Chapter 7 Fractions (Concepts)

Fractions and Related Terms

In earlier chapters, we have dealt with whole numbers ($0, 1, 2, ...$) and integers ($..., -2, -1, 0, 1, 2, ...$). These numbers represent 'complete' or 'whole' units. However, in real-life situations, we often need to represent or work with quantities that are only a part of a whole thing or a collection of things. For example, if you share a piece of cake, eat a few slices of a pizza, or describe a quantity that is less than a full unit, you are dealing with parts of a whole. To represent such parts mathematically, we use Fractions.

---

What is a Fraction?

A fraction is a number used to represent one or more equal parts of a whole. The 'whole' can be a single object (like an apple, a sheet of paper) or a group of objects (like a bunch of bananas, a set of pencils). The whole must be divided into a number of equal parts before we can talk about a fraction of it.

A fraction is written with two numbers separated by a horizontal line. The number above the line tells us how many parts we are considering, and the number below the line tells us the total number of equal parts the whole is divided into. The standard form of a fraction is:

• The number written above the fraction bar is called the Numerator. It counts the number of equal parts being taken, selected, or considered.
• The number written below the fraction bar is called the Denominator. It indicates the total number of equal parts into which the whole has been divided. An important rule is that the denominator can never be zero, as division by zero is not defined in mathematics.

Let's take the example of a pizza mentioned in the input. Suppose a round pizza is cut into $8$ equal slices. If you take and eat $3$ of these slices, the fraction of the pizza you have eaten is represented as $\frac{3}{8}$.

In this fraction $\frac{3}{8}$:

• The Numerator is $3$, because you took $3$ slices.
• The Denominator is $8$, because the whole pizza was divided into $8$ equal slices.

The fraction $\frac{3}{8}$ means $3$ parts out of $8$ equal parts.

---

Related Terms

Let's look at some specific terms related to fractions:

• Numerator: (Already defined) The top number in a fraction. It tells us how many of the equal parts we are considering. It can be any whole number (including 0 in some contexts, though for simple representation of parts of a whole, the numerator is usually a natural number or 0). For example, in $\frac{5}{12}$, the numerator is $5$.

• Denominator: (Already defined) The bottom number in a fraction. It tells us the total number of equal parts the whole is divided into. It must be a natural number (a counting number greater than 0). For example, in $\frac{5}{12}$, the denominator is $12$.

• Unit Fraction: A special type of fraction where the numerator is 1. A unit fraction represents a single equal part of the whole.

  Example: $\frac{1}{2}$ (representing one of two equal parts), $\frac{1}{4}$ (one of four equal parts), $\frac{1}{10}$ (one of ten equal parts), $\frac{1}{50}$ (one of fifty equal parts).

  Any fraction can be understood in terms of unit fractions. For example, $\frac{3}{8}$ can be thought of as the sum of three unit fractions of size $\frac{1}{8}$: $\frac{3}{8} = \frac{1}{8} + \frac{1}{8} + \frac{1}{8}$.

• Fractional Number: This term is sometimes used more broadly. In the context of elementary mathematics (like Class 6), it often refers to numbers that can be written in the form $\frac{p}{q}$, where $p$ is a whole number and $q$ is a natural number. This definition largely overlaps with the concept of a fraction representing parts of a whole. In higher classes, this concept evolves to include negative numbers and is covered under the term 'Rational Numbers'. For now, consider fractional numbers as representations of parts of a whole, typically positive.

---

Answer:

Representation of Fractions on a Number Line

We have learned how to represent whole numbers and integers on a number line. Fractions, which represent parts of a whole, can also be visually represented on a number line. The number line helps us to understand the value of a fraction and its position relative to whole numbers and other fractions.

---

Steps for Representation of Fractions on a Number Line

Representing Proper Fractions ($\frac{p}{q}$ where $p < q$)

Proper fractions are always less than 1 and greater than 0. So, they lie between 0 and 1 on the number line. To represent a proper fraction $\frac{p}{q}$ on a number line, where $p$ and $q$ are positive integers and $p < q$:

1. Draw a number line and mark the points representing $0$ and $1$.
2. Focus on the segment between $0$ and $1$. This segment represents one whole unit.
3. Divide this segment (from $0$ to $1$) into $q$ (the denominator) equal parts. The number of divisions will be $q$. The number of marks you make between $0$ and $1$ will be $q-1$. For example, if $q=2$, you make 1 mark; if $q=3$, you make 2 marks; if $q=5$, you make 4 marks.
4. Starting from $0$, count $p$ (the numerator) divisions towards $1$. The point at the end of the $p$-th division is the point representing the fraction $\frac{p}{q}$.

Let's illustrate with examples:

Example: Represent $\frac{1}{2}$ on the number line.

Here, $p=1$ and $q=2$. This is a proper fraction, so it's between 0 and 1.

1. Draw a number line and mark 0 and 1.
2. Divide the segment between 0 and 1 into $q=2$ equal parts. This requires making one mark exactly in the middle of 0 and 1.
3. Starting from 0, count $p=1$ part to the right. The first mark from 0 represents $\frac{1}{2}$.

Example: Represent $\frac{2}{3}$ on the number line.

Here, $p=2$ and $q=3$. This is a proper fraction, between 0 and 1.

