7. Fractions

Fractional Units and Equal Shares

Chapter 7: Fractions

Page 1 of 6: Fractional Units and Equal Shares

Concept Introduction

Imagine you and your friend order a large pizza. The pizza arrives as one big, complete circle—a whole. To share it, you cut it exactly down the middle into two equal pieces. Each piece is a fraction of the whole pizza. In this case, each of you gets 1/2 of the pizza.

Now, what if three more friends join you? To share the same pizza, you would need to cut it into five equal slices. Each person's share would now be 1/5 of the pizza. Notice that when more people share, each person's slice gets smaller. This simple idea is the key to understanding fractions: they are all about equal shares of a whole.

{{FORMULA: expr=Fraction = Numerator / Denominator | symbols=Numerator:Number of parts we have, Denominator:Total number of equal parts the whole is divided into}}

Definitions & Key Terms

The entire concept of fractions is built on a few simple terms. Understanding them clearly is the first step to mastering this chapter.

The Logic of Equal Shares

Understanding why 1/2 is bigger than 1/4, or why 1/5 is bigger than 1/9, comes from the idea of sharing. Let's break down the logic.

1. Start with a Whole: Let's think of one whole roti. This is our unit, represented by the number 1.

2. Divide and Share:

   • To get the fraction 1/5, you must divide the roti into 5 equal parts and take one part.
   • To get the fraction 1/9, you must divide the same roti into 9 equal parts and take one part.

3. Analyze the Denominator: The denominator tells you how many people (or parts) are sharing the whole.

   • A denominator of 5 means 5 people are sharing.
   • A denominator of 9 means 9 people are sharing.

4. Compare the Shares: If you share one roti among more people, each person will get a smaller piece. Sharing with 9 people means the slices have to be much smaller than if you only shared with 5 people.

5. The Conclusion: Therefore, the share 1/9 is smaller than the share 1/5.

{{KEY: type=concept | title=The Denominator Rule for Unit Fractions | text=When comparing two fractional units (fractions with 1 as the numerator), the fraction with the smaller denominator is the larger fraction. More shares mean smaller pieces!}}

Solved Examples

Example 1: Identifying a Fraction (Easy)

Given: A circle is divided into 8 equal parts, and 5 of those parts are coloured green.

To Find: What fraction of the circle is coloured green?

Solution:

1. First, identify the total number of equal parts the whole is divided into. This will be our denominator. The circle is divided into 8 equal parts. So, Denominator = 8.

2. Next, identify the number of parts we are considering (the coloured parts). This will be our numerator. There are 5 coloured parts. So, Numerator = 5.

3. Write the fraction by placing the numerator over the denominator.

   Fraction = Numerator / Denominator
   

   Fraction = 5/8
   

Final Answer: The fraction of the circle coloured green is 5/8.

Example 2: Comparing Fractional Units (Medium)

Given: Ajay has a chocolate bar. On Monday, he shares it equally among 4 friends (5 people in total including him). On Tuesday, he gets an identical chocolate bar and shares it equally just with his sister (2 people in total).

To Find: On which day did Ajay get a larger piece of chocolate?

Solution:

1. Determine the fraction of chocolate Ajay gets on Monday. The chocolate is shared among 5 people (Ajay + 4 friends). Ajay's share is one part out of five.

   Monday's Share = 1/5
   

2. Determine the fraction of chocolate Ajay gets on Tuesday. The chocolate is shared between 2 people (Ajay + his sister). Ajay's share is one part out of two.

   Tuesday's Share = 1/2
   

3. Compare the two fractional units, 1/5 and 1/2. Remember the rule: for unit fractions, the one with the smaller denominator is larger. Since 2 is smaller than 5, the fraction 1/2 is larger than 1/5.

   1/2 > 1/5
   

Final Answer: Ajay got a larger piece of chocolate on Tuesday.

Example 3: Finding the Weight of a Fractional Unit (Hard)

Given: A fruit seller has 1 kg of grapes. He divides them into 4 packets of equal weight to sell.

To Find: The weight of the grapes in each packet in kg.

Solution:

1. Identify the "whole". The whole is the total quantity of grapes. Whole = 1 kg.

2. Identify the number of equal parts the whole is divided into. This becomes the denominator. The grapes are divided into 4 equal packets.

3. The weight of one packet is one part of the whole. This is a fractional unit. The fraction representing one packet is 1/4 of the total weight.

4. Calculate the weight of one packet.

   Weight of one packet = 1/4 of 1 kg
   

   Weight = 1/4 kg
   

Final Answer: The weight of each packet of grapes is 1/4 kg.

Example 4: Understanding Parts of a Whole (Tricky)

Given: A long ribbon is cut into 6 equal pieces. The picture shows two of those pieces.

{{VISUAL: diagram: A drawing of two small, equal-length pieces of ribbon placed end-to-end, with a label underneath saying "These are 2 pieces out of 6".}}

To Find: What fraction of the original ribbon do these two pieces represent?

Solution:

1. Identify the fractional unit. The original ribbon was cut into 6 equal pieces. So, each single piece is 1/6 of the ribbon. The fractional unit is 1/6.

2. Count how many fractional units we have. We have 2 pieces of the ribbon.

3. Express the total amount as a collection of fractional units. We have 2 times the fractional unit 1/6.

   Total Fraction = 2 × (1/6)
   

4. The numerator represents the number of parts we have (2), and the denominator represents the total parts the whole was cut into (6).

   Fraction = 2/6
   

Final Answer: The two pieces represent 2/6 of the original ribbon.

Tips & Tricks

Common Mistakes to Avoid

Brain-Teaser Questions

1. A pizza is cut into a certain number of equal slices. My share is 1/6 of the pizza. My friend's share from an identical pizza is 1/8. If we both put our slices on a plate, what fraction of a single pizza do we have together? (Hint: Think in terms of adding slices, not fractions yet)

   💡 Answer: This is a trick question! The slices come from different pizzas that were cut differently. You cannot simply add them. To combine them, you would need to find a common way to slice both pizzas, a concept you will learn later. The question tests if you assume you can just add 1/6 and 1/8.

