Fractions
A fraction is a fair share of something. Learn to read, build, compare, add and subtract fractions — with pizzas, bars and number lines — and solve fraction word problems like a pro.
1What is a fraction?
A fraction is a part of a whole that has been divided into equal parts. We write it with two numbers and a line between them.
- Denominator (bottom)
- How many equal parts the whole is divided into.
- Numerator (top)
- How many of those parts we are talking about (shaded, eaten, used…).
If a shape is cut into parts of different sizes, the shaded part is not a simple fraction of the number of pieces. Always check the parts are equal. And read the question carefully: if 3 of 8 parts are shaded, the fraction not shaded is 5/8.
Fractions all around us
- 15 minutes = 15/60 = 1/4 of an hour; 20 minutes = 1/3; 30 minutes = 1/2; 45 minutes = 3/4.
- 25 paise = 1/4 of a rupee; 50 paise = 1/2 of a rupee.
- 250 m = 1/4 of a kilometre; 500 g = 1/2 of a kilogram.
- 6 months = 1/2 of a year; 3 months = 1/4 of a year.
A pizza is cut into 8 equal slices. Aarav eats 3 slices. What fraction of the pizza is left?
2Fraction of a collection
To find a fraction of a number: divide by the denominator (to find one part), then multiply by the numerator (to take that many parts).
- 2/3 of 24 → 24 ÷ 3 = 8 → 8 × 2 = 16.
- 3/4 of ₹120 → 120 ÷ 4 = 30 → 30 × 3 = ₹90.
- 5/9 of 36 children are girls → 36 ÷ 9 = 4 → 4 × 5 = 20 girls, so 36 − 20 = 16 boys.
2/5 of a number is 18. What is the number?
- 2 parts = 18, so 1 part (1/5) = 18 ÷ 2 = 9.
- The whole has 5 parts: 9 × 5 = 45.
- Check: 2/5 of 45 = 45 ÷ 5 × 2 = 18 ✓
What is 2/3 of 36?
3Types of fractions and mixed numbers
| Type | What it means | Examples |
|---|---|---|
| Unit | Numerator is 1 | 1/2, 1/7, 1/9 |
| Proper | Numerator < denominator (less than 1) | 2/3, 5/8, 9/10 |
| Improper | Numerator ≥ denominator (1 or more) | 5/4, 7/3, 8/8 |
| Mixed number | A whole number and a proper fraction | 1 1/4, 2 3/5 |
| Like | Same denominators | 2/7, 5/7, 6/7 |
| Unlike | Different denominators | 1/3, 2/5, 3/4 |
When the numerator equals the denominator, the fraction equals 1 whole: 4/4 = 9/9 = 1. When the numerator is 0, the fraction is 0: 0/5 = 0. A mixed number like 1 3/4 is not the same as 1/34 or 13/4 — it means 1 + 3/4.
Mixed numbers and improper fractions
Mixed → improper
Write 4 1/3 as an improper fraction.
- Multiply the whole number by the denominator: 4 × 3 = 12.
- Add the numerator: 12 + 1 = 13.
- Keep the same denominator: 13/3.
Improper → mixed
Write 37/8 as a mixed number.
- Divide: 37 ÷ 8 = 4 remainder 5.
- The quotient is the whole number (4); the remainder is the new numerator (5); the denominator stays 8.
- 37/8 = 4 5/8. Check: 4 × 8 + 5 = 37 ✓
Write 4 2/7 as an improper fraction.
4Equivalent fractions and simplest form
Equivalent fractions look different but are the same amount. Make them by multiplying (or dividing) the numerator and denominator by the same number.
5/6 = ?/42
- How did 6 become 42? 6 × 7 = 42.
- Do the same to the top: 5 × 7 = 35.
- So 5/6 = 35/42.
Simplest form (lowest terms)
A fraction is in its simplest form when the numerator and denominator have no common factor except 1. Divide both by their HCF to get there in one step.
Write 24/36 in simplest form.
- HCF of 24 and 36 is 12.
- 24 ÷ 12 = 2 and 36 ÷ 12 = 3.
- 24/36 = 2/3. (Dividing by 2 or 4 first also works — just keep going until you can't divide any more.)
12/18 and 4/6 are equal to 24/36, but they are not in simplest form. Examiners put these in the options. Also, adding the same number to top and bottom does not make an equivalent fraction: 1/2 ≠ 2/3.
Are 3/4 and 15/20 equivalent? Cross-multiply: 3 × 20 = 60 and 4 × 15 = 60. Equal products → equivalent fractions.
Find the missing number: 3/8 = ?/32
5Comparing and ordering fractions
Same denominator
Bigger numerator = bigger fraction. 5/9 > 4/9 (5 pieces of the same size beat 4).
Same numerator
Smaller denominator = bigger fraction, because the pieces are bigger. 3/4 > 3/7.
Both different
Make the denominators the same (use the LCM), or cross-multiply.
Which is greater: 2/3 or 3/5?