1. Draw a number line and mark 0 and 1.
2. Divide the segment between 0 and 1 into $q=3$ equal parts. This requires making two equally spaced marks between 0 and 1.
3. Starting from 0, count $p=2$ parts to the right. The second mark from 0 represents $\frac{2}{3}$.

Representing Improper Fractions or Mixed Numbers

Improper fractions are greater than or equal to 1. Mixed numbers also represent values greater than 1. To represent an improper fraction or a mixed number on a number line:

1. If you have an improper fraction, first convert it into a mixed number ($W \frac{P}{D}$). The whole number part ($W$) tells you which two whole numbers the fraction lies between (it lies between $W$ and $W+1$).
2. Draw a number line and mark the relevant whole numbers, including $W$ and $W+1$.
3. Focus on the segment between $W$ and $W+1$. This segment represents one whole unit.
4. Divide this segment (from $W$ to $W+1$) into $D$ (the denominator of the fractional part) equal parts.
5. Starting from $W$, count $P$ (the numerator of the fractional part) divisions towards $W+1$. The point at the end of the $P$-th division is the point representing the mixed number $W \frac{P}{D}$, which is also the improper fraction $\frac{N}{D}$.

Example: Represent $\frac{7}{4}$ on the number line.

First, convert the improper fraction $\frac{7}{4}$ to a mixed number. Divide $7$ by $4$:

$\begin{array}{r} 1\phantom{)} \\ 4{\overline{\smash{\big)}\,7\phantom{)}}} \\ \underline{-~4\phantom{)}} \\ 3\phantom{)} \end{array}$

Quotient is $1$, Remainder is $3$, Denominator is $4$. So, $\frac{7}{4} = 1\frac{3}{4}$.

The mixed number $1\frac{3}{4}$ tells us the fraction is between the whole numbers $1$ and $2$.

1. Draw a number line and mark the integers $0, 1, 2$, etc.
2. Consider the interval between $1$ and $2$.
3. Divide this interval into $D=4$ equal parts. This requires making $3$ equally spaced marks between $1$ and $2$.
4. Starting from $1$, count $P=3$ parts to the right. This third mark after $1$ represents $1\frac{3}{4}$.

We can also label the divisions between 1 and 2 as $1\frac{1}{4}, 1\frac{2}{4}, 1\frac{3}{4}$. The point $1\frac{3}{4}$ is our required fraction.

---

Answer:

Types Of Fractions

Fractions are numbers used to represent parts of a whole. They can be classified into different categories based on the relationship between their numerator and denominator, or based on their denominators when comparing multiple fractions.

---

Classification Based on Numerator and Denominator

Based on how the numerator compares to the denominator, fractions can be divided into three main types:

• Proper Fraction: A fraction in which the numerator is less than the denominator ($Numerator < Denominator$) is called a proper fraction. Proper fractions always represent a quantity that is less than one whole unit.

  Example: Consider the fraction $\frac{3}{4}$. The numerator $3$ is less than the denominator $4$. If you divide a cake into 4 equal parts and take 3, you have taken less than the whole cake. Hence, $\frac{3}{4}$ is a proper fraction.

  Other examples of proper fractions are $\frac{1}{2}, \frac{2}{5}, \frac{7}{10}, \frac{15}{16}, \frac{99}{100}$.

• Improper Fraction: A fraction in which the numerator is greater than or equal to the denominator ($Numerator \ge Denominator$) is called an improper fraction. Improper fractions represent a quantity that is equal to or greater than one whole unit.

  Example: Consider the fraction $\frac{5}{4}$. The numerator $5$ is greater than the denominator $4$. If you have a quantity represented by $\frac{5}{4}$, it means you have more than one whole unit (e.g., one whole cake divided into 4 parts, plus one more part from another similar cake). Hence, $\frac{5}{4}$ is an improper fraction.

  If the numerator is equal to the denominator, the fraction represents exactly one whole. Example: $\frac{4}{4} = 1, \frac{7}{7} = 1$. These are also considered improper fractions by definition.

  Other examples of improper fractions are $\frac{3}{2}, \frac{7}{5}, \frac{10}{3}, \frac{12}{12}, \frac{25}{8}$.

• Mixed Number (or Mixed Fraction): A mixed number is a way of expressing an improper fraction as a combination of a whole number and a proper fraction. Mixed numbers are always greater than $1$.

  Example: $1\frac{1}{2}$. This means $1$ whole unit plus $\frac{1}{2}$ of another unit. This is equivalent to $\frac{3}{2}$ as an improper fraction.

  Example: $3\frac{2}{5}$. This means $3$ whole units plus $\frac{2}{5}$ of another unit. This is equivalent to $\frac{17}{5}$ as an improper fraction.

  Other examples of mixed numbers are $2\frac{3}{4}, 5\frac{1}{8}, 10\frac{9}{10}$.

  Improper fractions and mixed numbers are different ways of writing the same value, and we can convert between them.

---

Conversion between Improper Fractions and Mixed Numbers

As improper fractions and mixed numbers represent the same value, it is often useful to convert from one form to the other.

Converting Improper Fraction to Mixed Number:

To convert an improper fraction $\frac{N}{D}$ (where $N \ge D$) to a mixed number, you need to divide the numerator ($N$) by the denominator ($D$).

• The Quotient of the division becomes the whole number part of the mixed number.
• The Remainder of the division becomes the numerator of the proper fraction part.
• The Denominator remains the same as the original denominator.