2. If 1/a is a smaller fraction than 1/b, what can you say for sure about the whole numbers a and b?

   💡 Answer: If 1/a is a smaller share than 1/b, it means the whole was divided into more pieces for 1/a. Therefore, the number a must be greater than the number b. (a > b).

3. Three friends, Ram, Sam, and Tim, equally share a large watermelon. So, each gets 1/3. Just as they are about to eat, two more friends, Ali and Joe, arrive. They decide to re-divide the same watermelon equally among all five of them. Does Ram's share of watermelon increase or decrease?

   💡 Answer: Ram's share decreases. Initially, his share was 1/3. After re-dividing for five people, his new share becomes 1/5. Since 1/3 is greater than 1/5, his share got smaller.

Mini Cheatsheet

Fractional Units as Parts of a Whole

Chapter 7: Fractions

Page 2 of 6: Fractional Units as Parts of a Whole

Concept Introduction

Imagine you have a large bar of chocolate to share equally with three of your friends. That means four people (you and three friends) need to get the same amount. What do you do? You break the chocolate bar into four equal pieces. Each of those equal pieces is a fraction of the whole bar.

In mathematics, a fraction represents a part of a whole. The 'whole' can be a single object, like our chocolate bar, a pizza, or a strip of paper. It can also be a collection of objects, like a bag of 10 marbles. The most important rule is that to create fractions, the whole must be divided into equal parts. Each equal part is a fractional unit. If our chocolate bar is broken into 4 equal pieces, the fractional unit is one-fourth (¼). If you get one piece, you have ¼ of the chocolate.

{{FORMULA: expr=Fraction = Numerator / Denominator | symbols=Numerator:Number of parts considered, Denominator:Total number of equal parts the whole is divided into}}

Definitions & Key Terms

Here are the basic building blocks for understanding fractions.

Understanding a Fraction's Structure

How is a fraction like ¾ actually built? It's not just "3 over 4". A better way to think about it is "3 times ¼". This helps us understand its size. Let's build this idea using a paper strip.

1. Start with the Whole. Take a single strip of paper. We will call its length "1 whole unit".

   [--------------------------------]  (1 Whole)
   

2. Create the Fractional Unit. Fold the strip into four equal parts. Each part is our fractional unit. Since there are four equal parts, the fractional unit is one-quarter, or ¼.

   [---------|---------|---------|---------]
      ¼         ¼         ¼         ¼
   

{{VISUAL: diagram: A paper strip shown in three stages. Stage 1: "1 Whole". Stage 2: The strip shown folded with creases, labeled "Folded into 4 equal parts". Stage 3: The strip unfolded, showing four segments, each labeled "¼".}}

3. Count the Fractional Units. Now, let's count these units. If we take one part, we have 1 × ¼ = ¼. If we take two parts, we have 2 × ¼ = ²/₄. If we take three parts, we have 3 × ¼ = ¾.

4. Express the Fraction. The fraction ¾ simply means we have collected three parts, where each part is one-fourth of the whole.

   [=========|=========|=========|---------]  (This shaded length is ¾)
   

This shows that the Denominator (4) sets the size of the piece (a quarter-sized piece), and the Numerator (3) sets the number of those pieces we have.

{{KEY: type=concept | title=The Power of the Unit Fraction | text=Every fraction is a multiple of a unit fraction. To understand ⅗, first understand ⅕. Then, simply take five of them! Reading ⅚ as "5 times ⅙" makes its size much clearer than just saying "five by six".}}

Solved Examples

Let's apply these concepts to solve some problems, from easy to tricky.

Example 1: Identifying a Simple Fraction (Easy)

Given: A circle is divided into 8 equal sectors, and 5 of them are coloured blue.

To Find: What fraction of the circle is coloured blue?

Solution:

1. First, identify the 'whole'. The whole is the entire circle.

2. Next, find the total number of equal parts the whole is divided into. The circle is divided into 8 equal sectors. This is our denominator.

   Denominator = 8
   

3. Now, count the number of parts we are interested in (the blue parts). There are 5 blue sectors. This is our numerator.

   Numerator = 5
   

4. Write the fraction by placing the numerator over the denominator.

   Fraction = Numerator / Denominator = 5/8
   

{{VISUAL: diagram: A circle divided into 8 equal pizza-like slices. 5 of the slices are shaded blue. A label points to the whole circle saying "1 Whole". A label points to one slice saying "Fractional Unit = ⅛". A label points to the shaded area saying "5 parts = ⅝".}}

Final Answer: The fraction of the circle coloured blue is ⅝.

Example 2: Expressing as Multiples of a Unit Fraction (Medium)

Given: A rectangle is divided into 6 equal parts. 4 parts are shaded.

To Find: Express the shaded portion as a multiplication of a fractional unit.

Solution:

1. Determine the total number of equal parts. The rectangle is divided into 6 equal parts.

2. This means the fractional unit is one part out of six.

   Fractional Unit = 1/6
   

3. Count how many of these fractional units are shaded. There are 4 shaded parts.

4. Therefore, the shaded fraction can be expressed as 4 collections of the unit fraction ⅙.

   Shaded Fraction = 4 × (1/6)
   

5. This is equal to the fraction ⁴/₆.

   4 × (1/6) = 4/6
   

Final Answer: The shaded portion is 4 times ⅙, or ⁴/₆.

Example 3: Word Problem with Fractions (Hard)

Given: Priya bakes a rectangular cake and cuts it into 12 equal square pieces for a party. During the party, 7 pieces are eaten.

To Find: What fraction of the cake is left?

