Fractions — Class 5 Mathematics (Samacheer Kalvi)
TN State Board (Samacheer Kalvi) Class 5 Mathematics, Term 3 Unit. Types of fractions, equivalent fractions, comparing and adding fractions, and conversion between decimal and common fractions.
1. The two numbers in a fraction
A fraction shows a part of a whole that has been divided into equal parts.
- The denominator, the bottom number, says how many equal parts the whole was cut into.
- The numerator, the top number, says how many of those parts you have.
In 3/4, the whole was cut into 4 equal parts and you have 3 of them.
The word equal is doing real work. Four pieces of different sizes are not quarters.
2. Proper, improper and mixed fractions
| Type | Rule | Examples |
|---|---|---|
| Proper | Numerator smaller than denominator | 3/4, 2/5, 1/8 |
| Improper | Numerator equal to or greater than denominator | 5/3, 7/4, 6/6 |
| Mixed | A whole number together with a proper fraction | 1 1/2, 2 3/4 |
A proper fraction is less than one whole. An improper fraction is one whole or more, which is why it can be rewritten as a mixed fraction.
Improper to mixed. Divide the numerator by the denominator. The quotient is the whole number and the remainder becomes the new numerator.
7/4: 7 ÷ 4 = 1 remainder 3, so 7/4 = 1 3/4.
Mixed to improper. Multiply the whole number by the denominator, add the numerator, and keep the same denominator.
2 3/5: (2 × 5) + 3 = 13, so 2 3/5 = 13/5.
3. Equivalent fractions
Different-looking fractions can mean the same amount. Half a pizza can be cut as 1/2, 2/4, 3/6 or 4/8 — the slices are thinner and more numerous, but the quantity is identical.
To find an equivalent fraction, multiply or divide both the numerator and the denominator by the same number.
- 1/2 = (1 × 3)/(2 × 3) = 3/6
- 8/12 = (8 ÷ 4)/(12 ÷ 4) = 2/3
Both parts must be treated the same way. Changing only the top changes the amount.
A fraction is in its simplest form when the numerator and denominator have no common factor left. 8/12 simplifies to 2/3.
4. Like and unlike fractions
Like fractions have the same denominator: 3/8, 5/8, 7/8.
Unlike fractions have different denominators: 1/2, 2/3, 3/5.
This distinction decides how you compare and add them, so check it first every time.
5. Comparing fractions
Like fractions. The parts are the same size, so simply compare the numerators.
3/8 < 5/8
Same numerator, different denominators. Think about the size of each part. The more pieces a whole is cut into, the smaller each piece must be, so a bigger denominator means a smaller fraction.
3/5 > 3/7
This is where the commonest mistake in the chapter lives. Seeing 1/3 and 1/2, students reason that 3 is bigger than 2, so 1/3 must be bigger. But a third of a chapati is smaller than half of it. 1/2 > 1/3.
Unlike fractions in general. Convert them to like fractions first, using a common denominator, then compare the numerators.
Compare 2/3 and 3/4. Use 12 as the common denominator: 2/3 = 8/12 and 3/4 = 9/12. Since 8 < 9, 2/3 < 3/4.
6. Adding and subtracting fractions
Like fractions are easy, because the parts are already the same size.
Add or subtract the numerators and keep the denominator unchanged.
- 3/8 + 2/8 = 5/8
- 7/9 − 4/9 = 3/9 = 1/3
Why does the denominator stay? Because you are counting eighths. Three eighths plus two eighths is five eighths — the type of part has not changed, only how many you have. Adding denominators to get 5/16 would be like saying three apples plus two apples equals five oranges.
Unlike fractions must first be made like, by finding a common denominator.
1/2 + 1/3: use 6, giving 3/6 + 2/6 = 5/6.
If the answer comes out improper, convert it to a mixed fraction: 4/5 + 3/5 = 7/5 = 1 2/5.
7. Fractions and decimals
A decimal is another way of writing a fraction whose denominator is 10, 100 or 1000.
The first place after the point is tenths and the second is hundredths.
Fraction to decimal. Divide the numerator by the denominator.
| Fraction | Decimal |
|---|---|
| 1/2 | 0.5 |
| 1/4 | 0.25 |
| 3/4 | 0.75 |
| 1/5 | 0.2 |
| 1/10 | 0.1 |
| 1/100 | 0.01 |
Decimal to fraction. Read the decimal aloud and write what you hear, then simplify.
- 0.5 is five tenths, 5/10 = 1/2
- 0.25 is twenty-five hundredths, 25/100 = 1/4
- 0.75 is seventy-five hundredths, 75/100 = 3/4
8. Worked examples
Example 1. Classify 7/4, 2/5 and 6/6.
Solution: 7/4 improper, 2/5 proper, 6/6 improper and equal to one whole.
Example 2. Write 9/4 as a mixed fraction.
Solution: 9 ÷ 4 = 2 remainder 1, so 2 1/4.
Example 3. Write 3 2/7 as an improper fraction.
Solution: (3 × 7) + 2 = 23, so 23/7.
Example 4. Simplify 8/12.
Solution: Divide both by 4 to get 2/3.
Example 5. Which is greater, 3/5 or 3/7?
Solution: The numerators match, and fifths are bigger than sevenths, so 3/5.
Example 6. Add 1/2 + 1/3.
Solution: Common denominator 6: 3/6 + 2/6 = 5/6.
Example 7. Subtract 5/6 − 1/4.
Solution: Common denominator 12: 10/12 − 3/12 = 7/12.
Example 8. Write 0.75 as a fraction in its simplest form.
Solution: 75/100, which simplifies to 3/4.
9. Practice
- In 4/9, name the numerator and the denominator.
- Is 8/5 proper or improper?
- Write 11/3 as a mixed fraction.
- Write 2 4/5 as an improper fraction.
- Simplify 15/20.
- Which is bigger, 1/2 or 1/3? Explain.
- Compare 2/3 and 3/4.
- Add 4/11 + 5/11.
- Add 1/4 + 1/6.
- Write 2/5 as a decimal, and 0.2 as a fraction in its simplest form.
10. Answers
- Numerator 4, denominator 9.
- Improper, since 8 is greater than 5.
- 11 ÷ 3 = 3 remainder 2, so 3 2/3.
- (2 × 5) + 4 = 14, so 14/5.
- Divide both by 5 to get 3/4.
- 1/2. The bigger the denominator, the smaller each part, so a half is bigger than a third.
- Common denominator 12: 8/12 and 9/12, so 2/3 < 3/4.
- 9/11.
- Common denominator 12: 3/12 + 2/12 = 5/12.
- 2/5 = 0.4, and 0.2 = 2/10 = 1/5.
11. Summary
- The denominator counts the equal parts; the numerator counts how many you have.
- Proper is less than one, improper is one or more, and mixed pairs a whole with a proper fraction.
- Improper to mixed: divide and keep the remainder as the new numerator.
- Mixed to improper: multiply the whole by the denominator and add the numerator.
- Multiply or divide both parts by the same number for an equivalent fraction.
- Like fractions share a denominator; unlike fractions do not.
- With like fractions the bigger numerator wins; with equal numerators the smaller denominator wins.
- Add and subtract like fractions on the numerators only, never on the denominators.
- Make unlike fractions like first, using a common denominator.
- Divide numerator by denominator to get a decimal; read a decimal as tenths or hundredths to get a fraction.
