Fractions are everywhere in our daily life – from dividing a pizza among friends to measuring ingredients for cooking. In Class 7, we study how to multiply and divide fractions, which are imp skills that help us solve many real-world problems. These notes will teach you step-by-step methods to work with fractions easily, along with historical facts about how great mathematicians like Brahmagupta developed these concepts in ancient India.
Multiplication of Fractions
Introduction to Fraction Multiplication
When we study mathematics, we often come across situations where we need to multiply fractions. Let me explain this with a simple example that will help you understand the concept better.
Aaron walks 3 kilometres in 1 hour. If we want to find the distance he covers in 5 hours, we simply multiply 5 × 3 = 15 km. This is quite straightforward when we work with whole numbers. But what happens when the distance per hour is a fraction?
Aaron’s tortoise walks 1/4 km in 1 hour. Now, to find the distance covered in 3 hours, we need to calculate 3 × 1/4 = 3/4 km. This is how we multiply a whole number with a fraction.
Multiplying Whole Number by Fraction
Let’s understand this concept with another example. Aaron covers a distance of 3 km in 1 hour. So, the distance covered by Aaron in 1/5 hours will be 1/5 × 3 km.
To calculate this, we divide 3 km into 5 equal parts, which gives us 3/5 km. Therefore, 1/5 × 3 = 3/5.
Similarly, the distance covered in 2/5 hours = 2/5 × 3 = 6/5 km.
Method for Multiplying Fraction by Whole Number
Here’s the step-by-step method: • First divide the multiplicand by the denominator of the multiplier • Then multiply the result by the numerator of the multiplier
Example: 2/5 × 3 = (3 ÷ 5) × 2 = 3/5 × 2 = 6/5
Examples of Fraction-Whole Number Multiplication
Example 1: A farmer distributes 2/3 acre to each of his 5 grandchildren. Total land distributed = 5 × 2/3 = 10/3 acres
Example 2: Internet costs ₹8 per hour. Cost for 1 1/4 hours = 5/4 × 8 = 5 × 2 = ₹10
Multiplying Two Fractions
Now let’s study how to multiply two fractions together. Remember our tortoise that walks 1/4 km in 1 hour? What if we want to find the distance it covers in 1/2 hour?
Distance in 1/2 hour = 1/2 × 1/4 km
To solve this, we divide 1/4 into 2 equal parts, which gives us 1/8. Therefore, 1/2 × 1/4 = 1/8.
Method for Multiplying Two Fractions
The process is similar to multiplying a fraction by a whole number: • First divide the multiplicand by the denominator of the multiplier • Then multiply the result by the numerator of the multiplier
Example: 3/4 × 2/5 = (2/5 ÷ 4) × 3 = 2/20 × 3 = 6/20 = 3/10
Connection between Area and Fraction Multiplication
There’s an interesting connection between area calculation and fraction multiplication. When we find the area of a rectangle with fractional sides, it equals the product of its sides.
For instance, a rectangle with sides 1/2 and 1/4 has an area of 1/8 square units. This demonstrates that 1/2 × 1/4 = 1/8.
Multiplying Numerators and Denominators
The general formula for multiplying fractions is: a/b × c/d = (a × c)/(b × d)
This formula was first stated by the great mathematician Brahmagupta in 628 CE.
Example: 5/12 × 7/18 = (5 × 7)/(12 × 18) = 35/216
This formula works for whole numbers too by writing them as fractions with denominator 1.
Multiplication of Fractions – Simplifying to Lowest Form
Before multiplying, we can cancel common factors to make our calculations easier.
Example: 12/7 × 5/24 = (12 × 5)/(7 × 24)
We can cancel the common factor 12: = (1 × 5)/(7 × 2) = 5/14
This process is called cancelling common factors.
Historical Note on Fraction Reduction
The process of reducing fractions to their lowest terms was called “apavartana” in ancient India. This concept was mentioned even in philosophical works by the Jaina scholar Umasvati around 150 CE.