Solution:

1. The 'whole' is the entire cake.

2. The total number of equal pieces the cake was cut into is 12. This will be the denominator for our fractions.

   Total Parts (Denominator) = 12
   

3. The number of pieces eaten is 7. So, the fraction of the cake eaten is ⁷/₁₂.

4. To find the fraction of cake left, we first need to find the number of pieces left.

   Pieces Left = Total Pieces - Pieces Eaten = 12 - 7 = 5
   

5. The number of pieces left is 5. This is the numerator for the fraction of cake that is left.

   Numerator (Left) = 5
   

6. Now, we form the fraction for the remaining cake.

   Fraction Left = (Pieces Left) / (Total Pieces) = 5/12
   

{{VISUAL: diagram: A rectangular grid representing a cake, divided into a 3x4 grid, making 12 small equal squares. 7 of the squares are greyed out and labeled "Eaten (⁷/₁₂)". The remaining 5 squares are white and labeled "Left (⁵/₁₂)".}}

Final Answer: The fraction of the cake left is ⁵/₁₂.

Example 4: Fractions of Different Shapes (Tricky)

Given: A square is divided into 4 equal triangles by its diagonals. A circle of the same size is divided into 4 equal quadrants. One part of the square (a triangle) and one part of the circle (a quadrant) are shaded.

To Find: Does the shaded triangle represent the same fraction of the square as the shaded quadrant represents of the circle?

Solution:

1. Analyze the square. The whole is the square. It is divided into 4 equal parts (triangles).

2. The shaded portion is 1 of these 4 equal parts.

   Fraction of Square Shaded = 1/4
   

3. Analyze the circle. The whole is the circle. It is divided into 4 equal parts (quadrants).

4. The shaded portion is 1 of these 4 equal parts.

   Fraction of Circle Shaded = 1/4
   

5. Compare the two fractions. Both fractions are ¼. Even though the shapes of the shaded parts are different (a triangle vs. a quadrant), they each represent the same fractional part of their respective wholes. The concept of a fraction depends on the number of equal parts, not the shape of those parts.

Final Answer: Yes, the shaded portions represent the same fraction (¼) for each of their respective wholes.

Tips & Tricks

Use these shortcuts to master fractions quickly.

Common Mistakes

Many students make these simple errors. Here’s how to avoid them.

Brain-Teaser Questions

1. Rohan eats half (½) of a cake. Then, his sister Shefali eats half of what is left. What fraction of the original cake did Shefali eat?

   💡 Answer: After Rohan ate ½, there was ½ of the cake left. Shefali ate half of the remaining half. Half of a half is a quarter (½ × ½ = ¼). So, Shefali ate ¼ of the original cake.

2. Look at a standard chessboard. It has 64 squares in an 8x8 grid. What fraction of the squares are black?

   💡 Answer: A chessboard has 64 squares in total. Half are white and half are black. So, there are 32 black squares. The fraction is ³²/₆₄. (You will later learn this simplifies to ½).

3. The NCERT textbook asks, "How many fractions lie between 0 and 1?" Can you name five fractions that are bigger than 0 but smaller than 1? Can there be a biggest fraction that is still less than 1?

   💡 Answer: There are infinite fractions between 0 and 1. Examples include ½, ¼, ¾, ⅕, ⁹/₁₀. There is no "biggest" fraction that is less than 1. You can always find a bigger one (e.g., ⁹⁹/₁₀₀ is big, but ⁹⁹⁹/₁₀₀₀ is even bigger and still less than 1).

Solved Numericals

This section provides extra practice with a clear, step-by-step format.

Hero Formula: Fraction = (Number of parts we are considering) / (Total number of equal parts in the whole)

Numerical Example 1

A farmer divides his rectangular field into 10 equal plots. He plants wheat in 6 of the plots. What fraction of the field is planted with wheat?

• GIVEN:

  • Total number of equal plots = 10
  • Number of plots with wheat = 6

• FORMULA: Fraction = (Parts with wheat) / (Total parts)

• SUBSTITUTION:

  Fraction = 6 / 10
  

• ANSWER: The fraction of the field planted with wheat is ⁶/₁₀.

Numerical Example 2

An egg tray contains 30 eggs. If 11 eggs are found to be broken, what fraction of the eggs are not broken?

• GIVEN:

  • Total number of eggs = 30
  • Number of broken eggs = 11

• FORMULA: First find the number of eggs that are not broken. Not Broken Eggs = Total Eggs - Broken Eggs Then use the fraction formula: Fraction = (Not Broken Eggs) / (Total Eggs)

• SUBSTITUTION:

  Number of Not Broken Eggs = 30 - 11 = 19
  

  Now substitute this into the fraction formula:

  Fraction = 19 / 30
  

• ANSWER: The fraction of eggs that are not broken is ¹⁹/₃₀.

Try It Yourself

1. In a class of 25 students, 14 are girls. What fraction of the class are boys?
2. A full fuel tank in a car holds 40 litres. If the gauge shows it is exactly ¾ full, how many litres of fuel are in the tank? (Hint: Find what ¼ of 40 is first).
3. What fraction of a day is 8 hours? (Hint: How many hours are in a whole day?)

Answer Key:

1. ¹¹/₂₅
2. 30 litres
3. ⁸/₂₄

Mini Cheatsheet

Measuring Using Fractional Units

Page 3: Measuring Using Fractional Units

Concept Introduction

Imagine you have a large bar of chocolate to share with your friends. If you want to give a piece to your friend Rohan, how do you describe the amount he gets? You can't just say "a piece," because that could be big or small!

First, you decide to break the chocolate bar into 8 equal squares. Now, each small square is a clear, standard size. This single, equal piece is our fractional unit. In this case, it's one-eighth (1/8) of the whole chocolate bar.

If you give Rohan 3 of these squares, he has 3 times the fractional unit, which means he has 3 times 1/8. We write this as the fraction 3/8. This simple idea—building bigger fractions by counting smaller, equal units—is the key to understanding what fractions really mean.

Definitions & Key Terms

Every fraction is built from a few simple parts. Understanding these parts is crucial for mastering fractions.

{{KEY: type=concept | title=The Golden Rule of Fractions | text=A fraction a/b can always be read as "a times 1/b". This helps you see that 5/9 is simply five "one-ninth" pieces put together.}}

The Logic: How Fractions are Built

Understanding fractions is like building with LEGOs. You start with one standard block (the fractional unit) and combine them to create something bigger.

