1. Chapter at a Glance
In Class 6 you learnt to add and subtract fractions. Now we multiply and divide them. How much is ¾ of 12 rotis? What is half of a quarter? How many ¼-litre glasses can be filled from 3 litres? Multiplication and division of fractions answer such everyday questions. You will also see why multiplying by a proper fraction makes a number smaller and dividing by it makes a number larger.
- Multiplying a fraction by a whole number and finding a fraction of a quantity.
- Multiplying two fractions using the area (rectangle) model.
- When a product is larger or smaller than the numbers multiplied.
- Reciprocals and dividing by a fraction.
- Word problems with fractions and mixed numbers.
- Brahmagupta's rules for multiplying and dividing fractions.
2. Multiplying a Fraction and a Whole Number
4 × ⅔ means ⅔ + ⅔ + ⅔ + ⅔ = 8/3. So multiply the numerator by the whole number and keep the denominator:
"⅔ of 12" means 12 divided into 3 equal parts, then 2 of those parts: 12 ÷ 3 × 2 = 8. In mathematics, the word "of" means multiply: ⅔ of 12 = ⅔ × 12 = 24/3 = 8.
A class has 40 students; ⅗ of them are girls. How many girls are there?
Solution: ⅗ × 40 = 120/5 = 24 girls.
3. Multiplying Two Fractions
What is ½ of ⅓? Take a square as one whole. Divide it into 3 vertical strips and shade one (⅓). Now divide the square into 2 horizontal strips and take half of the shaded strip. The doubly-shaded piece is 1 of 6 equal parts — so ½ × ⅓ = 1/6.
Cancel first to keep numbers small: 6/35 × 14/9 — cancel 6 and 9 by 3, 14 and 35 by 7: (2/5) × (2/3) = 4/15.
Mixed numbers
Convert mixed numbers to improper fractions before multiplying: 2½ × 1⅗ = 5/2 × 8/5 = 40/10 = 4.
4. Is the Product Bigger or Smaller?
With whole numbers, multiplying (by numbers bigger than 1) makes things bigger. With fractions:
- Multiplying by a fraction less than 1 gives a smaller result: ½ × 8 = 4 < 8.
- Multiplying by a fraction greater than 1 gives a bigger result: 3/2 × 8 = 12 > 8.
- Multiplying by 1 leaves the number unchanged.
- The product of two proper fractions is smaller than both: ½ × ⅓ = 1/6.
5. Dividing Fractions
Reciprocal: two numbers whose product is 1 are reciprocals. The reciprocal of a/b is b/a; the reciprocal of 5 is 1/5; the reciprocal of 1⅓ = 4/3 is 3/4. Zero has no reciprocal.
How many halves are in 3? Six, because 3 ÷ ½ = 6. Notice 3 × 2 = 6 — dividing by ½ is the same as multiplying by 2, the reciprocal of ½.
- 6 ÷ ⅔ = 6 × 3/2 = 9.
- ⅘ ÷ 2 = ⅘ × ½ = 2/5.
- ⅚ ÷ ⅖ = ⅚ × 5/2 = 25/12 = 2 1/12.
Dividing by a fraction less than 1 gives a larger result (how many small pieces fit in?); dividing by a number greater than 1 gives a smaller result.
How many ¼-litre glasses can be filled from 3½ litres of juice?
Solution: 3½ ÷ ¼ = 7/2 × 4 = 28/2 = 14 glasses.
A pinch of history
Brahmagupta (628 CE) stated the rule for multiplying fractions: multiply the numerators and multiply the denominators. For division, he said to swap the numerator and denominator of the divisor and then multiply — exactly the method we use today. Later, in Līlāvatī, Bhāskarāchārya (12th century) gave many lively word problems involving fractions.
6. Solved Problems (NCERT Style)
Ramu ate ⅓ of a cake and gave ½ of what was left to his sister. What fraction of the cake did the sister get?
Solution: Left after Ramu = 1 − ⅓ = ⅔. Sister got ½ × ⅔ = ⅓ of the cake.
Find the area of a rectangular plot 12½ m long and 8⅖ m wide.
Solution: 25/2 × 42/5 = 1050/10 = 105 m².
4⅕ kg of sweets is packed into boxes of ⅗ kg each. How many boxes are needed?
Solution: 21/5 ÷ 3/5 = 21/5 × 5/3 = 7. 7 boxes.
A car covers 7½ km in ⅙ hour. How far will it go in 1 hour at this speed?
Solution: 7½ ÷ ⅙ = 15/2 × 6 = 45 km.
From a swarm of bees, ⅕ went to a kadamba flower, ⅓ to a silindhra flower, and 3 times the difference of these two fractions flew to a kutaja flower. One bee was left flying in the air. How many bees were there?
Solution: ⅕ + ⅓ = 8/15. Difference = ⅓ − ⅕ = 2/15; 3 times = 6/15. Total so far = 14/15, so the remaining 1/15 is the 1 bee. The swarm had 15 bees.
