Working with Fractions

WORKING WITH FRACTIONS — Class 7 Ganita Prakash

8.1 MULTIPLICATION OF FRACTIONS

Understanding Multiplication with Fractions

**What is multiplication of fractions?**

Multiplication is used to find totals when we have equal groups or parts.

Just like multiplying whole numbers (5 × 3 = 15), we can multiply fractions the same way.

**Key Idea**: When distance or quantity is given as a fraction, multiply it by the number of times to find the total.

**Real-life Example 1**: Aaron walks 3 km in 1 hour. In 5 hours, he walks 5 × 3 = 15 km. This is just repeated addition: 3 + 3 + 3 + 3 + 3 = 15.

**Real-life Example 2**: Aaron's tortoise walks 1/4 km in 1 hour. In 3 hours, it walks 3 × (1/4) = 1/4 + 1/4 + 1/4 = 3/4 km.

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Multiplying a Whole Number by a Fraction

**Method**: To find (whole number) × (fraction), we:

1. **Divide** the whole number by the denominator of the fraction

2. **Multiply** the result by the numerator of the fraction

**Formula**: n × (a/b) = (n × a)/b OR (n/b) × a

**Step-by-Step Example 1**: 5 × (2/3)

Step 1: Divide 5 by 3 → 5/3

Step 2: Multiply by 2 → 2 × (5/3) = 10/3

Answer: 10/3 acres (from the farmer example with 5 grandchildren)

**Step-by-Step Example 2**: 1 hour of internet costs ₹8. How much does 1¼ hours cost?

Convert 1¼ to improper fraction: 1¼ = 5/4 hours

Cost = (5/4) × 8 = (5 × 8)/4 = 40/4 = 10

Answer: ₹10

**Key Rule**: When multiplying a fraction by a whole number, multiply the numerator by the whole number and keep the denominator the same.

(a/b) × n = (a × n)/b

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Multiplying a Fraction by Another Fraction

**Visual Method Using Unit Square**:

A unit square represents "1 whole"

To find (1/2) × (1/4), shade 1/4 of the square, then divide that into 2 equal parts

Count the final shaded area as a fraction of the whole

Result: 1/2 × 1/4 = 1/8

**Step-by-Step Example**: Find 3/4 × 2/5 (tortoise walks 2/5 km in 1 hour; how far in 3/4 hour?)

Step 1: Divide the whole into 5 rows and 4 columns = 5 × 4 = 20 equal parts

Step 2: Shade 2 out of 5 rows = 2/20 of the whole

Step 3: Take 3 of these shaded columns = 3 × 2/20 = 6/20

Step 4: Simplify = 6/20 = 3/10 km

**Connection to Area of Rectangle**:

If a rectangle has sides of length 1/2 unit and 1/4 unit, its area = 1/2 × 1/4 = 1/8 square units

**Important**: The area of a rectangle with fractional sides equals the product of its sides

This shows multiplication of fractions geometrically

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Multiplying Fractional Units (1/b × 1/d)

**Rule for Unit Fractions**:

When multiplying two unit fractions (fractions with numerator 1):

(1/b) × (1/d) = 1/(b × d)

The denominator is the product of the denominators

**Example**: (1/12) × (1/18) = 1/(12 × 18) = 1/216

**Why this works**:

The unit square is divided into 12 columns and 18 rows

Total parts = 12 × 18 = 216

Only 1 part is shaded

So the answer is 1/216

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Brahmagupta's Formula for Multiplying Fractions (628 CE)

**General Rule**:

**a/b × c/d = (a × c)/(b × d)**

Multiply the numerators together

Multiply the denominators together

Simplify if possible

**This formula works for**:

Whole numbers (write as fraction with denominator 1): 3 × (3/4) = (3/1) × (3/4) = 9/4

Fractions multiplied by fractions: (5/12) × (7/18) = 35/216

Mixed numbers (convert to improper fractions first)

**Step-by-Step Example**: (5/12) × (7/18)

Numerator: 5 × 7 = 35

Denominator: 12 × 18 = 216

Answer: 35/216

**Historical Note**: Brahmagupta's Brāhmasphuṭasiddhānta (628 CE) is one of the earliest texts to state this formula in general form.