1. Start with a Whole. Imagine one complete object, like a full pizza, a strip of paper, or the number 1 on a number line. This is our 'whole' or 'one unit'.

2. Divide into Equal Parts. We cut or divide the whole into a specific number of perfectly equal pieces. Let's say we divide it into b equal parts. The number b becomes the denominator of our fraction.

3. Identify the Fractional Unit. Each one of these new, smaller pieces is the fractional unit. Its value is 1/b. For example, if we cut a pizza into 8 slices, the fractional unit is one slice, or 1/8 of the pizza.

   {{VISUAL: diagram: A rectangular bar divided into 5 equal parts. The first part is labeled '1/5' (the fractional unit). The first three parts are shaded and bracketed together, labeled '3/5' or '3 times 1/5'.}}

4. Count the Units. Now, we simply count how many of these fractional units we want to talk about. Let's say we take a of these pieces. This count, a, becomes the numerator.

5. Combine to Form the Fraction. When we take a pieces, each of size 1/b, we are putting them together. The total amount is a times 1/b.

6. Write the Final Fraction. We write this combination in a compact form: a/b. So, 3 pieces of a pizza cut into 8 slices is 3 times 1/8, written as 3/8.

Solved Examples

Let's apply this concept to solve some problems, starting from easy and moving to more challenging ones.

Example 1: Identifying a Fraction from a Shape (Easy)

Given: A circle is divided into 8 equal sectors. 5 of these sectors are coloured blue.

To Find: The fraction representing the blue part of the circle.

Solution:

1. Identify the total number of equal parts. This will be the denominator. The circle is divided into 8 equal parts. So, the denominator is 8.

2. Identify the fractional unit. Each part represents 1/8 of the circle.

3. Count the number of parts being considered (the coloured parts). This will be the numerator. There are 5 blue sectors. So, the numerator is 5.

4. Write the fraction by placing the numerator over the denominator. The fraction is 5 on top of 8.

   Fraction = 5/8
   

Final Answer: The fraction representing the blue part is 5/8.

Example 2: Expressing a Quantity as a Fraction (Medium)

Given: A gardener has a bag of 12 seeds. He plants 7 of them.

To Find: What fraction of the seeds did the gardener plant? Express this in terms of its fractional unit.

Solution:

1. The 'whole' is the total number of seeds, which is 12. This defines our denominator. The fractional unit is 1/12.

2. The number of parts taken from the whole is the number of seeds planted, which is 7. This is our numerator.

3. The fraction is the number of seeds planted divided by the total number of seeds.

   Fraction = 7/12
   

4. To express this in terms of its fractional unit, we can say it is 7 times the fractional unit (1/12).

   7/12 = 7 × (1/12)
   

Final Answer: The gardener planted 7/12 of the seeds, which is 7 times the fractional unit of 1/12.

Example 3: Fractions on a Number Line (Hard)

Given: A number line where the distance from 0 to 1 is divided into 4 equal parts.

To Find: Mark the position of the fraction 7/4 on this number line.

Solution:

1. First, understand the fractional unit. The space between 0 and 1 is one whole unit, and it's divided into 4 parts. Therefore, each mark represents the fractional unit 1/4.

2. The first mark after 0 is 1/4, the second is 2/4, the third is 3/4, and the fourth mark is 4/4, which is exactly 1.

3. We need to find 7/4. This means we need to count 7 times the fractional unit of 1/4 starting from 0.

4. Let's count:

   • 1st mark: 1/4
   • 2nd mark: 2/4
   • 3rd mark: 3/4
   • 4th mark: 4/4 (at position 1)
   • 5th mark: 5/4
   • 6th mark: 6/4
   • 7th mark: 7/4

5. We can see that 7/4 is greater than 1. It is the third mark after the number 1.

Final Answer: To mark 7/4, you go past 1 to the third small marking after it. It is located at 1 + 3/4.

Example 4: A Word Problem with Fractional Units (Tricky)

Given: Anju reads 3/10 of a book on Monday and 4/10 of the same book on Tuesday.

To Find: What fraction of the book has Anju read in total? Express the total as a sum of its fractional units.

Solution:

1. Identify the fractional unit. The denominator is 10, so the whole book is thought of as being divided into 10 equal parts. The fractional unit is 1/10.

2. Express Monday's reading in terms of fractional units. Anju read 3/10, which means she read 3 parts of size 1/10. Monday = 1/10 + 1/10 + 1/10

3. Express Tuesday's reading in terms of fractional units. Anju read 4/10, which means she read 4 parts of size 1/10. Tuesday = 1/10 + 1/10 + 1/10 + 1/10

4. Calculate the total number of fractional units read. Total units = (Units from Monday) + (Units from Tuesday)

   Total units = 3 + 4 = 7
   

5. The total fraction is the total number of units (7) over the original denominator (10).

   Total Fraction = 7/10
   

Final Answer: Anju has read 7/10 of the book in total. This is the sum of seven fractional units of 1/10.

Tips & Tricks

Use these shortcuts to think about fractions more clearly and quickly.

Common Mistakes to Avoid

Many students make small errors when first learning fractions. Here’s what to watch out for.

Brain-Teaser Questions

1. A cake is cut into 12 equal slices. If 3/4 of the cake has been eaten, how many slices are left?

💡 Answer: 3/4 of 12 is (12 ÷ 4) × 3 = 3 × 3 = 9 slices eaten. So, 12 - 9 = 3 slices are left.

2. On a number line, a point 'A' is exactly halfway between 1/5 and 3/5. What fraction represents point 'A'?

💡 Answer: The fraction exactly in the middle of 1/5 and 3/5 is 2/5. You are just counting the fractional units: one-fifth, two-fifths, three-fifths.

3. I have a fraction where the numerator is 5. When I add this fraction to 2/7, I get a total of 1 (or 7/7). What is my original fraction?

💡 Answer: The total needed is 7/7. We already have 2/7. So we need 5 more "one-sevenths". The missing fraction is 5/7. The problem states the numerator is 5, which matches. So the fraction is 5/7.