- Multiply across: 2 × 5 = 10 (goes with 2/3) and 3 × 3 = 9 (goes with 3/5).
- 10 > 9, so 2/3 > 3/5.
Arrange in ascending order: 1/2, 2/3, 5/12, 3/4.
- LCM of 2, 3, 12 and 4 is 12. Change each to twelfths.
- 1/2 = 6/12, 2/3 = 8/12, 5/12 = 5/12, 3/4 = 9/12.
- Order the numerators: 5, 6, 8, 9.
- Ascending: 5/12 < 1/2 < 2/3 < 3/4.
Many students think 1/8 is bigger than 1/4 "because 8 is bigger". But 1/8 means sharing among 8 people — each gets less. Among 5/8, 5/6, 5/9 and 5/12, the smallest is 5/12.
Which fraction is the smallest?
6Adding and subtracting fractions
Like fractions
When the denominators are the same, add or subtract the numerators and keep the denominator. Simplify at the end.
- 5/9 − 2/9 = 3/9 = 1/3 (simplify by 3).
- 5/7 + 4/7 = 9/7 = 1 2/7 (an improper answer becomes a mixed number).
- 1 − 3/8: write 1 as 8/8. 8/8 − 3/8 = 5/8.
- 2 1/4 + 1 2/4: add wholes (3) and fractions (3/4) → 3 3/4.
3/11 + 5/11 is 8/11, not 8/22. And 1/3 + 1/4 is not 2/7 — the pieces are different sizes, so first make them the same size.
Unlike fractions (simple ones)
- Find a common denominator (the LCM of the denominators).
- Change each fraction into an equivalent fraction with that denominator.
- Add or subtract the numerators. Simplify.
- 1/2 + 1/4: 1/2 = 2/4 → 2/4 + 1/4 = 3/4.
- 2/3 + 2/9: 2/3 = 6/9 → 6/9 + 2/9 = 8/9.
- 3/4 − 1/3: LCM of 4 and 3 is 12 → 9/12 − 4/12 = 5/12.
- 1/3 + 1/4: 4/12 + 3/12 = 7/12.
Add: 2/3 + 1/6
7Fractions on a number line
On a number line, first find how many equal parts each whole is cut into — that is the denominator. Then count the jumps from 0.
Between 0 and 1 there are 4 small marks here, but 5 gaps. The denominator is the number of gaps (5). Counting marks gives the wrong answer 3/4.
8Fraction word problems
A ribbon is 40 m long. Meera uses 3/8 of it. How much ribbon is left?
- Used: 3/8 of 40 = 40 ÷ 8 × 3 = 15 m.
- Left: 40 − 15 = 25 m.
- Shortcut: the fraction left is 1 − 3/8 = 5/8, and 5/8 of 40 = 25 m ✓
Kabir had ₹240. He spent 1/4 on a book and 1/3 on a toy. How much money is left?
- Book: 1/4 of 240 = ₹60. Toy: 1/3 of 240 = ₹80.
- Spent: 60 + 80 = ₹140. Left: 240 − 140 = ₹100.
- Fraction way: 1/4 + 1/3 = 3/12 + 4/12 = 7/12 spent, so 5/12 is left; 5/12 of 240 = ₹100 ✓
A 16 L tank is 3/4 full. Then 3 1/2 L of water is used. How much water is left?
- Water in the tank: 3/4 of 16 = 16 ÷ 4 × 3 = 12 L.
- 12 − 3 1/2: take away 3 to get 9, then take away 1/2 more → 8 1/2 L.
What fraction of an hour is 40 minutes?
- 1 hour = 60 minutes, so 40 minutes = 40/60 of an hour.
- Simplify by the HCF 20: 40/60 = 2/3.
The word "fraction" comes from the Latin word fractus, meaning "broken". A fraction is a whole broken into equal pieces!
Ria had ₹240. She spent 3/8 of it. How much money does she have left?
9Watch and explore
10Quick recap
- A fraction = equal parts of a whole. Denominator: total equal parts. Numerator: parts taken.
- Fraction of a number: divide by the denominator, multiply by the numerator.
- If you know a part (2/5 = 18), find one part first (1/5 = 9), then the whole (45).
- Unit: numerator 1. Proper: less than 1. Improper: 1 or more. Mixed: whole + proper fraction.
- Like fractions: same denominator. Unlike: different denominators.
- Mixed → improper: multiply, add, keep. Improper → mixed: divide; remainder on top.
- Equivalent fractions: multiply or divide top and bottom by the same number.
- Simplest form: divide top and bottom by their HCF.
- Same denominator: bigger numerator wins. Same numerator: smaller denominator wins.
- Different both: make a common denominator or cross-multiply.
- Add/subtract like fractions: work on numerators, keep the denominator. Never add denominators.
- Unlike fractions: change to a common denominator first.
- On a number line, the denominator is the number of gaps in one whole.
- 15 min = 1/4 h, 20 min = 1/3 h, 30 min = 1/2 h, 45 min = 3/4 h.