Multiplication of Fractions
You already know how to add and subtract fractions. Now it is time to learn how to multiply them! Multiplication of fractions is actually simpler than addition and subtraction because you do not need to find a common denominator. You simply multiply the numerators together and the denominators together, and you have your answer.
Multiplication of fractions comes up in everyday life more often than you might think. When a recipe says "use 3/4 of a cup of sugar" and you want to make half the recipe, you need to find 1/2 of 3/4, which means 1/2 x 3/4. When you hear that 2/3 of the class passed a test, and 3/4 of those who passed got an A grade, the fraction of the class that got an A is 2/3 x 3/4. Sharing pizza, dividing land, calculating discounts, all of these involve multiplying fractions.
In this chapter for Class 7 NCERT Maths, we will learn how to multiply a fraction by a whole number, a fraction by another fraction, and a mixed number by a fraction. We will also learn a useful shortcut called cross-cancellation that makes multiplication faster. With plenty of solved examples and practice problems, you will become confident with fraction multiplication in no time.
What is Multiplication of Fractions?
A fraction represents a part of a whole. It is written as a/b, where a is the numerator (the number of parts you have) and b is the denominator (the total number of equal parts the whole is divided into). For example, 3/5 means 3 parts out of 5 equal parts.
Multiplication of fractions means finding a part of a part. When you multiply 1/2 by 3/4, you are finding "one-half of three-quarters". The word "of" in maths often means multiply.
Rule for Multiplying Fractions:
To multiply two fractions, multiply the numerators to get the new numerator, and multiply the denominators to get the new denominator.
a/b x c/d = (a x c) / (b x d)
Multiplying a Fraction by a Whole Number:
A whole number can be written as a fraction with denominator 1. So, to multiply a fraction by a whole number, multiply the numerator by the whole number and keep the denominator the same.
a/b x n = (a x n) / b
Cross-Cancellation (Simplifying Before Multiplying):
Before multiplying, check if any numerator and any denominator share a common factor. Cancel (divide) them by their common factor to make the numbers smaller. This gives the same answer but with easier calculations.
Multiplication of Fractions Formula
Multiplying Two Fractions:
a/b x c/d = (a x c) / (b x d)
Multiplying a Fraction by a Whole Number:
a/b x n = (a x n) / b
Multiplying Mixed Numbers:
Step 1: Convert each mixed number to an improper fraction.
Step 2: Multiply the improper fractions using the rule above.
Step 3: Convert the answer back to a mixed number if needed.
Mixed Number to Improper Fraction:
a b/c = (a x c + b) / c
Key Facts:
- Multiplying a fraction by 1 gives the same fraction: a/b x 1 = a/b
- Multiplying a fraction by 0 gives 0: a/b x 0 = 0
- Multiplying a fraction by its reciprocal gives 1: a/b x b/a = 1
- When you multiply a proper fraction by a proper fraction, the product is smaller than either fraction.
Types and Properties
There are several types of fraction multiplication problems you will encounter:
Type 1: Fraction x Fraction
Multiply the numerators and denominators directly. Example: 2/3 x 4/5 = (2 x 4) / (3 x 5) = 8/15. This is the most basic type. Think of it as finding 2/3 of 4/5. If you have 4/5 of a pizza and you eat 2/3 of that portion, you eat 8/15 of the whole pizza.
Type 2: Fraction x Whole Number
Write the whole number as a fraction with denominator 1, then multiply. Example: 3/7 x 5 = 3/7 x 5/1 = 15/7 = 2 1/7. Think of it as: if each person gets 3/7 of a cake, 5 people get 15/7 cakes in total.
Type 3: Mixed Number x Fraction
Convert the mixed number to an improper fraction first, then multiply. Example: 2 1/3 x 3/5 = 7/3 x 3/5 = 21/15 = 7/5 = 1 2/5.
Type 4: Mixed Number x Mixed Number
Convert both mixed numbers to improper fractions, then multiply. Example: 1 1/2 x 2 2/3 = 3/2 x 8/3 = 24/6 = 4.