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Simplifying Before Multiplying (Cancelling Common Factors)

**Method**: Before using Brahmagupta's formula, look for common factors between any numerator and any denominator, and divide them out.

**Example 1**: (12/7) × (5/24)

12 and 24 have a common factor of 12

(12/7) × (5/24) = (1 × 5)/(7 × 2) = 5/14

We divided 12 by 12 (= 1) and 24 by 12 (= 2)

**Example 2**: (14/15) × (25/42)

14 and 42 have a common factor of 14

25 and 15 have a common factor of 5

(14/15) × (25/42) = (1 × 5)/(3 × 3) = 5/9

We cancelled: 14÷14=1, 25÷5=5, 15÷5=3, 42÷14=3

**Why cancel?**: This makes the numbers smaller and easier to work with. It avoids having to simplify a large fraction at the end.

**A Pinch of History**: The process of reducing fractions to lowest terms is called **apavartana** in Sanskrit. This term was so well-known in ancient India that even philosopher Umasvati (c. 150 CE) used it as a simile in non-mathematical works!

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Order of Multiplication (Commutative Property)

**Rule**: 1/2 × 1/4 = 1/4 × 1/2 = 1/8

**In general**: a/b × c/d = c/d × a/b

**Why it works**:

A rectangle's area stays the same even if length and breadth are swapped

Brahmagupta's formula shows: (a × c)/(b × d) = (c × a)/(d × b) — multiplication is commutative

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Is the Product Always Greater?

**Important Discovery**:

When we multiply numbers, the product's relationship to the original numbers depends on whether those numbers are:

**Greater than 1** or **Between 0 and 1**

**Situation 1: Both numbers > 1**

Example: (4/3) × 4 = 16/3 ≈ 5.33

Product (16/3) > both (4/3) and 4

**Rule**: Product is **greater** than both numbers

**Situation 2: Both numbers between 0 and 1**

Example: (3/4) × (2/5) = 6/20 = 3/10

3/10 is less than both 3/4 and 2/5

**Rule**: Product is **less** than both numbers

**Situation 3: One number > 1, one between 0 and 1**

Example: (3/4) × 5 = 15/4 = 3.75

3.75 is less than 5 but greater than 3/4

**Rule**: Product is **between** the two numbers

**Key Conclusions**:

If one number is between 0 and 1, the product is **less** than the other number

If one number is greater than 1, the product is **greater** than the other number

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8.2 DIVISION OF FRACTIONS

Understanding Division as Inverse of Multiplication

**Basic Concept**:

Division can be rewritten as a multiplication problem

If 4 × ? = 12, then 12 ÷ 4 = ?

The question mark (?) is what we're looking for in division

**Dividend ÷ Divisor = Quotient**

**Connection**: Divisor × Quotient = Dividend

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The Reciprocal of a Fraction

**Definition**: The **reciprocal** of a/b is b/a

The reciprocal flips the numerator and denominator

Examples:

Reciprocal of 2/3 is 3/2

Reciprocal of 5/4 is 4/5

Reciprocal of 1/8 is 8/1 = 8

**Key Property**: When you multiply a fraction by its reciprocal, you get 1

(2/3) × (3/2) = 6/6 = 1

(5/7) × (7/5) = 35/35 = 1

a/b × b/a = 1 ✓

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Dividing Fractions Using Reciprocals

