Tag: Arithmetic

Introduction

The relationship between two expressions on either side of the equal to sign is represented by an equation in mathematics. One equal symbol and one variable are used in this kind of equation. Simple equations use arithmetic operations to balance the expressions.

A simple equation is an equation that shows the relationship between two expressions on both sides of the sign. Only one variable appears in these kinds of equations, either on the first side or the other side of the equal symbol. For instance, 83 = 5 – 4z. In the provided example, the variable is z. Simple equations use arithmetic operations to balance the expressions on both sides. Linear equations in one variable are also regarded as simple equations.

Equations

Equations are relationships between two or more expressions connected by the equals sign, or “=.” Variables, coefficients, and constants are the three components of an equation.

Variables: Variables are the names given to the symbols (typically English alphabets) that are assigned to an arbitrary, unknowable value.

Coefficients: The coefficients of a term are the numbers that are multiplied by a variable or the product of two variables in that term.

Constants: Constants are the numbers that are independent of variables.

Simple Equations

A type of equation known as a simple equation compares two linear expressions with just one variable in common. Several instances of basic/simple equations are

3x + 4 = 7

4x + 5 = 3x + 8

Since many of the situations, we encounter in real life can be formulated as simple equation problems, we can use simple equations to obtain the desired results in a variety of areas of life.

Simple Equations questions

Simple equation problems, which can be represented by a simple equation to find the value of something unknown based on some given conditions, are known as simple equation questions. One such example of applying simple equations to real-world situations is provided, 

Let’s say Amar and Bipin, two friends, are purchasing apples. Amar might have purchased 5 kg and Bipin 3 kg. If Amar paid Rs. 80 more than Bipin, we must determine the cost of a kg of apples. The following simple equation can be used to represent this situation:

5x = 3x + 80, where x is the price of 1 kg apples.

Solving Simple Equations

To answer questions involving simple equations, we change the equation so that the term with the variables is on one side of the equation and the term with constants is on the other. We then simplify both sides so that there is only one term on each side, one with variables and the other with constants.

The value of the variable is then obtained by simply multiplying the equation by the reciprocal of the coefficient.

Now, let’s look at some examples to help us better understand it.

Example: Solve the following simple equation, 5x – 20 = 3x + 60

Solution: Here we have 5x – 20 = 3x + 60

Adding 20 to both sides while subtracting 3x to move terms with variables to one side and constants to the other.

⇒ 5x – 20 + 20 – 3x = 3x + 60 + 20 – 3x

⇒ 5x – 3x = 60 + 20

⇒ 2x = 80

Dividing by 2 on both sides

⇒ x = 40

Simple Equation Problems

Simple equation problems are mathematical issues from the real world that are modelled by simple equations. We must first determine the number of arbitrary values present and their relationships to represent a given situation using a simple equation. If there is only one arbitrary value, it is easy to create a simple equation to describe it; however, if there are several arbitrary values, we must establish a direct relationship between them to do so.

Example: Determine whether the following scenario can be modelled as a simple equation or not. Amit is currently twice as older than his younger brother Sagar. The combined age of Amit and Sagar was 23, two years ago. Identify their current ages.

Solution: Since Amit’s age and Sagar’s age are arbitrary values, the only way we can depict this situation in a simple equation is if there is a direct correlation between their ages, which is implied by the first statement that Amit is currently twice as old as Sagar. As a result, we can express this as a simple equation problem.

Let Sagar’s present age be x years

And Amit’s present age be y years

Then, ATQ

In present, y = 2x

Also, two years ago, (x – 2) + (y – 2) = 23

Substituting y = 2x in the second equation,

⇒ x – 2 + 2x – 2 = 23

Equations in Everyday Life Examples

When a value for a quantity or identity is unknown in a real-world situation and cannot be determined by a simple mathematical operation, linear equations are used, such as when estimating future income, forecasting future profits, or figuring out mileage rates.

Here are a few real-world instances where applications of linear equations are used.

• Can be used to identify age-related problems.
• It is used to determine the distance, duration, and speed of a moving object.
• It is used to resolve problems involving money, percentages, etc.

Solved Examples

Example: Calculate the value of y from the equation:  – 5 = 6.

Solution: We will simplify the equation first by separating the variables and constants,

– 5 = 6

Add 5 on both sides,

⇒  – 5 + 5 = 6 + 5

⇒  = 11

Multiply by 3 on both sides,

⇒  x 3 = 11 x 3

⇒ 11y = 33

Divide by 11 on both sides,

⇒ y = 3

Summary

Simple equations are also known as linear equations when they contain multiple variables and can be resolved using a variety of techniques. To solve problems from daily life, such as how to measure an unknown length, etc., we use simple equations. The typical method of representing the relationship between variables is through simple equations. A simple equation is a linear equation that only has one variable. Simple equations were credited to Rene Descartes as their creator. One of the foundations of algebra is simple equations.

Frequently Asked Questions (FAQs)

1. What are Linear Equations?

Linear equations are the mathematical relations that relate two expressions of degree 1 with the equal to symbol.

2. What are Simple Equations?

Simple equations are linear equations that have only one variable. Simple equations can be solved easily and are very useful in many days to day life problems.

3. What are the different methods of Solving Simple Equations?

There are two ways that we can solve simple equations. The techniques are the systematic method and the trial-and-error method.

4. What is a Rational Expression?

A rational expression is expressed in terms of the fraction of two algebraic expressions, and it also belongs to the class of simple equations.

