the larger absolute value is smaller.
On the number line
Numbers increase from left to right. A number to the right is greater than a number to the left.
6. Operations on rational numbers
Addition
Same denominator: Add numerators, keep denominator. 1/5 + 2/5 = 3/5.
Different denominators: Find LCM, convert, then add. 1/4 + 1/6 = 3/12 + 2/12 = 5/12.
Subtraction
Same as addition but subtract numerators.
Multiplication
Multiply numerators together and denominators together. (-2/3) x (5/7) = -10/21.
Division
Multiply by the reciprocal. (5/6) / (-2/3) = (5/6) x (-3/2) = -15/12 = -5/4.
Properties
- Rational numbers are closed under addition, subtraction, and multiplication.
- Addition and multiplication are commutative and associative.
- 0 is the additive identity; 1 is the multiplicative identity.
- Every rational number has an additive inverse (-p/q) for (p/q).
7. Rational numbers between two rational numbers
There are infinitely many rational numbers between any two rational numbers.
Method to find rational numbers between two rationals
Method 1: Find the mean (average). The mean of two rationals always lies between them. Method 2: If a/b and c/d are two rationals, find (a+c)/(b+d) after making denominators equal. Method 3: Write equivalent fractions with larger denominators and count between them.
Example: Find three rational numbers between 1/3 and 2/3.
1/3 = 4/12, 2/3 = 8/12. Rational numbers: 5/12, 6/12 = 1/2, 7/12.
8. Representing rational numbers on the number line
To represent p/q on a number line:
- Divide the unit length into q equal parts.
- For positive p, count p parts to the right of 0.
- For negative p, count p parts to the left of 0.
9. Worked examples
Example 1: Express -12/18 in standard form.
Divide numerator and denominator by 6 (HCF of 12 and 18). -12/18 = -2/3.
Example 2: Compare -3/7 and -5/7.
Denominators are same. Compare numerators: -3 > -5. So -3/7 > -5/7.
Example 3: Add 3/8 + (-5/12).
LCM of 8 and 12 = 24. 3/8 = 9/24, -5/12 = -10/24. Sum = 9/24 + (-10/24) = -1/24.
Example 4: Multiply (-6/11) x (22/18).
(-6/11) x (22/18) = (-6 x 22)/(11 x 18) = -132/198 = -2/3.
Example 5: Find one rational number between 1/4 and 1/3.
Mean = (1/4 + 1/3)/2 = (3/12 + 4/12)/2 = (7/12)/2 = 7/24.
10. Common mistakes and how to fix them
| Mistake | Fix |
|---|---|
| Thinking -3/5 is greater than -2/5 | On a number line, -3 is left of -2, so -3/5 < -2/5 |
| Forgetting to convert to like denominators | Always use LCM to compare or add rationals |
| Not simplifying to standard form | Always reduce to lowest terms |
| Treating negative sign only on numerator | p/(-q) = -p/q = -(p/q). Standard form has positive denominator |
| Thinking only fractions are rational numbers | Integers, decimals (terminating/repeating) are also rational |
11. CBSE exam focus
| Question type | Marks | Frequency |
|---|---|---|
| Express in standard form | 2 marks | 1 question |
| Compare rational numbers | 2 marks | 1 question |
| Addition/subtraction of rationals | 2-3 marks | 1 question |
| Multiplication/division of rationals | 2-3 marks | 1 question |
| Find rational numbers between two given | 3 marks | 1 question |
12. Self-test
- Express 24/36 in standard form.
- Compare -7/15 and -9/20.
- Add: 5/12 + (-7/18).
- Multiply: (-9/14) x (28/27).
- Find three rational numbers between 1/5 and 2/5.
- Write four negative rational numbers greater than -1.
13. Answer key
- 24/36 = 2/3.
- LCM of 15 and 20 = 60. -7/15 = -28/60, -9/20 = -27/60. -28/60 < -27/60, so -7/15 < -9/20.
- LCM of 12 and 18 = 36. 5/12 = 15/36, -7/18 = -14/36. Sum = 1/36.
- (-9/14) x (28/27) = (-9 x 28)/(14 x 27) = -252/378 = -2/3.
- 1/5 = 4/20, 2/5 = 8/20. Between: 5/20 = 1/4, 6/20 = 3/10, 7/20.
- -3/4, -1/2, -1/4, -1/5 (any four between -1 and 0).
14. Quick revision
- Rational numbers = p/q where q is not zero.
- Standard form: denominator positive, numerator and denominator coprime.
- Positive > 0 > negative.
- For negative rationals with same denominator: compare numerators.
- Use LCM to compare or operate on different denominators.
- There are infinite rational numbers between any two rational numbers.
- Reciprocal is the multiplicative inverse (except 0).