**Method to Divide Fractions**:

1. Find the **reciprocal** of the divisor

2. **Multiply** the dividend by this reciprocal

**Brahmagupta's Division Formula**:

**a/b ÷ c/d = a/b × d/c = (a × d)/(b × c)**

**Step-by-Step Example 1**: 1 ÷ 2/3

Rewrite as multiplication: 2/3 × ? = 1

Reciprocal of 2/3 is 3/2

1 ÷ 2/3 = 1 × 3/2 = 3/2

Check: (2/3) × (3/2) = 6/6 = 1 ✓

**Step-by-Step Example 2**: 3 ÷ 2/3

Rewrite: 2/3 × ? = 3

We know (2/3) × (3/2) = 1

So we need to multiply by 3: (3/2) × 3 = 9/2

Check: (2/3) × (9/2) = 18/6 = 3 ✓

**Step-by-Step Example 3**: 1/5 ÷ 1/2

Rewrite: 1/2 × ? = 1/5

Reciprocal of 1/2 is 2/1 = 2

1/5 ÷ 1/2 = 1/5 × 2 = 2/5

Check: (1/2) × (2/5) = 2/10 = 1/5 ✓

**Step-by-Step Example 4**: 2/3 ÷ 3/5

Reciprocal of 3/5 is 5/3

2/3 ÷ 3/5 = 2/3 × 5/3 = (2 × 5)/(3 × 3) = 10/9

Check: (3/5) × (10/9) = 30/45 = 2/3 ✓

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Relationship Between Divisor, Dividend, and Quotient

**Key Discovery**: Unlike division of whole numbers, the quotient can be larger than the dividend!

**Example 1**: 6 ÷ 3 = 2

Quotient (2) < Dividend (6)

This is what we usually see with whole numbers

**Example 2**: 6 ÷ 1/4 = 24

6 × 4 = 24

Quotient (24) > Dividend (6)!

Why? Because we're dividing 6 into tiny pieces

**Example 3**: 1/8 ÷ 1/4 = 1/2

(1/8) × (4/1) = 4/8 = 1/2

Quotient (1/2) > Dividend (1/8)!

**Pattern**:

When divisor < 1 (a fraction), the quotient is **greater** than the dividend

When divisor > 1, the quotient is **less** than the dividend

When divisor = 1, the quotient = dividend

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8.3 SOME PROBLEMS INVOLVING FRACTIONS

Real-World Application Problems

**Example 3**: Making Tea

**Problem**: Leena made 5 cups of tea using 1/4 litre of milk. How much milk is in each cup?

**Solution**:

Total milk used in 5 cups = 1/4 litre

Milk per cup = 1/4 ÷ 5

Rewrite: 5 × (milk per cup) = 1/4

Using Brahmagupta's method:

Reciprocal of 5 is 1/5

Milk per cup = 1/5 × 1/4 = 1/20 litre

**Answer**: 1/20 litre per cup

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**Example 4**: Ancient Geometry (From Baudhāyana's Śhulbasūtra, c. 800 BCE)

**Problem**: Cover an area of 7½ square units with square bricks. Each brick has sides of 1/5 units. How many bricks are needed?

**Solution**:

Area of one brick = (1/5) × (1/5) = 1/25 square unit

Total area to cover = 7½ = 15/2 square units

Number of bricks needed:

(Number of bricks) × (Area of one brick) = Total area

Number of bricks = 15/2 ÷ 1/25

Using Brahmagupta's formula:

Reciprocal of 1/25 is 25

15/2 ÷ 1/25 = 15/2 × 25 = (15 × 25)/2 = 375/2 = 187.5 bricks

**Answer**: 187.5 or 187½ square bricks (historically important geometry problem!)

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Water Tank Problem

**Problem**: A tap fills 7/10 of a tank in 1 hour. How much gets filled in 3/4 hour?

**Solution**:

In 1 hour: 7/10 of tank

In 3/4 hour: (3/4) × (7/10)

= (3 × 7)/(4 × 10)

= 21/40 of the tank

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Land Division Problem

**Problem**: The government took 1/6 of Somu's land for a road. She gives half of the remaining land to her daughter Krishna. What fraction of the original land did Krishna get?

**Solution**:

Original land = 1 whole

Land taken by government = 1/6

Remaining land = 1 - 1/6 = 5/6

Krishna gets = 1/2 of remaining = (1/2) × (5/6) = 5/12 of original land

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Rectangle Area Problem

**Problem**: Find the area of a rectangle with sides 3¾ ft and 9⅗ ft.

**Solution**:

Convert to improper fractions:

3¾ = 15/4 ft

9⅗ = 48/5 ft

Area = length × width

Area = (15/4) × (48/5)

Cancel common factors: 15 and 5 have factor 5; 4 and 48 have factor 4

= (3/1) × (12/1) = 36 square feet

**Answer**: 36 square feet

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Sapling Distance Problem

**Problem**: Four saplings are planted in a row. The distance between any two consecutive saplings is 3/4 m. What is the distance between the first and last sapling?