Questions and Answers – Multiplication
1. Tenzin drinks 1/2 glass of milk daily. How many glasses does he drink in a week and in January?
Week: 7 × 1/2 = 7/2 = 3 1/2 glasses January: 31 × 1/2 = 31/2 = 15 1/2 glasses
2. A team makes 1 km canal in 8 days. What is their daily work and weekly work?
Daily work: 1/8 km Weekly work (5 days): 5 × 1/8 = 5/8 km
3. Three families share 5 litres of oil weekly. What is each family’s share per week and in 4 weeks?
Weekly share: 5 ÷ 3 = 5/3 litres per family 4 weeks: 4 × 5/3 = 20/3 litres
4. Moon sets 5/6 hour later each day. If it sets at 10 pm on Monday, what time will it set on Thursday?
Thursday is 3 days later Total delay: 3 × 5/6 = 15/6 = 2 1/2 hours Moon sets at 12:30 am on Thursday
5. Multiply and convert to mixed fractions:
| Expression | Calculation | Answer |
|---|---|---|
| 7 × 3/5 | 21/5 | 4 1/5 |
| 4 × 1/3 | 4/3 | 1 1/3 |
| 9/7 × 6 | 54/7 | 7 5/7 |
| 13/11 × 6 | 78/11 | 7 1/11 |
Product Comparison with Original Numbers
Understanding how the product compares to the original numbers is very imp:
• When both numbers are greater than 1: product is greater than both numbers • When both numbers are between 0 and 1: product is less than both numbers • When one number is between 0 and 1 and other is greater than 1: product is between the two numbers
Imp Rules for Product Comparison
• When one number is between 0 and 1: product is less than the other number • When one number is greater than 1: product is greater than the other number
Order of Multiplication
The order doesn’t matter in multiplication: a/b × c/d = c/d × a/b
This property follows from Brahmagupta’s formula.
8.2 Division of Fractions
Converting Division to Multiplication
Division can be converted to multiplication by finding what number we need to multiply with the divisor to get the dividend.
For example, 12 ÷ 4 = 3 can be written as 4 × ? = 12
The same technique applies to fraction division: 1 ÷ 2/3 becomes 2/3 × ? = 1
The answer is 3/2 because 2/3 × 3/2 = 1
Finding Reciprocals
To solve 2/3 × ? = 1, we need the reciprocal of 2/3.
The reciprocal of 2/3 is 3/2. In general, the reciprocal of a/b is b/a.
When a fraction is multiplied by its reciprocal, the result is always 1.
Division Method
To divide two fractions: • Find the reciprocal of the divisor • Multiply the dividend by this reciprocal
Formula: a/b ÷ c/d = a/b × d/c = (a × d)/(b × c)
Examples of Division
| Division | Conversion to Multiplication | Answer |
|---|---|---|
| 3 ÷ 2/3 | 3 × 3/2 | 9/2 |
| 1/5 ÷ 1/2 | 1/5 × 2/1 | 2/5 |
| 2/3 ÷ 3/5 | 2/3 × 5/3 | 10/9 |
Historical Note on Division
Brahmagupta first stated the division formula in 628 CE. Later, Bhaskara II clarified it in terms of reciprocals in 1150 CE.
Relationship between Dividend, Divisor and Quotient
This relationship is opposite to multiplication: • When divisor is greater than 1: quotient is less than dividend • When divisor is between 0 and 1: quotient is greater than dividend
8.3 Some Problems Involving Fractions
Problem Examples
Example 3: Leena uses 1/4 litre milk for 5 cups of tea. Milk per cup = 1/4 ÷ 5 = 1/4 × 1/5 = 1/20 litre
Example 4: Cover 7 1/2 square units with square bricks of side 1/5 units. Area of each brick = 1/5 × 1/5 = 1/25 square units Total area = 15/2 square units Number of bricks = 15/2 ÷ 1/25 = 15/2 × 25 = 375/2 bricks
Example 5: Four fountains fill a cistern in 1, 1/2, 1/4, and 1/5 days respectively. Combined rate per day = 1 + 2 + 4 + 5 = 12 times Time to fill together = 1/12 days
Fractional Relations
Problems involving finding fractions of geometric shapes require identifying the whole unit and finding fractional parts step by step.
