Division of Fractions
Now that you have mastered multiplying fractions, you are ready to learn how to divide them. Division of fractions might seem tricky at first, but there is a simple trick that makes it easy: to divide by a fraction, you multiply by its reciprocal. That is it! Once you understand this one idea, dividing fractions becomes as simple as multiplying them.
Division of fractions answers questions like: "How many 1/4-sized pieces can you cut from 3/4 of a pizza?" or "If you have 2/3 of a cake and want to share it equally among 4 people, how much does each person get?" These everyday situations are solved by dividing fractions.
In Class 7 NCERT Maths, you will learn to divide a whole number by a fraction, a fraction by a whole number, a fraction by another fraction, and mixed numbers by fractions. The key concept is the reciprocal (also called the multiplicative inverse). The reciprocal of 3/4 is 4/3 (you flip the fraction). Dividing by 3/4 is the same as multiplying by 4/3. This chapter will explain why this works and give you plenty of practice to become confident.
Let us start with a simple visual. Imagine you have 2 whole rotis and you want to cut them into pieces of 1/4 roti each. How many pieces will you get? You are dividing 2 by 1/4. Since each roti gives 4 quarter-pieces, 2 rotis give 8 pieces. So 2 / (1/4) = 8. Notice that dividing by a fraction less than 1 gives a bigger answer, not a smaller one. This might feel counterintuitive at first, but it makes perfect sense when you think about it: you are asking how many small pieces fit into a larger quantity. The smaller each piece, the more pieces you get.
Division of fractions is essential for many real-world calculations: finding speed when distance and time are fractions, distributing supplies, adjusting recipes, and solving word problems in exams. Mastering this topic will also prepare you for algebra, ratios, and proportions in higher classes. So let us dive in!
What is Division of Fractions?
Division of fractions means finding how many times one fraction fits into another, or splitting a fraction into equal parts.
The key idea behind dividing fractions is the reciprocal. The reciprocal of a fraction a/b is b/a (you swap the numerator and denominator). For example:
- Reciprocal of 3/5 is 5/3
- Reciprocal of 7 (which is 7/1) is 1/7
- Reciprocal of 1/4 is 4/1 = 4
Rule for Dividing Fractions:
To divide by a fraction, multiply by its reciprocal. This rule is sometimes remembered as "Keep, Change, Flip" (KCF):
- Keep the first fraction as it is.
- Change the division sign to multiplication.
- Flip the second fraction (take its reciprocal).
a/b / c/d = a/b x d/c = (a x d) / (b x c)
Why does this work? Dividing by a number is the same as multiplying by its inverse. Just as dividing by 2 is the same as multiplying by 1/2, dividing by 3/4 is the same as multiplying by 4/3. The reciprocal is the multiplicative inverse of a fraction.
Important Notes:
- The reciprocal of 0 does not exist. You cannot divide by 0, and 0/0 is undefined.
- The reciprocal of 1 is 1 (since 1/1 flipped is still 1/1).
- Every non-zero fraction has a unique reciprocal.
- The reciprocal of a proper fraction is always an improper fraction. For example, the reciprocal of 2/5 (proper) is 5/2 (improper).
Division of Fractions Formula
Dividing a Fraction by a Fraction:
a/b / c/d = a/b x d/c = (a x d) / (b x c)
Dividing a Fraction by a Whole Number:
a/b / n = a/b x 1/n = a / (b x n)
Dividing a Whole Number by a Fraction:
n / (a/b) = n x b/a = (n x b) / a
Remember: Keep, Change, Flip (KCF)
Keep the first fraction → Change / to x → Flip the second fraction
Key Facts:
- Any non-zero fraction divided by itself equals 1: (a/b) / (a/b) = 1
- A fraction divided by 1 equals the same fraction: (a/b) / 1 = a/b
- Division by zero is not defined.
- 0 divided by any non-zero fraction equals 0.
Types and Properties
There are several types of fraction division problems:
Type 1: Fraction / Fraction
Flip the second fraction and multiply. Example: 3/5 / 2/7 = 3/5 x 7/2 = 21/10 = 2 1/10. This asks: how many groups of 2/7 fit into 3/5?