Type 5: Fraction x Fraction with Cross-Cancellation
Before multiplying, cancel common factors between any numerator and any denominator. Example: 4/9 x 3/8. Here, 4 and 8 share a factor of 4 (4/4 = 1, 8/4 = 2), and 3 and 9 share a factor of 3 (3/3 = 1, 9/3 = 3). After cancellation: 1/3 x 1/2 = 1/6. Much easier!
Observation: When you multiply two proper fractions (both less than 1), the product is always smaller than either fraction. For example, 1/2 x 1/3 = 1/6, and 1/6 is smaller than both 1/2 and 1/3. This makes sense because you are finding a part of a part, which is always smaller.
However, when you multiply a fraction by a whole number greater than 1, the product is larger than the fraction. For example, 2/5 x 3 = 6/5 = 1 1/5, which is larger than 2/5. And when you multiply a fraction by a number equal to 1, the result stays the same: 3/7 x 1 = 3/7.
These observations help you estimate answers before calculating, which is a valuable skill for checking your work in exams.
Solved Examples
Example 1: Multiplying Two Simple Fractions
Problem: Find: 2/5 x 3/7
Solution:
Step 1: Multiply the numerators: 2 x 3 = 6
Step 2: Multiply the denominators: 5 x 7 = 35
Step 3: Write the product: 6/35
Step 4: Check if it can be simplified. 6 and 35 have no common factor other than 1.
Answer: 2/5 x 3/7 = 6/35
Example 2: Multiplying a Fraction by a Whole Number
Problem: Find: 3/8 x 4
Solution:
Step 1: Write 4 as 4/1.
Step 2: Multiply: 3/8 x 4/1 = (3 x 4) / (8 x 1) = 12/8
Step 3: Simplify: 12/8 = 3/2 = 1 1/2
Answer: 3/8 x 4 = 1 1/2
Example 3: Using Cross-Cancellation
Problem: Find: 5/12 x 8/15
Solution:
Step 1: Before multiplying, look for common factors to cancel.
Step 2: 5 (numerator of first) and 15 (denominator of second) share a factor of 5. Cancel: 5/5 = 1 and 15/5 = 3.
Step 3: 8 (numerator of second) and 12 (denominator of first) share a factor of 4. Cancel: 8/4 = 2 and 12/4 = 3.
Step 4: Now multiply the simplified fractions: 1/3 x 2/3 = 2/9
Answer: 5/12 x 8/15 = 2/9
Example 4: Multiplying Mixed Numbers
Problem: Find: 2 1/4 x 1 1/3
Solution:
Step 1: Convert mixed numbers to improper fractions: 2 1/4 = (2 x 4 + 1)/4 = 9/4 and 1 1/3 = (1 x 3 + 1)/3 = 4/3
Step 2: Multiply: 9/4 x 4/3
Step 3: Cross-cancel: 9 and 3 share factor 3 (9/3=3, 3/3=1). 4 and 4 share factor 4 (4/4=1, 4/4=1).
Step 4: Multiply after cancellation: 3/1 x 1/1 = 3
Answer: 2 1/4 x 1 1/3 = 3
Example 5: Word Problem: Recipe
Problem: A recipe calls for 2/3 of a cup of rice. Meera wants to make 3/4 of the recipe. How much rice does she need?
Solution:
Step 1: She needs 3/4 of 2/3 cup. "Of" means multiply.
Step 2: 3/4 x 2/3 = (3 x 2) / (4 x 3) = 6/12
Step 3: Simplify: 6/12 = 1/2
Answer: Meera needs 1/2 cup of rice.
Example 6: Word Problem: Area of a Rectangle
Problem: A rectangular garden is 5/6 m long and 3/4 m wide. Find its area.
Solution:
Step 1: Area = length x width = 5/6 x 3/4
Step 2: Cross-cancel: 3 and 6 share factor 3 (3/3=1, 6/3=2).
Step 3: Multiply: 5/2 x 1/4 = 5/8
Answer: The area of the garden is 5/8 m2.