Example: shaded region = 3/4 × 1/8 = 3/32 of the whole square
Historical Problem from Lilavati
Bhaskara II posed a problem about a miser giving a fraction of a dramma. The fraction given was: 1/2 × 2/3 × 3/4 × 1/5 × 1/16 × 1/4 = 1/1280 dramma
Since 1 dramma = 1280 cowrie shells, the miser gave exactly 1 cowrie shell.
Currency System in 12th Century India
The currency system was well-organized: • Gold coins (dinars/gadyanas) for large transactions • Silver coins (drammas/tankas) for everyday use • Copper coins (kasus/panas) for smaller transactions • Cowrie shells for the smallest transactions
Questions and Answers – Division
1. Evaluate these divisions:
| Expression | Calculation | Answer |
|---|---|---|
| 3 ÷ 7/9 | 3 × 9/7 | 27/7 |
| 14/4 ÷ 2 | 14/4 × 1/2 | 7/4 |
| 2/3 ÷ 2/3 | 2/3 × 3/2 | 1 |
2. Word problems:
(a) Maria used 8m lace, 1/4m per bag: 8 ÷ 1/4 = 32 bags (b) 1/2m ribbon for 8 badges: 1/2 ÷ 8 = 1/16m per badge (c) 1/6 kg flour per loaf, 5kg total: 5 ÷ 1/6 = 30 loaves
3. Flour for rotis: 1/4 kg for 12 rotis, for 6 rotis = 1/4 × 6/12 = 1/8 kg
4. Sridharacharya’s problem: 1 ÷ 1/6 + 1 ÷ 1/10 + 1 ÷ 1/13 + 1 ÷ 1/9 + 1 ÷ 1/2 = 6 + 10 + 13 + 9 + 2 = 40
5. Mira’s novel: 400 pages, read 1/5 + 3/10 = 1/2, remaining = 200 pages
6. Car petrol: 16 km per litre, 2 3/4 litres = 11/4 × 16 = 44 km
7. Travel time: Train 5 1/6 hours, plane 1/2 hour, savings = 31/6 – 1/2 = 14/3 hours
8. Cake sharing: 4/5 eaten, 1/5 remains, shared by 3 friends = 1/5 ÷ 3 = 1/15 each
Historical Development of Fractions
Fractions developed largely in India from 800 BCE onwards. The Shulbasutra texts used fractions for altar construction. Brahmagupta codified modern fraction rules in 628 CE. This theory was transmitted to Arabs and then to Europe by the 17th century. Now fractions are indispensable in modern mathematics.
Brahmagupta’s Contributions
Multiplication: “Product of numerators divided by product of denominators” Division: “Interchange numerator and denominator of divisor, then multiply”
Bhaskara II later clarified division in terms of reciprocals.
Geometric Interpretation
Bhaskara I described multiplication geometrically. Division of a square into rectangles shows fraction multiplication visually. This visual method helps understand that 1/5 × 1/4 = 1/20.
Imp Formulas
| Operation | Formula |
|---|---|
| Multiplication | a/b × c/d = (a × c)/(b × d) |
| Division | a/b ÷ c/d = a/b × d/c = (a × d)/(b × c) |
| Reciprocal | Reciprocal of a/b is b/a |
| Cancellation | Common factors can be cancelled before multiplication |
Imp Properties
• Multiplication order: a/b × c/d = c/d × a/b • Product comparison: Depends on whether numbers are greater than 1 or between 0 and 1 • Division comparison: When divisor is between 0 and 1, quotient is greater than dividend • Reciprocal property: a/b × b/a = 1
Download Free Mind Map from the link below
This mind map contains all important topics of this chapter
Visit our Class 7 Maths page for free mind maps of all Chapters