Mini Cheatsheet

Mixed Fractions

Page 4: Mixed Fractions

Concept Introduction

Imagine you're at a birthday party and there are 5 small pizzas to be shared equally among 2 friends. How much pizza does each friend get?

If you give each friend 2 whole pizzas, you will have used 4 pizzas. There is still 1 pizza left! You would cut that last pizza in half and give one half to each friend. So, each friend gets 2 whole pizzas and half a pizza. We write this as 2 ½ pizzas.

This number, 2 ½, is a mixed fraction. It combines a whole number (2) and a proper fraction (½). It's a convenient way to talk about quantities that are more than one whole but not an exact whole number. These situations arise from improper fractions—fractions where the numerator is larger than the denominator, like 5/2.

Definitions & Formulas

Understanding the language of mixed fractions is the first step to mastering them. Here are the key terms and the formulas that connect them.

{{FORMULA: expr=(Whole Number × Denominator) + Numerator | symbols=Conversion from Mixed to Improper Fraction}}

Logic: From Improper to Mixed and Back

Let's understand why these conversions work. It's all about grouping wholes.

Converting an Improper Fraction (e.g., 8/3) to a Mixed Fraction

An improper fraction like 8/3 means you have 8 pieces of something that was originally cut into 3 equal parts (thirds). How many whole things can you make?

1. Recall the meaning of a whole: We know that 3 thirds make one whole.

   3/3 = 1
   

2. Group the wholes: From our 8 thirds, we can take out one group of 3/3.

   8/3 = 3/3 + 5/3 = 1 + 5/3
   

3. Group again: We can take out another group of 3/3 from the remaining 5/3.

   8/3 = 3/3 + 3/3 + 2/3
   

4. Combine the wholes: We have two groups of 3/3, which means we have 2 wholes.

```
8/3 = 1 + 1 + 2/3
```

5. Write as a mixed fraction: This simplifies to 2 wholes and a remainder of 2/3.

```
8/3 = 2 ²/₃
```
This is exactly what division does: 8 ÷ 3 gives a quotient of 2 (the whole part) and a remainder of 2 (the numerator of the fractional part).

{{VISUAL: diagram: A number line from 0 to 3. The points 1, 2, and 3 are marked. The fraction 8/3, which is 2 ²/₃, is shown as a point two-thirds of the way between 2 and 3.}}

Converting a Mixed Fraction (e.g., 2 ²/₃) to an Improper Fraction

Now, let's reverse the process. How many "thirds" are in 2 ²/₃?

1. Break down the whole part: The number 2 means two complete wholes.

   2 = 1 + 1
   

2. Express wholes in terms of the fraction: Since our fractional part is in "thirds," we need to express each whole as thirds.

   1 = 3/3
   

3. Substitute the fractional wholes: Replace each 1 with 3/3.

   2 ²/₃ = (3/3 + 3/3) + 2/3
   

4. Count all the parts: Now, just add up all the numerators. You have 3 + 3 + 2 thirds.

   (3 + 3 + 2) / 3 = 8/3
   

   This is the same as the formula: (2 × 3) + 2 = 8, giving us the improper fraction 8/3.

Solved Examples

Example 1: Converting an Improper Fraction to a Mixed Fraction (Easy)

Given: The improper fraction 13/4.

To Find: Its equivalent mixed fraction.

Solution:

1. Divide the numerator (13) by the denominator (4).

   13 ÷ 4
   

2. Identify the quotient and the remainder. 4 goes into 13 three times (4 × 3 = 12), with 1 left over.

   • Quotient = 3
   • Remainder = 1

3. Construct the mixed fraction. The quotient becomes the whole number, the remainder becomes the new numerator, and the denominator stays the same.

   Whole Number = 3
   New Numerator = 1
   Denominator = 4
   

Final Answer:

13/4 = 3 ¹/₄

Example 2: Converting a Mixed Fraction to an Improper Fraction (Medium)

Given: The mixed fraction 5 ³/₇.

To Find: Its equivalent improper fraction.

Solution:

1. Use the conversion formula: (Whole Number × Denominator) + Numerator.

2. Multiply the whole number (5) by the denominator (7).

   5 × 7 = 35
   

3. Add the original numerator (3) to this result.

   35 + 3 = 38
   

4. This result (38) is the new numerator. The denominator stays the same (7).

Final Answer:

5 ³/₇ = 38/7

Example 3: Word Problem Application (Hard)

Given: A baker uses 21 cups of flour to bake 5 identical cakes.

To Find: How much flour is in each cake? Express the answer as a mixed fraction.

Solution:

1. Set up the problem as a division. The total amount of flour (21 cups) is divided by the number of cakes (5). This gives the fraction 21/5.

   Flour per cake = 21/5 cups
   

2. This is an improper fraction. To express it as a mixed fraction, divide the numerator by the denominator.

   21 ÷ 5
   

3. Find the quotient and remainder. 5 goes into 21 four times (5 × 4 = 20), with 1 left over.

   • Quotient = 4
   • Remainder = 1

4. Form the mixed fraction. The whole part is 4, the fractional part is ¹/₅.

Final Answer:

Each cake uses 4 ¹/₅ cups of flour.

{{KEY: type=concept | title=Interpreting Fractions as Division | text=Remember that the fraction a/b is another way of writing the division a ÷ b. This is key to solving word problems. "21 cups for 5 cakes" immediately translates to 21/5 cups per cake.}}

Example 4: Comparison of Fractions (Tricky)

Given: Two fractions, 19/6 and 3 ¹/₃.

To Find: Which fraction is greater?

Solution:

1. To compare them easily, both fractions should be in the same format. Let's convert the mixed fraction 3 ¹/₃ into an improper fraction.

2. Apply the conversion formula: (Whole × Denominator) + Numerator.

   (3 × 3) + 1 = 9 + 1 = 10
   

3. The improper fraction is 10/3.

4. Now we must compare 19/6 and 10/3. To do this, we need a common denominator. The least common multiple (LCM) of 6 and 3 is 6.

