Fractions and Decimals

Fractions and decimals are essential concepts in mathematics which serve as the foundation for many mathematical operations. In this chapter, we will explore what fractions and decimals are, how they work and their practical applications in our daily lives.

Fraction

A fraction is a representation of a part of a larger or smaller collection. It allows us to express values that are not whole numbers.

For example, if you have a cake and you plan to share it with a friend, each slice of the cake can be represented as a fraction. You and your friends (2 persons) eat 1 slice which is half of the cake.

Sum of each part of the cake gives the whole number.

For example, if you have a cake and you plan to share it with your two friends, each slice of the cake can be represented as a fraction. You and your two friends (3 persons) eat 1 slice which is one-third of the cake.

For example, if you have a cake and you plan to share it with your five friends, each slice of the cake can be represented as a fraction. You and your five friends (6 persons) eat 1 slice which is one-sixth of the cake.

Here, 1/2, 1/3 and 1/6 are fractions.

Parts of a Fraction

A fraction is composed of two fundamental components:

→ Numerator:The numerator is the number at the top of the fraction that indicates how many parts the fraction has.

→ Denominator:The denominator is the number at the bottom of the fraction that indicates the total number of equal parts the whole is divided into.

Examples:

→ In the fraction38, "3" is the numerator (the fraction has one part) and "8" is the denominator (the whole is divided into two equal parts).

Types of Fractions

Various types of fractions are as follows:

a) Unit Fractions: These are fractions with "1" as their numerator.

Examples: 1/2, 1/3, 1/4, 1/6, 1/7, 1/10, 1/13, 1/99, 1/100, 1/999, 1/1000, etc.

b) Proper Fractions: Proper fractions are those where the numerator is smaller than the denominator.

Examples: 1/2, 2/3, 5/8, 12/13, 23/24, 49/50, 51/52, 99/100, 100/101, etc.

c) Improper Fractions: In contrast, improper fractions have a numerator greater than the denominator.

Examples: 3/2, 4/3, 7/5, 9/7, 11/8, 13/9, 17/13, 17/15, 49/47, 101/100, etc.

d) Mixed Numbers: These are a combination of a whole number and a proper fraction.

Examples: 1¹⁄₂, 2²⁄₃, 3⁴⁄₅, 5³⁄₇, 3³⁄₈, 1³⁄₁₁, 2⁴⁄₁₃, 3¹⁷⁄₁₉, 5¹⁹⁄₂₁, 3²⁹⁄₂₈, etc.

Lowest Form of Fraction

The lowest form of the fraction is the method in which the denominator and numerator have a common factor as one. Reducing a fraction to its simplest form involves ensuring that the numerator and denominator share 1 as common factors (numbers that evenly divide both).

For example, let's consider the fraction 18/27.

Both 18 and 27 can be divided by 9. When we divide 18 by 9, we get 2. When we divide 27 by 9, we get 3.
Therefore, 

Lowest form of the fraction 18/27= 2/3

Equivalent Fractions

Equivalent fractions are different fractions that express the same value when reduced to their lowest form. To find equivalent fractions, it is necessary to multiply or divide the numerator and denominator of a fraction by the same value.

For example, the fractions 25/15, 15/9, 10/6 and 20/12 are equivalent fractions because when reduced to their lowest form, are all equal to 5/3.

Comparing the Fractions with the Same Denominator

Like fractions are groups of two or more fractions that have exactly the same denominator or the fractions that have the same numbers in the denominators.

For example: 1/7, 2/7, 3/7, 4/7, 5/7, 6/7,19/7, 37/7, etc. are like fractions.

The fraction with a larger number of numbers shall be greater and the fraction with a smaller number of numbers shall be less if the denominators are equal.

For example:

Unlike fractions are groups of two or more fractions that have different denominators of fractions that have different numbers in the denominators.

For example: 1/11, 2/13, 3/5, 7/12, 5/17, etc. are unlike fractions.

Addition of Fractions

Adding fractions involves combining two or more fractions and it depends on two key scenarios:

1. When the fractions have the same denominators i.e. Like Fraction.
2. When the fractions have different denominators i.e. Unlike Fractions.

Addition of Like Fractions (Same Denominators)

To add like fractions, follow these steps:

Step 1: Ensure that the denominators are the same.
Step 2: Add the numerators and place the result over the common denominator.
Step 3: Simplify the resulting fraction to obtain the final sum.
Step 4: If needed, convert the fraction into the lowest form.

Example: What is the sum of fractions 3/8 and 7/8?

a) 1/2
b) 1/4
c) 3/4
d) 5/4

Answer: d) 5/4

Explanation: When adding 3/8 and 7/8, both fractions share the same denominator (8), so simply add the numerators: 3 + 7 = 10. Therefore, the addition is 10/8 and its lowest form is 5/4. Addition of fractions 3/8 and 7/8 is shown:

Addition of Unlike Fractions (Different Denominators)

To add unlike fractions, follow these steps: 
Step 1: Examine the denominators of the fractions whether fractions have the same or different denominators.
Step 2: Convert unlike fractions into like fractions by aligning the denominators of the fractions by determining the least common multiple (LCM) of the denominators and rationalising them to get the same denominators.
Step 3: Sum the numerators of the fractions while keeping the denominator the same.
Step 4: Simplify the resulting fraction to obtain the final sum.
Step 5: If needed, simplify the fraction into the lowest form.