Type 2: Fraction / Whole Number
Write the whole number as a fraction with denominator 1, flip it (so it becomes 1/n), and multiply. Example: 4/5 / 3 = 4/5 x 1/3 = 4/15. Think of it as: if 4/5 of a cake is shared equally among 3 people, each person gets 4/15 of the cake.
Type 3: Whole Number / Fraction
Write the whole number as a fraction, flip the divisor fraction, and multiply. Example: 6 / (3/4) = 6/1 x 4/3 = 24/3 = 8. This asks: how many 3/4-sized portions are in 6 wholes?
Type 4: Mixed Number / Fraction
Convert the mixed number to an improper fraction, then divide using the KCF method. Example: 2 1/2 / 1/4 = 5/2 / 1/4 = 5/2 x 4/1 = 20/2 = 10.
Type 5: Mixed Number / Mixed Number
Convert both mixed numbers to improper fractions, then use KCF. Example: 3 1/3 / 1 2/3 = 10/3 / 5/3 = 10/3 x 3/5 = 30/15 = 2.
Observation: When you divide a number by a proper fraction (less than 1), the result is always larger than the original number. For example, 6 / (1/2) = 12, which is larger than 6. This makes sense: you are asking how many half-portions fit into 6, and the answer is 12.
Solved Examples
Example 1: Dividing a Fraction by a Fraction
Problem: Find: 3/4 / 2/5
Solution:
Step 1: Keep the first fraction: 3/4
Step 2: Change / to x
Step 3: Flip the second fraction: 2/5 becomes 5/2
Step 4: Multiply: 3/4 x 5/2 = (3 x 5) / (4 x 2) = 15/8
Step 5: Convert to mixed number: 15/8 = 1 7/8
Answer: 3/4 / 2/5 = 15/8 = 1 7/8
Example 2: Dividing a Fraction by a Whole Number
Problem: Find: 5/6 / 3
Solution:
Step 1: Write 3 as 3/1. Now divide: 5/6 / 3/1
Step 2: Flip the second fraction: 3/1 becomes 1/3
Step 3: Multiply: 5/6 x 1/3 = 5/18
Answer: 5/6 / 3 = 5/18
Example 3: Dividing a Whole Number by a Fraction
Problem: Find: 8 / (2/3)
Solution:
Step 1: Write 8 as 8/1. Now divide: 8/1 / 2/3
Step 2: Flip the second fraction: 2/3 becomes 3/2
Step 3: Multiply: 8/1 x 3/2 = 24/2 = 12
Answer: 8 / (2/3) = 12. There are 12 groups of 2/3 in 8.
Example 4: Dividing Mixed Numbers
Problem: Find: 3 1/2 / 1 1/4
Solution:
Step 1: Convert to improper fractions: 3 1/2 = 7/2 and 1 1/4 = 5/4
Step 2: Divide: 7/2 / 5/4
Step 3: Flip and multiply: 7/2 x 4/5 = 28/10
Step 4: Simplify: 28/10 = 14/5 = 2 4/5
Answer: 3 1/2 / 1 1/4 = 2 4/5
Example 5: Word Problem: Sharing Pizza
Problem: Ananya has 3/4 of a pizza. She wants to share it equally among 3 friends. How much pizza does each friend get?
Solution:
Step 1: We need to divide 3/4 by 3: 3/4 / 3
Step 2: Write 3 as 3/1, flip to get 1/3.
Step 3: Multiply: 3/4 x 1/3 = 3/12 = 1/4
Answer: Each friend gets 1/4 of the pizza.
Example 6: Word Problem: Ribbon Cutting
Problem: A ribbon is 7/8 metre long. It is cut into pieces, each 1/4 metre long. How many pieces can be cut?
Solution:
Step 1: Divide the total length by the length of each piece: 7/8 / 1/4
Step 2: Flip and multiply: 7/8 x 4/1 = 28/8 = 7/2 = 3 1/2
Answer: 3 full pieces can be cut (with 1/2 of a piece remaining).