Example 7: Word Problem: Fraction of a Group
Problem: In a class of 40 students, 3/5 are girls. Of the girls, 1/4 play basketball. How many girls play basketball?
Solution:
Step 1: Number of girls = 3/5 of 40 = 3/5 x 40 = (3 x 40)/5 = 120/5 = 24
Step 2: Girls who play basketball = 1/4 of 24 = 1/4 x 24 = 24/4 = 6
Answer: 6 girls play basketball.
Alternative: 3/5 x 1/4 x 40 = 3/20 x 40 = 120/20 = 6. Same answer!
Example 8: Multiplying a Fraction by its Reciprocal
Problem: What is 7/9 x 9/7?
Solution:
Step 1: 9/7 is the reciprocal of 7/9.
Step 2: Multiply: 7/9 x 9/7 = (7 x 9) / (9 x 7) = 63/63 = 1
Answer: 7/9 x 9/7 = 1. A fraction multiplied by its reciprocal always equals 1.
Example 9: Word Problem: Sharing Pizza
Problem: Aarav has 3/4 of a pizza. He gives 2/3 of what he has to his sister. How much of the whole pizza does his sister get?
Solution:
Step 1: Sister gets 2/3 of 3/4 = 2/3 x 3/4
Step 2: Cross-cancel: 3 and 3 cancel to give 1 and 1.
Step 3: Multiply: 2/1 x 1/4 = 2/4 = 1/2
Answer: Aarav's sister gets 1/2 of the whole pizza.
Example 10: Word Problem: Pocket Money
Problem: Sneha gets Rs. 500 as pocket money. She saves 2/5 of it. Of the savings, she puts 1/2 in a piggy bank. What fraction of her total pocket money goes into the piggy bank? How much money is that?
Solution:
Step 1: Fraction saved = 2/5. Fraction into piggy bank = 1/2 of 2/5 = 1/2 x 2/5 = 2/10 = 1/5
Step 2: Amount in piggy bank = 1/5 of Rs. 500 = 500/5 = Rs. 100
Answer: 1/5 of her pocket money goes into the piggy bank, which is Rs. 100.
Real-World Applications
Multiplication of fractions is used widely in everyday life:
Cooking and Recipes: When you want to make half a recipe or double a recipe, you multiply each ingredient quantity by the desired fraction. If a recipe needs 3/4 cup of flour and you want to make 2/3 of the recipe, you calculate 2/3 x 3/4 = 1/2 cup. This is probably the most common real-life use of fraction multiplication for students your age.
Shopping and Discounts: A "1/4 off" sale means you pay 3/4 of the original price. If a shirt costs Rs. 800 and there is a 1/4 discount, the price you pay is 3/4 x 800 = Rs. 600. Understanding fraction multiplication helps you quickly calculate how much you save during sales.
Area Calculations: When the length and width of a rectangle are given as fractions, you multiply them to find the area. A plot that is 3/4 km long and 2/5 km wide has area = 3/4 x 2/5 = 6/20 = 3/10 km2. Architects and builders frequently work with fractional measurements.
Probability: In probability (which you will study in detail later), the probability of two independent events both happening is the product of their individual probabilities. If the probability of rain on Monday is 1/3 and on Tuesday is 1/4, the probability of rain on both days is 1/3 x 1/4 = 1/12.
Science: In science experiments, fractional quantities are multiplied. If a plant grows 3/8 cm per day, in 5 days it grows 5 x 3/8 = 15/8 = 1 7/8 cm. Chemical solutions, medicine dosages, and experiment measurements often involve fractions.
Time and Work: If a worker can finish 1/5 of a job in one day, then in 3 days the worker completes 3 x 1/5 = 3/5 of the job. Work problems in maths and real life regularly use fraction multiplication.
Maps and Scale: Maps use scale factors that are fractions. If a map has a scale of 1/50000, then a distance of 3 cm on the map represents 3 x 50000 = 150000 cm = 1.5 km in real life. Understanding fraction multiplication helps in reading maps and models.