Example: What is the sum of fractions 3/2 and 7/3?

a) 5
b) 5/2
c) 5/3
d) 5/4

Answer: a) 5

Explanation: Addition of fractions 3/2 and 7/3 is shown as

Decimal

A decimal is a number expressed in the scale of tens. The term "Decimal" originates from Latin and signifies "based on 10" denoting a system rooted in tens.

For example:

Representation of Decimal Number

Decimals are numbers that have a whole number part and a fractional part separated by a decimal point, shown as

The first digit after the decimal represents the tenth place. The next digit after the decimal represents the hundredth place. Decimal place values are shown in the figure as

A decimal point (.) signifies the division between these two parts. The digits that follow the decimal point represent values smaller than one.

In order to divide a decimal number by 10, move the decimal point to the left by one place.

Examples of decimal numbers:

(1) 3/10 = 0.3

(2) 7/10 = 0.7

(3) 17/10 = 1.7

In order to divide a decimal number by 100, move the decimal point to the left by two places.

Examples of decimal numbers:

(1) 3/100 = 0.03

(2) 17/100 = 0.17

(3) 137/100 = 1.37

In order to divide a decimal number by 1000, move the decimal point to the left by three places.

Examples of decimal numbers:

(1) 7/1000 = 0.007

(2) 17/1000 = 0.017

(3) 137/1000 = 0.137

(4) 2937/1000 = 2.937

Decimals using division method

Fractions are converted into decimals using the division method.

The steps to convert any fraction into a decimal number are as follows:

Step 1: In a fraction "p/q", "p" is the numerator and "q" is the denominator. Take "p" as the dividend and "q" as the divisor.

Step 2: Divide the numerator (p) by the denominator (q) using division. Add the decimal point and as many zeros at the end of the dividend to find the different number of decimal places. This division will yield a decimal quotient.

Step 3: The result of the division is the decimal equivalent of the fraction "p/q" This is the decimal representation of the fraction.

For example, if you want to convert the fraction 1/2 into a decimal:

Step 1: In a fraction "1/2", "1" is the numerator and "2" is the denominator. "1" is the dividend and "2" is the divisor.

Step 2: Divide the numerator (1) by the denominator (2) using division. Add the decimal point and one zero at the end of the dividend to find one digit after the decimal point. This division will yield a decimal quotient of 0.5, shown in

Step 3: The result of the division is 0.5 which is the decimal equivalent of the fraction 1/2.

Some more examples of decimal numbers:

(1) 2/4 = 1/2 = 0.5

(2) 3/2 = 1.5

(3) 5/2 = 2.5

(4) 1/4 = 0.25

(5) 3/4 = 0.75

Example: What is the decimal number of the fraction 7100?

a) 0.7
b) 0.07
c) 0.007
d) 0.0007

Answer: b) 0.07

Explanation: Decimal number of the fraction 7/100 is shown as:

Addition and Subtraction of Decimal Numbers

Addition of decimal numbers is a mathematical operation used to combine numbers that have decimal points. The process is similar to adding whole numbers but you must pay attention to the placement of the decimal point.

Steps for adding decimal numbers are as follows:

→ Align the decimals.
→ Add the digits.
→ Place the decimal point at the correct place to get the decimal number after addition.

For example, the addition of 7.64 and 5.23 is shown as

Subtraction of decimal numbers is a mathematical operation used to find the difference between numbers that have decimal points. The process is similar to subtracting whole numbers but it is important to pay attention to the placement of the decimal point.

Steps forsubtracting decimal numbersare as follows:

→ Align the decimals.
→ Subtract the digits.|
→ Place the decimal point at the correct place to get the decimal number after subtraction.

For example, the subtraction of 7.51 from 11.93 is shown as:

Multiplication and Division of Decimal Numbers

Multiplication of decimal numbers is a mathematical operation used to find the product of numbers that have decimal points. The process is similar to multiplying whole numbers but you must pay attention to the placement of the decimal point in the final answer.

Steps formultiplyingdecimal numbersare as follows:

→ Multiply the numbers out of the original numbers and don't count the decimal points.
→ Count how many digits are after the decimal point.
→ Finally, place the decimal point from the right of the digit to the product to get the decimal number after multiplication.

For example, the multiplication of 2.25 and 1.3 is shown as

Division of decimal numbers is a mathematical operation used to find the quotient when one decimal number is divided by another. The process is similar to long division with whole numbers but it involves paying attention to the decimal point placement in both the dividend and divisor.

Steps fordividing decimal numbersare as follows:

→ Divide the whole number portion of the dividend by the divisor.
→ Position the decimal point in the quotient directly above the decimal point in the dividend. Then, proceed to bring down the digit in the tenth place.
→ Continue the division process by bringing down the other digits one at a time in sequence. Keep dividing until you reach a remainder of 0.
→ The placement of decimals in the quotient should align with the decimals in the dividend throughout the process.
→ The quotient is the answer of the division.

→ For example, the division of 11.6 by 2 is shown as

The quotient is the answer of the division as 11.6 2 = 5.8