📚 Class 8 – CBSE Mathematics
Topic: Rational Numbers
Goal: Understand what a rational number is, how to spot it, and how to add, subtract, multiply, and divide them.
1. What is a Rational Number?
- A rational number is any number that can be written as a fraction
[ \frac{p}{q} ] where p and q are integers (whole numbers) and q ≠ 0. - In other words: a number that can be expressed as the ratio of two whole numbers.
Everyday Analogy
Think of a pizza that is cut into 8 equal slices.
- If you eat 3 slices, the amount you ate is 3/8 of the pizza – a rational number.
- If you eat 5 slices, that's 5/8.
Both are fractions of the whole pizza.
2. Where Do Rational Numbers Appear?
| Context | Example | Rational? |
|---|---|---|
| Money | ₹ 7.50 | Yes → 750/100 |
| Time | 2 hours 15 minutes | Yes → 135/60 |
| Speed | 48 km/hr | Yes → 48/1 |
| Fraction of a whole | ¼ of a cake | Yes → 1/4 |
Tip: Any number that can be expressed as a whole number divided by another whole number (not zero) is rational.
3. Types of Rational Numbers
| Type | How it Looks | Example |
|---|---|---|
| Whole number | Numerator = whole number, denominator = 1 | 5 → 5/1 |
| Proper fraction | Numerator < Denominator | 3/4 |
| Improper fraction | Numerator > Denominator | 9/4 |
| Decimal with a finite number of digits | Ends after a few places | 0.75 = 75/100 |
| Repeating decimal | Some digits repeat forever | 0.333… = 1/3 |
All the above are rational.
4. How to Check if a Decimal is Rational
- Finite decimal (ends after a few digits) → Rational.
- Repeating decimal (e.g., 0.142857142857…) → Rational.
- Non‑repeating infinite decimal (like π, e) → Not rational.
5. Operations with Rational Numbers
5.1 Addition & Subtraction
- Find a common denominator (the same bottom number).
- Change the fractions so they have that denominator.
- Add or subtract the numerators.
- Simplify if possible.
Example
Add ( \frac{2}{3} + \frac{1}{6} )
- Common denominator = 6.
- Convert: ( \frac{2}{3} = \frac{4}{6} ).
- Add: ( \frac{4}{6} + \frac{1}{6} = \frac{5}{6} ).
Result: ( \frac{5}{6} ) (already simple).
5.2 Multiplication
- Multiply the numerators together.
- Multiply the denominators together.
- Simplify.
Example
( \frac{3}{4} \times \frac{2}{5} = \frac{3 \times 2}{4 \times 5} = \frac{6}{20} = \frac{3}{10} ).
5.3 Division
- Flip the second fraction (take the reciprocal).
- Multiply as in the previous step.
Example
( \frac{5}{6} \div \frac{2}{3} )
- Flip: ( \frac{2}{3} \rightarrow \frac{3}{2} ).
- Multiply: ( \frac{5}{6} \times \frac{3}{2} = \frac{15}{12} = \frac{5}{4} ).
6. Simplifying Fractions
- Find the greatest common divisor (GCD) of numerator and denominator.
- Divide both by the GCD.
Example
Simplify ( \frac{18}{24} )
- GCD of 18 and 24 is 6.
- Divide: ( 18 ÷ 6 = 3 ); ( 24 ÷ 6 = 4 ).
- Result: ( \frac{3}{4} ).
7. Converting Between Forms
| From | To | How |
|---|---|---|
| Fraction → Decimal | Divide numerator by denominator. | ( \frac{3}{4} = 0.75 ). |
| Decimal → Fraction | Write as a fraction with denominator 10ⁿ (n = digits). | 0.2 = 2/10 = 1/5. |
| Fraction → Percentage | Multiply by 100%. | ( \frac{1}{4} = 25% ). |
8. Quick Practice Problems
- Add ( \frac{7}{9} + \frac{2}{3} ).
- Subtract ( \frac{5}{8} - \frac{1}{4} ).
- Multiply ( \frac{4}{7} \times \frac{3}{5} ).
- Divide ( \frac{9}{10} \div \frac{3}{4} ).
- Simplify ( \frac{42}{56} ).
Answers (for self‑checking)
- ( \frac{13}{9} ) → ( 1 \frac{4}{9} ).
- ( \frac{3}{8} ).
- ( \frac{12}{35} ).
- ( \frac{6}{5} ) → ( 1 \frac{1}{5} ).
- ( \frac{3}{4}