Answer:

Converting Mixed Number to Improper Fraction:

To convert a mixed number $W \frac{P}{D}$ (where $W$ is the whole number, $P$ is the numerator of the proper fraction, and $D$ is the denominator) into an improper fraction, follow these steps:

1. Multiply the whole number part ($W$) by the denominator ($D$).
2. Add the numerator of the proper fraction ($P$) to the result of the multiplication. This sum becomes the numerator of the improper fraction.
3. Keep the original denominator ($D$).

Answer:

---

Classification Based on Denominators (for sets of fractions)

When we have two or more fractions, we can classify them based on their denominators:

• Like Fractions: A group of two or more fractions that have the same denominator are called like fractions. They represent parts of a whole that have been divided into the same number of equal pieces.

  Example: $\frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5}$. All these fractions have the denominator $5$.

  Other examples: $\frac{3}{7}, \frac{5}{7}, \frac{6}{7}$; $\frac{11}{13}, \frac{5}{13}, \frac{9}{13}$.

• Unlike Fractions: A group of two or more fractions that have different denominators are called unlike fractions. They represent parts of a whole that have been divided into different numbers of equal pieces.

  Example: $\frac{1}{2}, \frac{2}{3}, \frac{3}{4}$. The denominators are $2, 3,$ and $4$, which are all different.

  Other examples: $\frac{2}{5}, \frac{4}{7}, \frac{1}{6}$; $\frac{8}{11}, \frac{3}{10}, \frac{5}{12}$.

---

Answer:

---

Answer:

Equivalent Fractions

Fractions are used to represent parts of a whole. Sometimes, different fractions can represent the exact same part of a whole. These fractions are called Equivalent Fractions. They may look different because they have different numerators and denominators, but they have the same value.

Understanding equivalent fractions is important for comparing fractions, adding and subtracting unlike fractions, and simplifying fractions.

---

Finding Equivalent Fractions

We can obtain equivalent fractions from a given fraction by performing the same operation (either multiplication or division) on both the numerator and the denominator by the same non-zero number.

The principle behind finding equivalent fractions is that multiplying or dividing both the numerator and the denominator by the same non-zero number is equivalent to multiplying or dividing the fraction by $1$. For example, $\frac{2}{2} = 1, \frac{3}{3} = 1$, etc. Multiplying any number by $1$ does not change its value.

Method 1: By Multiplication

To find an equivalent fraction by multiplication, multiply both the numerator and the denominator of the given fraction by the same natural number (any counting number like 2, 3, 4, ...). We do not multiply by 1 because it would just give the original fraction back.

Let $\frac{p}{q}$ be a fraction. If $k$ is any natural number other than 1, then an equivalent fraction is $\frac{p \times k}{q \times k}$.

Example: Find three equivalent fractions of $\frac{1}{2}$.

We can multiply the numerator and denominator by 2, 3, 4, and so on.

Multiplying by 2:

Multiplying by 3:

Multiplying by 4:

So, $\frac{1}{2}, \frac{2}{4}, \frac{3}{6}, \frac{4}{8}$, and so on, are all equivalent fractions. They all represent the same amount, which is half of a whole, as shown visually below:

Method 2: By Division

To find an equivalent fraction by division, divide both the numerator and the denominator of the given fraction by the same common factor (a number that divides both the numerator and the denominator exactly). This process is also known as simplifying a fraction.

Let $\frac{p}{q}$ be a fraction. If $k$ is a common factor of $p$ and $q$ (and $k \neq 0$), then an equivalent fraction is $\frac{p \div k}{q \div k}$.

Example: Find some equivalent fractions of $\frac{12}{18}$ by division.

First, find the common factors of 12 and 18. The factors of 12 are 1, 2, 3, 4, 6, 12. The factors of 18 are 1, 2, 3, 6, 9, 18. The common factors (other than 1) are 2, 3, and 6.

Dividing by the common factor 2:

Dividing by the common factor 3:

Dividing by the common factor 6 (which is the HCF of 12 and 18):

So, $\frac{12}{18}, \frac{6}{9}, \frac{4}{6}, \frac{2}{3}$ are equivalent fractions. The fraction $\frac{2}{3}$ is the simplest form of $\frac{12}{18}$ because the only common factor of 2 and 3 is 1. This is covered in the next section (Reduction of a Fraction).

---

Checking for Equivalent Fractions

To check if two given fractions are equivalent, we can use a method called cross-multiplication. Two fractions $\frac{a}{b}$ and $\frac{c}{d}$ are equivalent if the product of the numerator of the first fraction and the denominator of the second fraction is equal to the product of the denominator of the first fraction and the numerator of the second fraction.

Conversely, if $a \times d = b \times c$, then $\frac{a}{b} = \frac{c}{d}$.

Let's understand why this works. If $\frac{a}{b} = \frac{c}{d}$, we can make their denominators the same by multiplying the first fraction by $\frac{d}{d}$ and the second by $\frac{b}{b}$.

Since $b \times d = d \times b$ (due to commutative property of multiplication), the denominators of the new fractions $\frac{a \times d}{b \times d}$ and $\frac{c \times b}{d \times b}$ are the same ($bd$). If the original fractions were equivalent, their equivalent fractions with the same denominator must also be equal in numerator.

This implies that their numerators must be equal:

Example: Are $\frac{2}{3}$ and $\frac{8}{12}$ equivalent fractions?

We will use the cross-multiplication method.