5. The fraction 19/6 already has the denominator 6. Convert 10/3 to an equivalent fraction with a denominator of 6.

   (10 × 2) / (3 × 2) = 20/6
   

6. Compare the numerators of the two fractions: 19/6 and 20/6. Since 20 > 19, the fraction 20/6 is greater.

Final Answer:

20/6 > 19/6, so 3 ¹/₃ is greater than 19/6.

Tips & Tricks

Use these shortcuts to speed up your calculations during exams.

Common Mistakes

Many students make these simple errors. See if you can spot the difference!

Brain-Teaser Questions

1. A recipe calls for 4 ½ cups of sugar. You only have a ¹/₂ cup measuring scoop. How many scoops will you need to add?

   💡 Answer: First, convert 4 ½ to an improper fraction: (4 × 2) + 1 = 9. So, 9/2 cups are needed. Since each scoop is ¹/₂, you will need 9 scoops.

2. I am a mixed fraction. My whole part is 5. My denominator is 7. When I am converted to an improper fraction, my numerator is 39. What is my original fractional part?

   💡 Answer: Let the mixed fraction be 5 ˣ/₇. Converting this gives ((5 × 7) + x) / 7 = (35 + x) / 7. We are told this equals 39/7. So, 35 + x = 39, which means x = 4. The fractional part was ⁴/₇.

3. What is the next number in this sequence? 1 ¹/₅, 2 ²/₅, 3 ³/₅, ...

   💡 Answer: The whole number part is increasing by 1 each time (1, 2, 3, ...). The fractional part's numerator is also increasing by 1 (¹, ², ³, ...), while the denominator stays fixed at 5. The next term would be 4 ⁴/₅.

Mini Cheatsheet

Equivalent Fractions — Part 1

Chapter 7: Fractions

Page 5 of 6: Equivalent Fractions — Part 1

Concept Introduction

Imagine you and your friend buy two identical medium pizzas. You are very hungry and eat half of your pizza. Your friend cuts their pizza into four equal slices and eats two of them. Who ate more pizza?

Let's think about it. You ate 1/2 of your pizza. Your friend ate 2/4 of theirs. Even though the numbers look different, a quick look shows you both ate the exact same amount! The empty space in both pizza boxes is identical.

This is the magic of equivalent fractions. They are fractions that look different but represent the exact same value or amount of a whole. Understanding them is like having a secret code to describe the same quantity in many different ways.

{{FORMULA: expr=a/b = (a × n) / (b × n) | symbols=a:numerator, b:denominator, n:any non-zero whole number}}

Definitions & Formulas

Let's formally define the key terms we'll be using on this page.

The Logic: Why Are They "Equivalent"?

How can 1/2 be the same as 2/4? It seems like they have different numbers. The secret lies in visualizing what the fractions represent. Let's use a simple paper strip to understand this.

1. Start with a Whole. Imagine a single strip of paper. This represents one whole, or the number 1.

2. Creating Halves. Fold this strip exactly in the middle and open it. You now have two equal parts. Shading one of these parts represents the fraction 1/2. The shaded portion is one part out of two total parts.

3. Creating Fourths. Now, take an identical, new strip of paper. Fold it in the middle, and then fold it in the middle again. When you open it, you will see four equal parts.

4. Comparing the Shaded Areas. If you shade two of these four parts, you are representing the fraction 2/4.

5. The "Aha!" Moment. Place the two paper strips side-by-side. You will see that the shaded length of 1/2 on the first strip is exactly the same as the shaded length of 2/4 on the second strip.

{{VISUAL: diagram: A fraction wall showing the equivalence of 1/2, 2/4, 3/6, and 4/8 on stacked, coloured bars.}}

This visual proof shows that 1/2 = 2/4. What did we actually do mathematically? To go from 1/2 to 2/4, we multiplied both the numerator and the denominator by 2.

(1 × 2) / (2 × 2) = 2/4

This is the fundamental rule: Multiplying or dividing the numerator and the denominator of a fraction by the same non-zero number gives an equivalent fraction. You're not changing the fraction's value, just the number of pieces it's described with.

{{KEY: type=concept | title=The Golden Rule of Equivalence | text=To find an equivalent fraction, whatever you do to the numerator, you MUST do the exact same thing to the denominator. This keeps the fraction's value balanced and unchanged.}}

Solved Examples

Let's practice finding and checking equivalent fractions with some examples, from easy to tricky.

Example 1: Finding the Next Three Equivalent Fractions (Easy)

Given: The fraction 2/3.

To Find: The next three equivalent fractions.

Solution:

1. To find the first equivalent fraction, we can multiply both the numerator and denominator by 2.

   (2 × 2) / (3 × 2) = 4/6
   

2. To find the second one, let's multiply the original fraction's numerator and denominator by 3.

   (2 × 3) / (3 × 3) = 6/9
   

3. For the third one, we can multiply the original fraction's numerator and denominator by 4.

   (2 × 4) / (3 × 4) = 8/12
   

Final Answer:

The next three equivalent fractions for 2/3 are 4/6, 6/9, and 8/12.

Example 2: Checking for Equivalence (Medium)

Given: Two fractions, 5/8 and 15/24.

To Find: Are these fractions equivalent?

Solution:

1. We need to see if we can get from the first fraction to the second by multiplying the numerator and denominator by the same number.

2. Let's look at the numerators. How do we get from 5 to 15? We multiply by 3.

   5 × 3 = 15
   

3. Now, let's check if the same multiplier works for the denominators. Does multiplying 8 by 3 give us 24?

   8 × 3 = 24
   

4. Since we multiplied both the numerator and the denominator by the same number (3) to get the second fraction, they are equivalent.

Final Answer:

Yes, 5/8 and 15/24 are equivalent fractions.

Example 3: Finding a Missing Value (Hard)

Given: The equivalence 4/9 = ? / 36.

To Find: The missing numerator.

Solution:

1. First, look at the parts we know: the denominators. We need to figure out what number 9 was multiplied by to get 36.

2. We can find this by dividing 36 by 9.

   36 ÷ 9 = 4
   

3. So, the denominator was multiplied by 4. To keep the fractions equivalent, we must do the same to the numerator.

4. Multiply the original numerator 4 by the same number, 4.

   4 × 4 = 16
   

5. The missing numerator is 16.

Final Answer:

The missing numerator is 16. The complete equivalence is 4/9 = 16/36.