7. Master Formula Table
| Operation | Rule | Example |
|---|---|---|
| Whole × fraction | n × a/b = na/b | 5 × 3/4 = 15/4 |
| Fraction of quantity | "of" = × | 2/5 of 30 = 12 |
| Fraction × fraction | ac/bd | 2/3 × 4/5 = 8/15 |
| Reciprocal | a/b ↔ b/a | 7/9 ↔ 9/7 |
| Division | a/b ÷ c/d = a/b × d/c | 3/4 ÷ 1/2 = 3/2 |
| Mixed numbers | convert to improper first | 1½ = 3/2 |
| × proper fraction | result smaller | ¼ × 20 = 5 |
| ÷ proper fraction | result larger | 20 ÷ ¼ = 80 |
8. Master Summary Table
| Topic | Key idea | Exam focus |
|---|---|---|
| Fraction × whole | Repeated addition | Fraction of a quantity |
| Fraction × fraction | Area model | Cancelling first |
| Size of product | Depends on whether factor < 1 | Estimation |
| Reciprocal | Product = 1 | Find reciprocals |
| Division | Multiply by reciprocal | "How many fit?" problems |
| History | Brahmagupta, Bhāskara | Indian contributions |
9. Mind Map
- "of" means ×
- Top × top, bottom × bottom
- Area model
- Reciprocal
- Keep–change–flip
- × proper → smaller
- ÷ proper → larger
- Sharing, packing
- Area, speed
- Līlāvatī puzzles
10. Tips and Tricks
- Cancel common factors across numerators and denominators before multiplying.
- Estimate first: 2⅞ × 4⅒ ≈ 3 × 4 = 12.
- For "how many … fit into …" questions, divide.
- Flipping the first fraction instead of the divisor.
- Multiplying mixed numbers part by part: 2½ × 3½ is not 6¼.
- Making denominators common before multiplying — not needed.
11. Important One-liners
- "Of" in fraction problems means multiply.
- Product of fractions = product of numerators over product of denominators.
- The reciprocal of a/b is b/a.
- A number times its reciprocal is 1.
- 0 has no reciprocal.
- To divide by a fraction, multiply by its reciprocal.
- Multiplying by a proper fraction makes a positive number smaller.
- Dividing by a proper fraction makes a positive number larger.
- Brahmagupta gave the rules for multiplying and dividing fractions.
12. Exam-pattern Questions with Detailed Answers
Find ¾ of 36.
- 24
- 27
- 30
- 9
Show answer
36 ÷ 4 = 9; 9 × 3 = 27.
Answer: (B) 27
Write the reciprocal of 2⅗.
Show answer
2⅗ = 13/5, so its reciprocal is 5/13.
Answer: 5/13
Find ⅖ × 5/8.
Show answer
Cancel 5: (2 × 1)/(1 × 8) = 2/8 = ¼.
Answer: ¼
Evaluate 3¾ ÷ 1¼.
Show answer
15/4 ÷ 5/4 = 15/4 × 4/5 = 3.
Answer: 3
Without calculating, which is larger: 15 × ⅞ or 15 ÷ ⅞?
Show answer
Multiplying by ⅞ (less than 1) makes 15 smaller; dividing by ⅞ makes it larger. So 15 ÷ ⅞ is larger.
Answer: 15 ÷ ⅞
A rope 18 m long is cut into pieces of 2¼ m each. How many pieces are made?
- 6
- 8
- 9
- 40
Show answer
18 ÷ 9/4 = 18 × 4/9 = 8.
Answer: (B) 8
Rehana spends ⅓ of her pocket money on books and ¼ of the rest on snacks. What fraction is left?
Show answer
After books: ⅔ left. Snacks: ¼ × ⅔ = 1/6. Left = ⅔ − 1/6 = 4/6 − 1/6 = ½.
Answer: ½
The product of two fractions is 14/15. One of them is 7/9. Find the other.
Show answer
Other = 14/15 ÷ 7/9 = 14/15 × 9/7 = 18/15 = 6/5.
Answer: 6/5
A tank is ⅝ full. After using 30 litres, it is ⅜ full. What is its capacity?
Show answer
30 L = ⅝ − ⅜ = 2/8 = ¼ of the capacity. Capacity = 30 ÷ ¼ = 120 L.
Answer: 120 L
A worker finishes ⅖ of a job in 6 days. At the same rate, how long will the whole job take?
Show answer
Whole job = 6 ÷ ⅖ = 6 × 5/2 = 15 days.
Answer: 15 days
Simplify (⅔ + ¾) × 12/17.
Show answer
⅔ + ¾ = 8/12 + 9/12 = 17/12. Then 17/12 × 12/17 = 1.
Answer: 1
Half of a number is added to one-third of it, and the result is 25. Find the number.
Show answer
½n + ⅓n = 5n/6 = 25, so n = 25 ÷ 5/6 = 25 × 6/5 = 30.
Answer: 30
13. Quick Revision Summary
- "Of" means multiply; a fraction of a quantity = (quantity ÷ denominator) × numerator.
- a/b × c/d = ac/bd; cancel first; convert mixed numbers to improper fractions.
- Reciprocal of a/b is b/a; a/b ÷ c/d = a/b × d/c.
- × proper fraction → smaller; ÷ proper fraction → larger.
- Brahmagupta and Bhāskarāchārya gave these rules and many puzzles.