First fraction: $\frac{a}{b} = \frac{2}{3}$ (where $a=2, b=3$).

Second fraction: $\frac{c}{d} = \frac{8}{12}$ (where $c=8, d=12$).

Calculate the cross-products:

Since the cross-products are equal ($24 = 24$), the fractions $\frac{2}{3}$ and $\frac{8}{12}$ are equivalent.

---

Answer:

---

Answer:

Reduction of a Fraction

Fractions often appear in different equivalent forms. For example, $\frac{1}{2}, \frac{2}{4}, \frac{3}{6}, \frac{4}{8}$ all represent the same quantity. While these are all correct, it is common practice to express a fraction in its simplest form. Reducing a fraction means simplifying it to its lowest terms.

A fraction is said to be in its lowest terms or simplest form when its numerator and denominator have no common factor other than $1$. This means you cannot divide both the numerator and the denominator by any number other than $1$ and still get whole numbers.

---

Reducing a Fraction to its Lowest Terms

To reduce a fraction to its lowest terms, you need to find the greatest common factor (GCF) of the numerator and the denominator and divide both by this GCF. Dividing by the GCF ensures you reach the lowest terms in a single step.

Method 1: Using the Highest Common Factor (HCF) / Greatest Common Factor (GCF)

Find the HCF of the numerator and the denominator. Then divide both the numerator and the denominator by their HCF.

Answer:

Method 2: By Repeated Division (Cancellation Method)

If finding the HCF is difficult, you can repeatedly divide the numerator and the denominator by any common factor you find, until there are no common factors left other than 1.

Answer:

---

Answer:

---

Converting Fractions to Equivalent Like Fractions

Like fractions are fractions that have the same denominator. For example, $\frac{1}{7}, \frac{3}{7},$ and $\frac{6}{7}$ are like fractions. Fractions with different denominators, such as $\frac{2}{3}$ and $\frac{3}{4}$, are called unlike fractions.

Converting unlike fractions into equivalent like fractions is a crucial step for comparing, adding, or subtracting them.

Method for Converting to Like Fractions

The goal is to find a common denominator for all the fractions and then create equivalent fractions with that new denominator. The best common denominator to use is the Least Common Multiple (LCM) of the original denominators.

1. Find the LCM of the denominators of all the given fractions.
2. For each fraction, find the factor by which its denominator must be multiplied to get the LCM. (Factor = LCM / Original Denominator).
3. Multiply both the numerator and the denominator of the fraction by this factor to get the equivalent like fraction.

Answer:

Comparison of Fractions

Just like we compare whole numbers or integers to see which is larger or smaller, we also need to compare fractions. Comparing fractions means determining whether one fraction is greater than, less than, or equal to another fraction. The method for comparison depends on the type of fractions being compared, specifically whether they have the same denominator or different denominators.

---

Comparing Like Fractions

Like fractions are fractions that have the same denominator. Comparing like fractions is straightforward because the whole is divided into the same number of equal parts for each fraction. Therefore, to compare like fractions, we only need to compare their numerators.

Rule: When comparing two or more like fractions, the fraction with the larger numerator represents a greater part of the whole, and is therefore the greater fraction.

Example: Compare $\frac{3}{8}$ and $\frac{5}{8}$.

These are like fractions because both have the same denominator, $8$. We compare their numerators, $3$ and $5$.

Since $5$ is greater than $3$ ($5 > 3$), the fraction with the numerator $5$ is greater than the fraction with the numerator $3$.

Visually, if you divide a whole into 8 equal parts, taking 5 parts is more than taking 3 parts.

---

Comparing Unlike Fractions

Unlike fractions are fractions that have different denominators. To compare unlike fractions, we cannot directly compare their numerators because the wholes are divided into different numbers of equal parts. To compare them, we first need to convert them into equivalent like fractions. This means finding equivalent fractions for each of the given fractions such that they all have the same denominator.

The easiest way to find a common denominator is to find the Least Common Multiple (LCM) of the original denominators. This LCM will be the new common denominator for the equivalent fractions.

Steps to Compare Unlike Fractions:

1. Find the LCM of the denominators of the given fractions.
2. Convert each fraction into an equivalent fraction with the LCM as the new denominator. To do this, multiply both the numerator and the denominator of each fraction by the number that makes the original denominator equal to the LCM.
3. Compare the numerators of the equivalent like fractions. The fraction with the larger numerator is the greater fraction.

Example: Compare $\frac{2}{3}$ and $\frac{3}{4}$.

These are unlike fractions because their denominators are $3$ and $4$, which are different.

1. Find the LCM of the denominators $3$ and $4$. The multiples of 3 are 3, 6, 9, 12, 15, ... The multiples of 4 are 4, 8, 12, 16, ... The least common multiple is 12. So, LCM$(3, 4) = 12$.
2. Convert each fraction into an equivalent fraction with a denominator of 12.
   • For $\frac{2}{3}$: To make the denominator 12, we multiply 3 by 4. So, we must multiply the numerator by 4 as well.

     $\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}$

   • For $\frac{3}{4}$: To make the denominator 12, we multiply 4 by 3. So, we must multiply the numerator by 3 as well.

     $\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$

3. Now we compare the equivalent like fractions: $\frac{8}{12}$ and $\frac{9}{12}$.

   Comparing the numerators, we have $8$ and $9$. Since $9 > 8$, the fraction $\frac{9}{12}$ is greater than $\frac{8}{12}$.