Sports Statistics: A cricket team wins 3/4 of its matches. Out of 20 matches, the number of wins is 3/4 x 20 = 15 matches. Sports analysts use fractions to calculate win rates, batting averages, and performance metrics.
Key Points to Remember
- To multiply two fractions, multiply numerators together and denominators together: a/b x c/d = (a x c)/(b x d).
- To multiply a fraction by a whole number, treat the whole number as a fraction with denominator 1.
- To multiply mixed numbers, first convert them to improper fractions, then multiply.
- Cross-cancellation (simplifying before multiplying) makes calculations easier and avoids large numbers.
- A fraction multiplied by its reciprocal always equals 1.
- Any fraction multiplied by 0 is 0. Any fraction multiplied by 1 is the same fraction.
- The product of two proper fractions (both less than 1) is always smaller than either fraction.
- The word "of" in word problems usually means multiplication.
- Always simplify your final answer to the lowest terms.
Practice Problems
- Find: 4/7 x 5/9
- Find: 3/8 x 6
- Find: 7/10 x 5/14 (use cross-cancellation)
- Find: 1 2/5 x 2 1/3
- Riya has 5/6 of a chocolate bar. She eats 1/2 of it. What fraction of the whole bar did she eat?
- A farmer uses 3/4 of his land for crops. Of the crop land, 2/5 is used for wheat. What fraction of the total land is used for wheat?
- Find the area of a rectangle with length 7/8 m and width 4/7 m.
- Amit gets Rs. 600 as pocket money. He spends 2/3 of it. Of what he spends, 1/4 goes to stationery. How much does he spend on stationery?
Frequently Asked Questions
Q1. How do you multiply two fractions?
To multiply two fractions, multiply the numerators (top numbers) together to get the new numerator, and multiply the denominators (bottom numbers) together to get the new denominator. For example, 2/3 x 4/5 = (2x4)/(3x5) = 8/15. Then simplify the result if possible.
Q2. What does 'of' mean in fraction problems?
'Of' means multiply. When a problem says '1/2 of 3/4', it means 1/2 x 3/4 = 3/8. This is one of the most important things to remember in fraction word problems. Whenever you see 'of' between a fraction and a number, multiply them.
Q3. How do you multiply a fraction by a whole number?
Write the whole number as a fraction with denominator 1, then multiply as usual. For example, 3/5 x 4 = 3/5 x 4/1 = 12/5 = 2 2/5. You are essentially multiplying the numerator by the whole number and keeping the denominator the same.
Q4. What is cross-cancellation in fraction multiplication?
Cross-cancellation is a shortcut where you simplify before multiplying. You can divide any numerator and any denominator by their common factor. For example, in 4/9 x 3/8, you can cancel the 4 and 8 (both divisible by 4: 4->1, 8->2) and the 3 and 9 (both divisible by 3: 3->1, 9->3). This gives 1/3 x 1/2 = 1/6, which is much easier to calculate.
Q5. How do you multiply mixed numbers?
First, convert each mixed number to an improper fraction. Then multiply the improper fractions as usual. Finally, convert the answer back to a mixed number if needed. For example, 2 1/3 x 1 1/2 = 7/3 x 3/2 = 21/6 = 7/2 = 3 1/2.
Q6. Is the product of two proper fractions always smaller than the original fractions?
Yes! When you multiply two proper fractions (both less than 1), the product is always smaller than either of the original fractions. For example, 1/2 x 1/3 = 1/6, and 1/6 is less than both 1/2 and 1/3. This makes sense because you are finding a fraction of a fraction, which is a smaller part.
Q7. 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 3/5 is 5/3. A fraction multiplied by its reciprocal always equals 1: 3/5 x 5/3 = 15/15 = 1. The reciprocal is especially important in division of fractions.
Q8. Can you multiply more than two fractions at once?
Yes! Multiply all the numerators together and all the denominators together. For example, 1/2 x 2/3 x 3/4 = (1x2x3)/(2x3x4) = 6/24 = 1/4. You can also cross-cancel before multiplying to make the calculation easier.