Example 4: Real-World Equivalence Problem (Tricky)

Given: A recipe for one cake needs 3/4 litre of milk. A baker is making a much larger batch and uses 12/16 litre of milk.

To Find: Did the baker use the correct proportion of milk for the larger batch, or did they make a mistake?

Solution:

1. The problem is asking if the fraction 3/4 is equivalent to the fraction 12/16.

2. Let's see if we can transform 3/4 into 12/16. We'll start with the numerators. How do we get from 3 to 12? We multiply by 4.

   3 × 4 = 12
   

3. Now, we must apply the same multiplication to the denominator. Let's see if we get the correct new denominator, which is 16.

   4 × 4 = 16
   

4. Since multiplying both the top and bottom of 3/4 by 4 gives us 12/16, the fractions are equivalent. The baker used the correct proportion.

Final Answer:

Yes, the baker used the correct proportion of milk because 3/4 is equivalent to 12/16.

Tips & Tricks

Use these shortcuts to master equivalent fractions faster.

Common Mistakes

Many students make these simple errors. Look out for them!

Brain-Teaser Questions

Ready for a challenge? Try these questions that require a bit more thinking.

1. Find a fraction equivalent to 3/7 where the sum of the numerator and the denominator is 60.
2. An artist painted 15/25 of a large canvas blue. What is the simplest form of the fraction of the canvas that is not painted blue?
3. Are the fractions 4/6 and 6/9 equivalent? Prove your answer without using cross-multiplication.

💡 Answer 1: The original fraction is 3/7. The sum of its numerator and denominator is 3 + 7 = 10. We need a sum of 60. Since 60 ÷ 10 = 6, we must multiply both parts of the original fraction by 6. (3 × 6) / (7 × 6) = 18/42. (Check: 18 + 42 = 60). The fraction is 18/42.

💡 Answer 2: If 15/25 is painted blue, then the part not painted is 25/25 - 15/25 = 10/25. To simplify 10/25, we find the greatest common factor, which is 5. (10 ÷ 5) / (25 ÷ 5) = 2/5. So, 2/5 of the canvas is not painted blue.

💡 Answer 3: Yes, they are equivalent. We can prove this by simplifying both fractions to their simplest form. For 4/6, the common factor is 2: (4 ÷ 2) / (6 ÷ 2) = 2/3. For 6/9, the common factor is 3: (6 ÷ 3) / (9 ÷ 3) = 2/3. Since both fractions simplify to the same fraction (2/3), they are equivalent to each other.

Mini Cheatsheet

Here's a quick summary of everything on this page. Screenshot this for your last-minute revision!

Equivalent Fractions — Part 2

Chapter 7: Fractions

Page 6 of 6: Equivalent Fractions — Part 2

Concept Introduction

Imagine you and your friend buy two pizzas of the same size for a movie night. You cut your pizza into 2 large slices and eat 1 of them. Your friend, who likes smaller pieces, cuts their pizza into 8 slices and eats 4. Who ate more pizza?

Even though you ate 1/2 and your friend ate 4/8, a quick look shows you both ate the exact same amount! This is the magic of equivalent fractions. They are fractions that look different because they use different numbers, but they represent the very same value or portion of a whole. Just like ₹1 is the same as 100 paise, 1/2 is the same as 4/8. Understanding this helps us compare, add, and subtract fractions easily.

{{FORMULA: expr=a/b = (a × k) / (b × k) | symbols=a:numerator, b:denominator, k:any non-zero number}}

Definitions & Formulas

This table breaks down the key terms we'll use to master equivalent fractions.

The Logic Behind Equivalence

Why does multiplying or dividing the top and bottom of a fraction by the same number keep its value unchanged? The logic comes from the idea of splitting and grouping.

1. Start with a fraction. Let's take the fraction 1/2. This means we have 1 part out of 2 total equal parts.

   {{VISUAL: diagram: A circle split into two equal halves, with one half shaded, representing 1/2.}}

2. Choose a multiplier. Let's pick the number 3. We will multiply both the numerator and the denominator by 3.

(1 × 3) / (2 × 3)

3. Interpret the multiplication. Multiplying the denominator 2 by 3 means we are splitting each of the original 2 parts into 3 smaller, equal pieces. So, the total number of pieces in the whole is now 2 × 3 = 6.

4. Interpret the numerator. Multiplying the numerator 1 by 3 means that our original single shaded part is also split into 3 smaller pieces. So, the number of shaded parts is now 1 × 3 = 3.

5. Form the new fraction. We now have 3 shaded parts out of a total of 6 parts. This gives us the new fraction 3/6.

6. Conclusion. The total shaded area hasn't changed, only the number of slices it's made of. Therefore, the value remains the same. This shows that 1/2 is equivalent to 3/6. The same logic applies to division, which is like grouping smaller pieces into larger ones.

{{KEY: type=concept | title=The Multiplicative Identity | text=Multiplying or dividing the numerator and denominator by the same number k is the same as multiplying the fraction by k/k. Since any number divided by itself (like 3/3 or 5/5) is equal to 1, you are essentially multiplying the fraction by 1. Multiplying by 1 never changes a number's value, which is why the fraction remains equivalent.}}

Solved Examples

Example 1: Finding Equivalent Fractions (Easy)

Given: The fraction 3/5.

To Find: The first three equivalent fractions of 3/5.

Solution:

1. To find equivalent fractions, we'll use the Golden Rule: multiply the numerator and denominator by the same whole number (starting with 2, then 3, then 4).

2. First equivalent fraction (multiply by 2):

(3 × 2) / (5 × 2) = 6/10

3. Second equivalent fraction (multiply by 3):

(3 × 3) / (5 × 3) = 9/15

4. Third equivalent fraction (multiply by 4):

(3 × 4) / (5 × 4) = 12/20

Final Answer:

The first three equivalent fractions of 3/5 are 6/10, 9/15, and 12/20.