   $\frac{9}{12} > \frac{8}{12}$

Since $\frac{9}{12}$ is equivalent to $\frac{3}{4}$ and $\frac{8}{12}$ is equivalent to $\frac{2}{3}$, we can conclude that:

Comparing Fractions with the Same Numerator

A special rule applies when comparing unlike fractions (fractions with different denominators) that happen to have the same numerator.

Rule: If two fractions have the same numerator, the fraction with the smaller denominator is the greater fraction.

Reasoning

The denominator of a fraction tells us into how many equal parts a whole has been divided. A smaller denominator means the whole is divided into fewer, and therefore larger, parts. A larger denominator means the whole is divided into more, and therefore smaller, parts. If you take the same number of parts (the numerator), taking the larger parts will give you a greater overall value.

For example, let's compare $\frac{1}{2}$ and $\frac{1}{3}$.

• Both fractions have the same numerator, 1.
• The denominators are 2 and 3.
• Comparing the denominators, we see that $2 < 3$.
• The fraction with the smaller denominator, $\frac{1}{2}$, represents one part of a whole that was divided into only two parts. The fraction $\frac{1}{3}$ represents one part of a whole divided into three parts. The part from the first fraction is larger.

Therefore, $\frac{1}{2} > \frac{1}{3}$. Think about sharing a pizza: half of a pizza is a larger share than one-third of the same pizza.

Answer:

Comparing a Fraction with 1

We can quickly compare a fraction to the whole number 1 based on whether it is a proper or improper fraction:

• If the fraction is a proper fraction (numerator < denominator), it represents a part less than a whole. So, a proper fraction is always less than 1.

  Example: $\frac{3}{5} < 1$, $\frac{99}{100} < 1$.

• If the fraction is an improper fraction (numerator $\ge$ denominator), it represents a quantity equal to or greater than a whole.
  • If the numerator is equal to the denominator, the fraction is equal to 1.

    Example: $\frac{5}{5} = 1$, $\frac{10}{10} = 1$.

  • If the numerator is greater than the denominator, the fraction is greater than 1.

    Example: $\frac{7}{4} > 1$ (because $\frac{7}{4} = 1\frac{3}{4}$), $\frac{11}{6} > 1$.

---

Answer:

Addition & Subtraction of Fractions

Now that we know what fractions are and how to compare them, let's learn how to perform the basic arithmetic operations of addition and subtraction with fractions. The method we use depends on whether the fractions are like fractions (same denominator) or unlike fractions (different denominators).

---

Addition and Subtraction of Like Fractions

Adding or subtracting like fractions is similar to adding or subtracting whole numbers of the same unit. For example, adding 3 apples and 2 apples gives 5 apples. Similarly, adding 3 tenths and 2 tenths gives 5 tenths.

Rule: To add or subtract like fractions, we simply add or subtract their numerators and keep the denominator the same.

Let $\frac{a}{c}$ and $\frac{b}{c}$ be two like fractions (where $c \neq 0$).

For Addition:

For Subtraction:

Example: Add $\frac{3}{7} + \frac{2}{7}$.

These are like fractions with denominator 7. We add the numerators and keep the denominator.

The sum is $\frac{5}{7}$. Since 5 and 7 have no common factors other than 1, the fraction is already in its lowest terms.

Example: Subtract $\frac{5}{9} - \frac{2}{9}$.

These are like fractions with denominator 9. We subtract the numerators and keep the denominator.

The result is $\frac{3}{9}$. We should always check if the resulting fraction can be reduced to its lowest terms. The numerator 3 and the denominator 9 have a common factor of 3.

So, $\frac{5}{9} - \frac{2}{9} = \frac{1}{3}$.

---

Addition and Subtraction of Unlike Fractions

Unlike fractions have different denominators. We cannot directly add or subtract their numerators because the parts of the whole are of different sizes. To add or subtract unlike fractions, we must first convert them into equivalent like fractions. Once they have the same denominator, we can add or subtract them using the rules for like fractions.

Steps:

1. Find the Least Common Multiple (LCM) of the denominators of the unlike fractions. This LCM will be the common denominator.
2. Convert each unlike fraction into an equivalent fraction with the LCM as the denominator.
3. Now that the fractions are like fractions, add or subtract their numerators and keep the common denominator.
4. Simplify the resulting fraction to its lowest terms if possible.

Example: Add $\frac{1}{2} + \frac{1}{3}$.

These are unlike fractions (denominators 2 and 3).

1. Find the LCM of 2 and 3. LCM(2, 3) = 6. The common denominator will be 6.
2. Convert each fraction to an equivalent fraction with denominator 6.
   • For $\frac{1}{2}$: Multiply numerator and denominator by 3 (since $2 \times 3 = 6$).

     $\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}$

   • For $\frac{1}{3}$: Multiply numerator and denominator by 2 (since $3 \times 2 = 6$).

     $\frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6}$

3. Now add the equivalent like fractions: $\frac{3}{6} + \frac{2}{6}$. Add numerators ($3+2=5$) and keep the denominator ($6$).

   Sum $= \frac{3}{6} + \frac{2}{6} = \frac{3 + 2}{6} = \frac{5}{6}$

4. Simplify the result $\frac{5}{6}$. The only common factor of 5 and 6 is 1. So, it is already in lowest terms.

Thus, $\frac{1}{2} + \frac{1}{3} = \frac{5}{6}$.