Example 7: Word Problem: Filling Containers
Problem: A jug holds 5/6 of a litre of juice. Each glass holds 1/3 of a litre. How many glasses can be filled from the jug?
Solution:
Step 1: Divide: 5/6 / 1/3
Step 2: Flip and multiply: 5/6 x 3/1 = 15/6 = 5/2 = 2 1/2
Answer: 2 full glasses can be filled (with half a glass remaining).
Example 8: Dividing Zero by a Fraction
Problem: Find: 0 / (5/7)
Solution:
Step 1: 0 divided by any non-zero number is 0.
Step 2: 0 / (5/7) = 0 x 7/5 = 0
Answer: 0 / (5/7) = 0
Example 9: Word Problem: Distance and Speed
Problem: A snail covers 3/10 km in 3/5 of an hour. What is the snail's speed in km per hour?
Solution:
Step 1: Speed = Distance / Time = (3/10) / (3/5)
Step 2: Flip and multiply: 3/10 x 5/3
Step 3: Cross-cancel: 3 and 3 cancel, 5 and 10 cancel (5/5=1, 10/5=2)
Step 4: Multiply: 1/2 x 1/1 = 1/2
Answer: The snail's speed is 1/2 km per hour.
Example 10: Word Problem: Distributing Land
Problem: A farmer has 4 1/2 acres of land. He divides it into plots of 3/4 acre each. How many plots does he get?
Solution:
Step 1: Convert: 4 1/2 = 9/2
Step 2: Divide: 9/2 / 3/4 = 9/2 x 4/3
Step 3: Cross-cancel: 9 and 3 (9/3=3), 4 and 2 (4/2=2)
Step 4: Multiply: 3/1 x 2/1 = 6
Answer: The farmer gets 6 plots.
Real-World Applications
Division of fractions is used in many real-life situations:
Cooking and Baking: If a recipe serves 8 people and uses 3/4 cup of butter, but you want to make it for 4 people (half the recipe), you divide 3/4 by 2 to get 3/8 cup of butter per serving adjustment. Recipes constantly require dividing fractional quantities. If you have 2 1/2 cups of flour and each batch of cookies needs 3/4 cup, you divide 5/2 by 3/4 to find you can make 3 1/3 batches.
Sharing and Distribution: When you share 2/3 of a cake equally among 4 people, each person gets (2/3) / 4 = 2/12 = 1/6 of the cake. Fair sharing of fractional quantities is a common real-life application. In schools, if 3/5 of a project is to be completed equally by 3 students, each student does (3/5) / 3 = 1/5 of the project.
Measurement and Construction: If you have a rope that is 5/2 metres long and you need pieces of 1/4 metre each, you divide to find how many pieces you can cut: (5/2) / (1/4) = 10 pieces. Tailors divide fabric lengths by pattern piece sizes, which are often fractions.
Speed, Distance, and Time: Speed = Distance / Time. When both distance and time are fractions, you divide fractions to find speed. For example, if you walk 3/4 km in 1/2 hour, your speed is (3/4) / (1/2) = 3/2 = 1.5 km/hr. This is a very common application in physics and everyday travel calculations.
Unit Pricing: If 3/4 kg of apples costs Rs. 60, the price per kg is 60 / (3/4) = 60 x 4/3 = 80. So apples cost Rs. 80 per kg. Shopkeepers and consumers use this type of calculation regularly.
Land and Area: If a farmer has 3/4 of an acre and wants to divide it into plots of 1/8 acre each, the number of plots is (3/4) / (1/8) = 6. Land distribution problems commonly involve fraction division.
Time Management: If a task takes 1/3 of an hour and you have 2 2/3 hours available, how many times can you complete the task? Divide: (8/3) / (1/3) = 8 times. Planning activities within a time budget often requires dividing fractions.
Key Points to Remember
- To divide by a fraction, multiply by its reciprocal (flip the second fraction and multiply).