Introduction

Although factors and multiples are entirely different concepts, they are related. To determine the factors, we divide the given number by another number, whereas multiples of the given number can be obtained by multiplying the given number by any other number. Multiplication is involved in both ideas. To obtain a given number, we multiply two numbers; the two numbers we multiplied are referred to as the obtained number’s factors.

For example, 4 x 5 = 20. Therefore, 20 is a multiple of 4 and 5, and 4 and 5 are factors of 20.

The number that is the factor of two or more numbers is referred to as the common factor. GCD (Greatest Common Divisor) and HCF (Highest Common Factor) are terms that relate to this idea.

The common multiple is the number that is a multiple of two or more other numbers. The Least Common Multiple, or LCM, is related to this idea. Different divisibility criteria can be used to determine whether a given number is divisible by another without actually conducting the division operation.

Factors

A number must divide completely, leaving no remainder, to be the factor of any other number. In other words, we can also say that the divisor is a factor of the dividend if a number (the dividend) is exactly divisible by any other number (the divisor), leaving no remainder.

For Example: Let’s take the number 36, if we check for factors of 36, we have

36 = 1 x 36 = 2 x 18 = 3 x 12 = 4 x 9 = 6 x 6

Properties of Factors

• If a division of a number by another number leaves no remainder, then that second number is said to be the factor of the first number.
• A number can only have a finite number of factors.
• Prime numbers are those that only have themselves and the number 1 as factors.
• Composite numbers are those that have more than two factors.
• Finding a number’s factors involves using division.
• The obtained factors are always less than the initial number.

Multiples

Multiples are numbers created by multiplying the given number by integers. The multiplication table shows the multiples of a given number.

Properties of Multiples

• The results of multiplying an integer by a given number are referred to as the given number’s multiples.
• There are an infinite number of multiples of a number.
• Finding a number’s multiples requires the use of multiplication.
• The multiples of a given number exceed or are equal to that number.
• Every number is a multiple of itself.

Difference between Factors and Multiples

Some differences between factors and multiples are given in the table below:

Factors	Multiples
Factors are exact divisors of a number.	Multiple has the number as its exact divisor.
Factors of a number are finite.	Multiples of a number are infinite.
Factors are obtained by division.	Multiples are obtained by multiplication.
Factors of a number are always less than or equal to the number itself.	Multiples of a number are always greater than or equal to the number itself.

Common Factors and HCF

A common factor is any factor that two or more numbers share.

For example, take 35 and 42

Factors of 35 = 1, 5, 7, 35

Factors of 42 = 1, 2, 3, 6, 7, 14, 21, 42

Both 35 and 42 have some factors such as 1 and 7 that are common to both, these are known as common factors of 35 and 42.

Now, in this case in the list of common factors, 7 is the largest number, or we can also call it the highest common factor, i.e., HCF.

Thus, HCF or the highest common factor of a set of numbers is defined as the largest number that divides all the numbers in the given set of numbers.

Common Multiples and LCM.

Common multiples are those multiples that are shared by two or more different numbers.

For example, take 6 and 8

Some multiples of 6 are, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60 etc.

Some multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64, 72, 80 etc.

Here, both 6 and 8 share some common multiples such as 24, 48 and infinitely many more. These are known as common multiples of 6 and 8.

In this case, in the list of common multiples, 24 is the smallest, or we can also call it the least common multiple, i.e., LCM.

Thus, LCM or least common multiple of a set of numbers is defined as the smallest number that is a multiple of or is divisible by all the numbers in the given set of numbers.

Solved Examples

Question: Find the list of factors of 36.

Solution: We know that 1 and the number itself, i.e., 36, are the two trivial factors, so we will start dividing by the next number.

36 ÷ 2 = 18, Thus, 2 and 18 are two more factors of 36, moving to the next number

36 ÷ 3 = 12, Thus, 3 and 12 are two more factors of 36, moving to the next number

36 ÷ 4 = 9, Thus, 4 and 9 are two more factors of 36, moving to the next number

Clearly, 36 is not divisible by 5 since it doesn’t have 5 or 0 in the unit place, moving to the next number

36 ÷ 6 = 6, Thus, 6 is the final factor of 36.

Factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.

Summary

This article provides insight into the topic of Factors and Multiples, while also shining a light on the concept of common factors and common multiples. To completely divide another number without leaving any remainder, a number must be the factor of that other number. The multiples are the results of multiplying the given number by integers. Common multiples are those multiples that are shared by two or more different numbers. A common factor exists for two or more different numbers.

Frequently Asked Questions (FAQs)

1. What are Factors and Multiples?

Ans. Factors of a number are defined as the number that divides the given number completely and evenly without leaving any remainder.

Multiples on the other hand are the numbers obtained by multiplying the given number by different integers.

2. What are Prime Numbers?

Ans. Prime numbers are defined as numbers greater than 1 that have only 2 factors, i.e., 1 and the number itself. Some examples of prime numbers are 2, 3, 5, 7, 11, etc.

3. What is the Fundamental Theorem of Arithmetic?

Ans. The fundamental theorem of arithmetic states that every number can be broken into the product of some prime numbers, also known as its prime factors. This product is unique to a number and cannot ever change no matter how you find it, only the order of the product changes.

4. What is the Relationship between HCF and LCM of two Numbers?

Ans. The relationship between is defined as: The product of HCF and LCM of two numbers is equal to the product of the two numbers.

HCF (a, b) × LCM (a, b) = a × b