Example: Subtract $\frac{3}{4} - \frac{1}{6}$.

These are unlike fractions (denominators 4 and 6).

1. Find the LCM of 4 and 6. Multiples of 4: 4, 8, 12, 16, ... Multiples of 6: 6, 12, 18, ... LCM(4, 6) = 12. The common denominator will be 12.
2. Convert each fraction to an equivalent fraction with denominator 12.
   • For $\frac{3}{4}$: Multiply numerator and denominator by 3 (since $4 \times 3 = 12$).

     $\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$

   • For $\frac{1}{6}$: Multiply numerator and denominator by 2 (since $6 \times 2 = 12$).

     $\frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12}$

3. Now subtract the equivalent like fractions: $\frac{9}{12} - \frac{2}{12}$. Subtract numerators ($9-2=7$) and keep the denominator ($12$).

   Difference $= \frac{9}{12} - \frac{2}{12} = \frac{9 - 2}{12} = \frac{7}{12}$

4. Simplify the result $\frac{7}{12}$. The only common factor of 7 and 12 is 1. So, it is already in lowest terms.

Thus, $\frac{3}{4} - \frac{1}{6} = \frac{7}{12}$.

---

Adding and Subtracting Mixed Numbers

Mixed numbers consist of a whole number part and a fractional part. To add or subtract mixed numbers, you can use either of the following methods:

Method 1: Convert to Improper Fractions

Convert each mixed number into an improper fraction. Then, add or subtract the improper fractions using the methods described above (find a common denominator if they are unlike fractions). Finally, convert the result back to a mixed number if required.

Method 2: Add/Subtract Whole Parts and Fractional Parts Separately

Add or subtract the whole number parts separately. Then, add or subtract the fractional parts separately (convert them to like fractions first if needed). Combine the results. Be careful with subtraction when the first fraction is smaller than the second; you might need to 'borrow' from the whole number part.

Example (Addition): Add $2 \frac{1}{3} + 1 \frac{1}{2}$.

Method 1 (Using Improper Fractions):

• Convert mixed numbers to improper fractions:

  $2 \frac{1}{3} = \frac{(2 \times 3) + 1}{3} = \frac{6 + 1}{3} = \frac{7}{3}$

  $1 \frac{1}{2} = \frac{(1 \times 2) + 1}{2} = \frac{2 + 1}{2} = \frac{3}{2}$

• Add the improper fractions: $\frac{7}{3} + \frac{3}{2}$. Denominators are 3 and 2. LCM(3, 2) = 6.

  $\frac{7}{3} = \frac{7 \times 2}{3 \times 2} = \frac{14}{6}$

  $\frac{3}{2} = \frac{3 \times 3}{2 \times 3} = \frac{9}{6}$

  Sum $= \frac{14}{6} + \frac{9}{6} = \frac{14 + 9}{6} = \frac{23}{6}$

• Convert the improper fraction result back to a mixed number: Divide 23 by 6. $23 \div 6$.

  $\begin{array}{r} 3\phantom{)} \\ 6{\overline{\smash{\big)}\,23\phantom{)}}} \\ \underline{-~18\phantom{)}} \\ 5\phantom{)} \end{array}$

  Quotient = 3, Remainder = 5. So, $\frac{23}{6} = 3\frac{5}{6}$.

Thus, $2 \frac{1}{3} + 1 \frac{1}{2} = 3 \frac{5}{6}$.

Method 2 (Separating Parts):

The problem is $2\frac{1}{3} + 1\frac{1}{2} = (2 + \frac{1}{3}) + (1 + \frac{1}{2})$.

• Add the whole number parts: $2 + 1 = 3$.
• Add the fractional parts: $\frac{1}{3} + \frac{1}{2}$. These are unlike fractions. LCM(3, 2) = 6.

  $\frac{1}{3} + \frac{1}{2} = \frac{1 \times 2}{3 \times 2} + \frac{1 \times 3}{2 \times 3} = \frac{2}{6} + \frac{3}{6} = \frac{2 + 3}{6} = \frac{5}{6}$

• Combine the results of the whole number sum and the fractional part sum:

  Total sum $= 3 + \frac{5}{6} = 3\frac{5}{6}$

Both methods give the same result.

Example (Subtraction): Calculate $3 \frac{1}{4} - 1 \frac{2}{3}$.

Method 1 (Using Improper Fractions):

• Convert mixed numbers to improper fractions:

  $3 \frac{1}{4} = \frac{(3 \times 4) + 1}{4} = \frac{12 + 1}{4} = \frac{13}{4}$

  $1 \frac{2}{3} = \frac{(1 \times 3) + 2}{3} = \frac{3 + 2}{3} = \frac{5}{3}$

• Subtract the improper fractions: $\frac{13}{4} - \frac{5}{3}$. Denominators are 4 and 3. LCM(4, 3) = 12.

  $\frac{13}{4} = \frac{13 \times 3}{4 \times 3} = \frac{39}{12}$

  $\frac{5}{3} = \frac{5 \times 4}{3 \times 4} = \frac{20}{12}$

  Difference $= \frac{39}{12} - \frac{20}{12} = \frac{39 - 20}{12} = \frac{19}{12}$

• Convert the improper fraction result back to a mixed number: Divide 19 by 12. $19 \div 12$.