Example 2: Checking for Equivalence (Medium)

Given: Two fractions, 4/7 and 12/21.

To Find: Determine if the two fractions are equivalent.

Solution:

1. We can check for equivalence by seeing if we can get from one fraction to the other by multiplication. Let's start with 4/7.

2. Ask: What number do we multiply the numerator 4 by to get 12? The answer is 3, because 4 × 3 = 12.

3. Now, apply the same multiplier to the denominator 7:

7 × 3 = 21

4. This matches the denominator of the second fraction. Since we can multiply both the numerator and the denominator of 4/7 by the same number (3) to get 12/21, the fractions are equivalent.

Final Answer:

Yes, 4/7 and 12/21 are equivalent fractions.

Example 3: Finding a Missing Value (Hard)

Given: The equivalence 5/8 = ? / 32.

To Find: The missing numerator.

Solution:

1. First, look at the denominators to find the relationship between the two fractions. We need to find the number that multiplies 8 to give 32.

2. Perform the division to find the multiplier:

32 ÷ 8 = 4

3. The multiplier is 4. According to the Golden Rule, we must multiply the numerator of the first fraction by the same number.

4. Apply the multiplier to the numerator 5:

5 × 4 = 20

5. Therefore, the missing numerator is 20.

Final Answer:

The missing numerator is 20, making the equivalence 5/8 = 20/32.

Example 4: Real-World Application (Tricky)

Given: In Class 6A, 18 out of 30 students are boys. In Class 6B, 21 out of 35 students are boys.

To Find: Which class has a greater fraction of boys, or are they the same?

Solution:

1. First, write the fraction of boys for each class.

   • Class 6A: 18/30
   • Class 6B: 21/35

2. To compare these fractions, it's best to reduce them to their simplest form. We do this by dividing the numerator and denominator by their greatest common factor (GCF).

3. Simplify the fraction for Class 6A. The GCF of 18 and 30 is 6.

(18 ÷ 6) / (30 ÷ 6) = 3/5

4. Simplify the fraction for Class 6B. The GCF of 21 and 35 is 7.

(21 ÷ 7) / (35 ÷ 7) = 3/5

5. Both fractions simplify to 3/5. This means the original fractions 18/30 and 21/35 are equivalent.

Final Answer:

Both classes have the same fraction of boys (3/5).

Solved Numericals

This section provides worked-out problems typical of CBSE exams, focusing on the core calculation skills for this topic.

Hero Formula(s):

• To find a "larger" equivalent fraction: a/b = (a × k) / (b × k)
• To find the "simplest form": a/b = (a ÷ k) / (b ÷ k) where k is the Greatest Common Factor (GCF).

Numerical 1

GIVEN: A water bottle contains 3/4 litres of water.

FORMULA: a/b = (a × k) / (b × k)

SUBSTITUTION: We need to express this quantity in terms of twelfths of a litre (i.e., with a denominator of 12).

• First, find the multiplier k by seeing how the denominator changes from 4 to 12.
• k = 12 ÷ 4 = 3
• Now, apply this multiplier to the numerator.
• New numerator = 3 × 3 = 9
• So, 3/4 = 9/12.

ANSWER: The water bottle contains 9/12 litres of water.

Numerical 2

GIVEN: A baker used 12 kg of flour from a 16 kg bag.

FORMULA: a/b = (a ÷ k) / (b ÷ k)

SUBSTITUTION: We want to find the fraction of flour used in its simplest form.

• The initial fraction is 12/16.
• To simplify, we need the Greatest Common Factor (GCF) of 12 and 16.
• Factors of 12: 1, 2, 3, 4, 6, 12
• Factors of 16: 1, 2, 4, 8, 16
• The GCF (k) is 4.
• Now, divide both the numerator and denominator by 4.
• 12 ÷ 4 = 3
• 16 ÷ 4 = 4

ANSWER: The baker used 3/4 of the bag of flour.

Try It Yourself

1. Check if 7/9 and 28/36 are equivalent fractions. (Yes/No)
2. Find the missing denominator: 2/3 = 14 / ?
3. A team won 15 out of 25 games. What fraction of games did they win, in simplest form?

(Answers are at the very end of the page)

Tips & Tricks

Use these shortcuts to check for equivalence quickly and accurately.

Common Mistakes

Many students make these simple errors. Pay close attention to avoid them!

Brain-Teaser Questions

1. I am a fraction equivalent to 4/7. My denominator is 15 more than my numerator. What fraction am I?

💡 Answer: The equivalent fractions of 4/7 are 8/14, 12/21, 16/28, etc. The difference between the denominator and numerator in 4/7 is 7-4 = 3. In an equivalent fraction (4×k)/(7×k), the difference will be 7k - 4k = 3k. We are told this difference is 15. So, 3k = 15, which means k = 5. The fraction is (4×5)/(7×5) = 20/35.

2. A large rectangle is shaded with the fraction 48/64. Without changing the total shaded area, you want to redraw the lines so there are only 8 parts in total. What will the new fraction be?

💡 Answer: This is a simplification problem. You need to find a fraction equivalent to 48/64 with a denominator of 8. We need to find k such that 64 ÷ k = 8. So, k = 64 ÷ 8 = 8. Now, divide the numerator by the same number: 48 ÷ 8 = 6. The new fraction is 6/8.

3. Priya says that to find a fraction equivalent to 2/3, you can add the numerator and denominator to themselves, like (2+2)/(3+3) = 4/6. Her friend says this is wrong because you can only multiply. Who is correct and why?

💡 Answer: Priya is correct in her result (4/6 is equivalent to 2/3), but her reasoning is flawed and will not always work. Adding the numbers to themselves is the same as multiplying by 2: 2+2 = 2×2 and 3+3 = 3×2. So her method accidentally worked in this specific case. Her friend is correct about the general rule: only multiplication or division by the same non-zero number is the universally correct method.

Mini Cheatsheet

Answer Key for Try It Yourself: 1. Yes | 2. 21 | 3. 3/5