- Remember the KCF method: Keep the first fraction, Change division to multiplication, Flip the second fraction.
- The reciprocal of a/b is b/a. A number multiplied by its reciprocal equals 1.
- To divide a fraction by a whole number n, multiply by 1/n. Example: 3/4 / 5 = 3/4 x 1/5 = 3/20.
- To divide a whole number by a fraction a/b, multiply by b/a. Example: 6 / (2/3) = 6 x 3/2 = 9.
- For mixed numbers, convert to improper fractions first, then apply KCF.
- Dividing by a proper fraction (less than 1) gives a result larger than the original number. This is because you are finding how many small parts fit into the number.
- Dividing by an improper fraction (greater than 1) gives a result smaller than the original number.
- Division by zero is not defined. The reciprocal of zero does not exist.
- Zero divided by any non-zero fraction is zero: 0 / (a/b) = 0.
- A fraction divided by itself always equals 1: (a/b) / (a/b) = 1.
- Always simplify the final answer to its lowest terms. Use cross-cancellation before multiplying to keep numbers small.
- Check your answer by multiplying the quotient by the divisor. If you get the dividend, the answer is correct.
Practice Problems
- Find: 5/8 / 3/4
- Find: 7/12 / 7
- Find: 9 / (3/5)
- Find: 2 2/3 / 1 1/3
- A 3/4 m long ribbon is cut into pieces of 1/8 m each. How many pieces are there?
- A tank has 5/6 litres of water. Each cup holds 1/6 litre. How many cups can be filled?
- Arjun walks 4/5 km in 2/3 of an hour. What is his speed in km/hr?
- A farmer distributes 6 3/4 kg of seeds equally into bags of 3/4 kg each. How many bags does he fill?
Frequently Asked Questions
Q1. How do you divide fractions?
To divide fractions, keep the first fraction the same, change the division sign to multiplication, and flip the second fraction (take its reciprocal). Then multiply as usual. For example, 3/4 / 2/5 = 3/4 x 5/2 = 15/8 = 1 7/8. This method is called Keep, Change, Flip (KCF).
Q2. What is the reciprocal of a fraction?
The reciprocal of a fraction a/b is b/a. You flip the numerator and denominator. For example, the reciprocal of 2/3 is 3/2, and the reciprocal of 5 (which is 5/1) is 1/5. When you multiply a fraction by its reciprocal, the result is always 1.
Q3. Why do you flip and multiply when dividing fractions?
Dividing by a number is the same as multiplying by its inverse. Just as 10 / 2 = 10 x (1/2) = 5, dividing by 3/4 means multiplying by 4/3 (the reciprocal of 3/4). The reciprocal undoes the fraction, turning division into the simpler operation of multiplication.
Q4. How do you divide a whole number by a fraction?
Write the whole number as a fraction over 1, then flip the divisor and multiply. For example, 6 / (2/3) = 6/1 x 3/2 = 18/2 = 9. The answer tells you how many groups of 2/3 fit into 6.
Q5. How do you divide mixed numbers?
First, convert both mixed numbers to improper fractions. Then use KCF (Keep, Change, Flip) to divide. For example, 2 1/2 / 1 1/4 = 5/2 / 5/4 = 5/2 x 4/5 = 20/10 = 2.
Q6. Why is the answer bigger when you divide by a fraction less than 1?
When you divide by a number less than 1, you are asking how many small parts fit into a larger quantity, so the answer is bigger. For example, 3 / (1/2) = 6 because six halves make 3 wholes. The smaller the fraction you divide by, the larger the result.
Q7. Can you divide by zero in fractions?
No. Division by zero is never allowed, whether with whole numbers, fractions, or any other numbers. The reciprocal of 0 does not exist because 0/1 flipped would be 1/0, which is undefined.
Q8. What is the difference between multiplying and dividing fractions?
When multiplying fractions, you multiply numerators and denominators directly. When dividing, you first flip the second fraction (take the reciprocal) and then multiply. Multiplication makes the result smaller (for proper fractions), while division by a proper fraction makes the result larger.