  $\begin{array}{r} 1\phantom{)} \\ 12{\overline{\smash{\big)}\,19\phantom{)}}} \\ \underline{-~12\phantom{)}} \\ 7\phantom{)} \end{array}$

  Quotient = 1, Remainder = 7. So, $\frac{19}{12} = 1\frac{7}{12}$.

Thus, $3 \frac{1}{4} - 1 \frac{2}{3} = 1 \frac{7}{12}$.

Note: Method 2 for subtraction of mixed numbers can be slightly more complex if the first fraction is smaller than the second, requiring 'borrowing' from the whole number. The improper fraction method is often simpler and less prone to errors for beginners in subtraction.

---

Answer:

---

Answer:

Multiplication & Division of Fractions

Unlike addition and subtraction, multiplication and division of fractions do not require a common denominator. They have their own distinct rules, which are generally simpler than those for addition and subtraction, especially for unlike fractions.

---

Multiplication of Fractions

Multiplying fractions is a straightforward process. When you multiply a fraction by another fraction, you are essentially finding a 'fraction of a fraction'. For example, $\frac{1}{2} \times \frac{1}{4}$ means finding half of one-fourth.

Rule: To multiply two or more fractions, multiply all the numerators together to get the new numerator, and multiply all the denominators together to get the new denominator. Simplify the resulting fraction to its lowest terms if possible.

Let $\frac{a}{b}$ and $\frac{c}{d}$ be two fractions (where $b \neq 0, d \neq 0$).

Example: Multiply $\frac{2}{3} \times \frac{4}{5}$.

Multiply the numerators: $2 \times 4 = 8$.

Multiply the denominators: $3 \times 5 = 15$.

The resulting fraction is $\frac{8}{15}$. The common factors of 8 and 15 are only 1. So, it is already in its lowest terms.

Example: Multiply $\frac{3}{4} \times \frac{2}{9}$.

Multiply the numerators: $3 \times 2 = 6$.

Multiply the denominators: $4 \times 9 = 36$.

The resulting fraction is $\frac{6}{36}$. This can be simplified. The HCF of 6 and 36 is 6.

So, $\frac{3}{4} \times \frac{2}{9} = \frac{1}{6}$.

Simplifying Before Multiplication (Cancellation)

To make multiplication easier and avoid simplifying large numbers at the end, you can cancel out common factors between any numerator and any denominator *before* multiplying. This is possible because multiplication allows us to rearrange factors.

Example: Multiply $\frac{3}{4} \times \frac{2}{9}$.

Look for common factors between numerator and denominator pairs (across fractions). 3 (numerator of first fraction) and 9 (denominator of second fraction) have a common factor of 3. 2 (numerator of second fraction) and 4 (denominator of first fraction) have a common factor of 2.

Now the remaining numbers are $\frac{1}{4}$ and $\frac{2}{3}$. We can further cancel 2 from the denominator 4 and the numerator 2.

Now multiply the reduced numerators and denominators:

This method is very efficient.

Multiplying a Fraction by a Whole Number:

To multiply a fraction by a whole number, treat the whole number as a fraction with a denominator of 1 (e.g., $5 = \frac{5}{1}$). Then multiply the fractions as usual.

Example: Multiply $5 \times \frac{3}{10}$.

Rewrite 5 as $\frac{5}{1}$.

Simplify the result $\frac{15}{10}$: HCF of 15 and 10 is 5.

The improper fraction $\frac{3}{2}$ can be converted to a mixed number: $1\frac{1}{2}$.

Using cancellation from the start:

This is much quicker.

'of' Means Multiplication:

In word problems involving fractions, the word 'of' usually indicates multiplication.

Example: What is $\frac{1}{4}$ of $20$?

This means calculate $\frac{1}{4} \times 20$.

So, $\frac{1}{4}$ of $20$ is $5$.

Multiplying Mixed Numbers:

To multiply mixed numbers, first convert them into improper fractions. Then, multiply the improper fractions using the rule for multiplying fractions. Finally, convert the result back to a mixed number if needed.

Example: $1\frac{1}{2} \times 2\frac{1}{3}$.

Convert to improper fractions: $1\frac{1}{2} = \frac{(1 \times 2) + 1}{2} = \frac{3}{2}$. $2\frac{1}{3} = \frac{(2 \times 3) + 1}{3} = \frac{7}{3}$.

Multiply the improper fractions:

Convert to mixed number: $\frac{7}{2} = 3\frac{1}{2}$.

---

Reciprocal of a Fraction

The concept of a reciprocal is essential for understanding fraction division. The reciprocal of a non-zero fraction is obtained by simply swapping its numerator and its denominator. It is also known as the multiplicative inverse.

If $\frac{a}{b}$ is a non-zero fraction, its reciprocal is $\frac{b}{a}$.

Example: The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$.

Example: The reciprocal of $5$. First, write 5 as a fraction: $\frac{5}{1}$. The reciprocal is $\frac{1}{5}$.

Example: The reciprocal of $1\frac{1}{4}$. First, convert to improper fraction: $1\frac{1}{4} = \frac{5}{4}$. The reciprocal is $\frac{4}{5}$.

An important property: The product of a fraction and its reciprocal is always 1.

---

Division of Fractions

Dividing by a fraction is the same as multiplying by its reciprocal. This is the key rule for fraction division.

Rule: To divide a fraction by another non-zero fraction, multiply the first fraction by the reciprocal of the second fraction (the divisor).