**Solution**:

4 saplings → 3 gaps between them

Each gap = 3/4 m

Total distance = 3 × (3/4) = 9/4 = 2¼ m

**Answer**: 2¼ metres

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Comparison Problem

**Problem**: Which is heavier: 12/15 of 500 grams OR 3/20 of 4 kg?

**Solution**:

First quantity: (12/15) × 500

= (12 × 500)/15

= 6000/15

= 400 grams

Second quantity: (3/20) × 4000 (converting 4 kg to 4000 g)

= (3 × 4000)/20

= 12000/20

= 600 grams

**Answer**: 3/20 of 4 kg is heavier (600 g > 400 g)

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KEY FORMULAS SUMMARY

**Multiplying Whole by Fraction**: n × (a/b) = (n × a)/b

**Brahmagupta's Multiplication Formula**: (a/b) × (c/d) = (a × c)/(b × d)

**Reciprocal**: Reciprocal of a/b = b/a

**Division of Fractions**: (a/b) ÷ (c/d) = (a/b) × (d/c) = (a × d)/(b × c)

**Commutative Property**: (a/b) × (c/d) = (c/d) × (a/b)

**Unit Fractions**: (1/b) × (1/d) = 1/(b × d)

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IMPORTANT RULES TO REMEMBER

✓ **Before multiplying fractions, cancel common factors** between any numerator and denominator

✓ **Convert mixed numbers to improper fractions** before multiplying or dividing

✓ **To divide by a fraction, multiply by its reciprocal**

✓ **The product of a fraction and its reciprocal is always 1**

✓ **When multiplying by a fraction < 1, the product is smaller than the original number**

✓ **When dividing by a fraction < 1, the quotient is larger than the original number**

✓ **Always simplify your final answer to lowest terms**

✓ **Check division answers by multiplying back**: Divisor × Quotient should = Dividend

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HISTORICAL CONNECTIONS

**Brahmagupta (628 CE)**: Mathematician who gave the general formula for multiplying and dividing fractions in his work **Brāhmasphuṭasiddhānta**. These formulas are still used today exactly as he stated them!

**Apavartana** (Ancient Sanskrit Term): The process of reducing fractions to lowest terms. This term was so important it was used even in philosophical texts.

**Baudhāyana's Śhulbasūtra (c. 800 BCE)**: One of humanity's oldest geometry texts, containing practical problems about covering areas with square tiles using fractions.

**Umasvati (c. 150 CE)**: Jaina scholar who used the concept of reducing fractions as a simile in philosophical works, showing how important this concept was in ancient Indian mathematics and culture.

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COMMON MISTAKES TO AVOID

❌ **Mistake 1**: Multiplying denominators when you should simplify first

Right: (12/7) × (5/24) → cancel 12 and 24 → 5/14

Wrong: (12/7) × (5/24) = 60/168 (doesn't simplify nicely)

❌ **Mistake 2**: Forgetting to flip the fraction when dividing

Right: (2/3) ÷ (3/5) = (2/3) × (5/3) = 10/9

Wrong: (2/3) ÷ (3/5) = (2/3) × (3/5) = 6/15 ✗

❌ **Mistake 3**: Not converting mixed numbers to improper fractions

Right: 1¼ × (2/3) = (5/4) × (2/3) = 10/12 = 5/6

Wrong: 1¼ × (2/3) = 1 × (2/3) + ¼ × (2/3) (makes it complicated)

❌ **Mistake 4**: Forgetting to check if a fraction can be simplified

Wrong: Leaving answer as 6/20 instead of 3/10

Right: Always reduce to lowest terms

❌ **Mistake 5**: Assuming product is always larger

Wrong: (1/2) × (1/3) should give something > 1/2

Right: (1/2) × (1/3) = 1/6, which is smaller than both 1/2 and 1/3