Let $\frac{a}{b}$ and $\frac{c}{d}$ be two fractions, with $\frac{c}{d} \neq 0$ (so $c \neq 0, d \neq 0$).

Example: Divide $\frac{2}{3}$ by $\frac{1}{4}$.

The first fraction is $\frac{2}{3}$. The second fraction (divisor) is $\frac{1}{4}$.

Find the reciprocal of the divisor $\frac{1}{4}$: The reciprocal of $\frac{1}{4}$ is $\frac{4}{1}$.

Now, multiply the first fraction by the reciprocal of the second fraction:

Perform the multiplication:

The result is the improper fraction $\frac{8}{3}$. Convert to a mixed number: $\frac{8}{3} = 2\frac{2}{3}$.

Dividing a Whole Number by a Fraction:

To divide a whole number by a fraction, multiply the whole number by the reciprocal of the fraction.

Example: Divide 6 by $\frac{2}{3}$.

Multiply 6 by the reciprocal of $\frac{2}{3}$ (which is $\frac{3}{2}$).

Perform the multiplication (using cancellation):

So, $6 \div \frac{2}{3} = 9$. This means there are 9 groups of $\frac{2}{3}$ in 6 whole units.

Dividing a Fraction by a Whole Number:

To divide a fraction by a whole number, multiply the fraction by the reciprocal of the whole number. Remember that the reciprocal of a whole number 'a' is $\frac{1}{a}$ (since $a = \frac{a}{1}$).

Example: Divide $\frac{3}{4}$ by 5.

Multiply $\frac{3}{4}$ by the reciprocal of 5 (which is $\frac{1}{5}$).

Perform the multiplication:

So, $\frac{3}{4} \div 5 = \frac{3}{20}$. This means if you divide $\frac{3}{4}$ of something into 5 equal parts, each part will be $\frac{3}{20}$ of the original whole.

Dividing Mixed Numbers:

To divide mixed numbers, first convert them into improper fractions. Then, divide the improper fractions using the rule for dividing fractions (multiply the first by the reciprocal of the second). Finally, convert the result back to a mixed number if needed.

Example: $2\frac{1}{4} \div 1\frac{1}{2}$.

Convert to improper fractions: $2\frac{1}{4} = \frac{(2 \times 4) + 1}{4} = \frac{9}{4}$. $1\frac{1}{2} = \frac{(1 \times 2) + 1}{2} = \frac{3}{2}$.

Now divide the improper fractions:

Multiply (using cancellation):

Convert to mixed number: $\frac{3}{2} = 1\frac{1}{2}$.

---

Answer:

Uses of Fractions

Fractions are not just abstract mathematical concepts; they are extremely useful in our daily lives and are applied in countless real-world situations. Any time we talk about a part of something, whether it's a physical object, a quantity, or a measurement, we are likely using fractions or concepts related to them.

---

Real-Life Applications of Fractions

Here are some common areas where fractions are used:

• Cooking and Recipes: This is one of the most common places you encounter fractions. Recipes use fractions to specify amounts of ingredients like cups, teaspoons, tablespoons, etc. For example, a recipe might call for $\frac{1}{2}$ cup of milk, $\frac{3}{4}$ teaspoon of baking powder, or $2\frac{1}{4}$ cups of flour. Adjusting recipes for more or fewer servings also involves multiplying or dividing fractions.

• Sharing and Distribution: When you share things equally among a group of people, fractions naturally come into play. If a pizza is cut into 8 slices and shared among 4 people, each person gets $\frac{2}{8}$ (or $\frac{1}{4}$) of the pizza. If a parent divides property among their children, each child might get a fraction of the total property, like $\frac{1}{3}$ or $\frac{1}{4}$.

• Measurement: Fractions are crucial for measurements that fall between whole units. For instance, measuring length (e.g., a piece of cloth is $3\frac{1}{2}$ metres long, a table is $\frac{3}{4}$ metres wide), weight (e.g., $1\frac{1}{4}$ kilograms of vegetables), or volume (e.g., a bottle containing $\frac{1}{2}$ litre of juice).

• Time: We use fractions when referring to parts of an hour, day, or year. For example, 'half an hour' is $\frac{1}{2}$ hour, 'quarter of an hour' is $\frac{1}{4}$ hour, 'three quarters of an hour' is $\frac{3}{4}$ hour. Phrases like 'quarter past three' refer to $3$ and $\frac{1}{4}$ hours past a reference time (like midnight or noon). Half a day is $\frac{1}{2}$ of 24 hours, which is 12 hours.

• Shopping and Discounts: Sale advertisements often mention discounts as fractions, such as "$\frac{1}{2}$ price off" or "$\frac{1}{4}$ discount". This means the price is reduced by that fraction of the original price.

• Data Representation and Statistics: Fractions are used to describe proportions or parts of a group in surveys, reports, or statistics. For example, "$\frac{3}{5}$ of the students surveyed said they enjoy reading" tells you that out of every 5 students, 3 enjoy reading.

• Sports: In sports, distances, times, or scores can involve fractions (e.g., running $\frac{1}{2}$ lap, $3 \frac{1}{2}$ rounds of a competition).

• Construction and Engineering: Precise measurements and specifications in building and design often involve fractions.

In all these situations, fractions provide a way to express quantities that are not whole numbers, allowing for more precise measurements, fair sharing, and clear communication about parts of a whole.

---

Answer:

---

Answer: