Category: Maths Notes PPT

Introduction Lines

Introduction lines are defined as a collection of infinitesimal points. Having two arrowheads, a line can be extended in either direction. A segment, on the other hand, is a finite space between two defined points. It cannot be extended in either direction. It is a subset of the line and is used to measure its length. It has been observed students often confuse introduction lines with segments. The basic difference between the introduction lines and segment is that the former can be extended while the length of the latter is fixed. This difference can be seen in class 3 lines and point angles chapter.

Types of Lines

Depending on its use, a line has four types. They are as follows:

• Vertical Line: It is a line that exists in a vertical position. In terms of direction, it runs from the North to the South. In simpler terms, it is a line running from the top of the page to the bottom of the page.

• Horizontal Line: It is a line that exists in a horizontal position. In terms of direction, it runs from the East to the West. In simpler terms, it is a line running from the left side of the page to the right side of the page.

• Parallel Line: When two lines run in the same direction without intersecting or meeting each other they are called parallel lines. These lines can be extended infinitesimally, yet they will never meet at any point.

• Perpendicular Line: Two straight lines are said to be perpendicular to each other when they intersect each other at a 90-degree angle. The measure of their angle is also known as a right angle. The point at which they intersect is called their point of intersection. Two perpendicular lines form four right angles at their point of intersection.

With the lines and angles introduction, it became easy to build shapes of variable lengths.

What is a Ray?

A Ray is a part of the line. It has a fixed starting point and can be extended in only one direction. It has no endpoint. Rays play an important role in the introduction to angles. One of the interesting things about ray is that the mathematical notation of a ray requires the latter or the end letter in the front followed by the next point. For clarity, see the below diagram, and the notation that follows. 

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Now, in the above diagram you see a ray AB, but while expressing it in mathematical notation we write [overline{BA}] with an arrow sign at the top. 

Introduction to Angles

An Angle is the space that is formed by two rays that have the same vertex or endpoint. The two rays can be extended in the direction of the pointed arrow at their end. 

Types of Angles

Depending on its use, an angle can have a number of types. Some of them are as follows:

• Acute Angle: It is an angle made by two rays that are 0° to 90° apart from each other. The angle value is always less than 90°.

• Right Angle: It is an angle made by two rays that are perpendicular to each other. The measure of this angle is always 90°.

• Obtuse Angle: It is an angle made by two rays that are 90° to 180° apart from each other. The measure of this angle is always greater than 90°.

These are the three primary angles. Some of the other types of angles include;

• Straight Angle: This is an angle whose measure is equal to 180°. It looks like a straight line and has both its arms pointing in opposite directions.

• Reflex Angle: This is a type of angle whose value is always the greater one between the two possible angles made by a pair of rays. Its range lies between 180° to 360°.

• Complete Angle: It is the angle that completes one turn in either clockwise or anti-clockwise direction and returns back to its starting position. The measure of this angle is 360°.

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The introduction to angles has made it easier to determine how far apart, angularly, two lines or rays are. Also, introduction to angles provides important insight into various properties of angles which will be discussed later through solved examples and FAQs.

Solved Examples

Example – 1. Look at the figure given below from class 3 line and point angles syllabus. Identify all the angles in it.

Solution: There are two types of angles present in the given figure. They are Acute angle and the Right Angle. ∠ACD and DCB are acute angles with a measure less than 90°. ∠ACB is the right angle.

 

Value of ∠DCB is 30°. Then, the measure of angle ACD is 90 – 30 = 60°. ∠ACD and ∠DCB are also complementary angles as their sum adds up to 90°.

Example – 2 From the figure illustrated below find the value of ∠MOP, if ∠MOQ or ∠y=100°.

Now sum of angles ∠MOP and ∠MOQ is 180°, using this piece of information, we can easily find the value of 2x or ∠MOP.

As, ∠MOP + ∠MOQ = 180°

So, 2x + 100 = 180°

On solving the above equation, you will get 2x or ∠MOP as 80°. 

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You can find more such examples in class 3 line and point angles chapter, which will help you to brush up the concept. Also, concepts discussed in class 3 line and point angles will stay with you in the next clases, as it forms the foundation.

What are Logarithms? 

Logarithms are the alternative ways of processing the methods of exponentials. Let’s understand logarithms with an example. We know 2 to the power 3 is equal to 8. This is called an exponential equation.

Let us now assume that you are asked “4 raised to which power is 64?”

The answer replied by you will be 3.

This situation can be explained as a logarithm equation log[_{4}] (64) =3

From this, we can say that log base three of twenty-seven is three. 

We can clearly understand from the exponential equation and logarithm equation that there is some relationship between them.

Define Logarithm

A logarithm is used to raise the power of a number to get a certain number. 

log[_{2}] 8 = 3

Logarithm Properties

In logarithm, we will learn about some properties which will help us solve the logarithm equations. We know the logarithm equation has the same relationship with the exponential equation. It also has some similarity between the properties of the logarithm to exponential.

The following are the properties:

Property of Product in Logarithm

It is the sum of the numbers of logarithms

For example,

Solve the logarithm for log[_{7}] (3x):

We can see that inside the bracket there are two variables, 3 and x. Now we will use the product rule to solve the logarithm

log[_{a}](3x) = log[_{a}](3) + log[_{a}](x)

We can also simplify two variables of a logarithm into a single logarithm by using the property of the product. Keep in your mind that the bases of logarithm should be the same in two variables to use the property of the product while simplifying the variables into one.

For example,

We cannot use the property of the product to simplify a logarithm which is 

log[_{4}](8) + log[_{6}](x)

Property of Quotient

It helps you to find the difference between the two variables which are in the form denominator and numerator. Now, let us see an example for our better understanding,

By solving the logarithm of log[_{5}] [frac{r}{4}] :

We are going to write this logarithm equation into different forms by using the property of quotients.

log[_{5}][frac{r}{4}] = log[_{5}](r) – log[_{5}](4)

Now, we will understand how to condense the different forms of a logarithm into dividend and divisor forms. 

log[_{3}](4) – log[_{3}](h) = log[_{3}][frac{4}{h}]

When we simplify the different form of a logarithm into dividend and divisor form, the bases of logarithm should be the same.

We don’t use the property of quotients when the bases of logarithm are not the same.

Property of Power

This property helps us to know that the power of the log is the exponent times the logarithm of the base of the power.

Let’s see an example,

By solving log[_{3}](x[^{2}]) :  

Here we will convert the single logarithm into multiple logarithms by using the property of power of logarithm.

log[_{3}](x[^{2}]) = 2.log[_{3}](x) = 2log[_{3}](x)

Now we will convert a multiple of a logarithm into a single logarithm.

Let’s take the help of the property of quotient to convert multiple logarithms into a single logarithm

5.log[_{3}](9) = log[_{3}](9[^{5}]) = log[_{3}](59049)

The Inverse of Property of Logarithms

From the definition and few examples above, we have understood that the exponential equation and logarithm equation follows the same kind of relationship as function. That means the exponential equation’s inverse is logarithm equations.

When two inverses are written in the form of equations, they equal to y.

So,

f(y)= a[^{y}] and g(y)= log[_{b}]y

This means: 

f.g= b[^{log_{b}y}] = y and g.f = log[_{b}] b[^{y}] = y 

These are called the inverse properties of Logarithm. 

Application of Logarithms

In this world of modern technology, people are always finding ways to do things in simpler and easier ways. Therefore, people invented calculators and logarithms to make mathematical equations easier to solve. 

So, let us find some more advantages of learning logarithms:

• Logarithms are used in various fields of science and many other industries.

• Logarithms help to find the pH value in chemistry because the value for pH can be small, so we use the logarithm to have a range for using it for small numbers.

• Logarithms are widely used in banking.

• Logarithms are used to find the half-life of radioactive material.

• It allows us to find out the earthquake’s intensity.

• Even in the field of medicine or engineering, we can see some usage of logarithm and its properties.

An Overview of Mathematical Reasoning

Do you remember, how as a middle-school grader you were often given worksheets to solve, that had questions like, “all odd numbers are prime numbers: true or false”? These statements could be true or false. These questions formed the basis of what is today known as mathematical reasoning. It is the branch of mathematics that deals with the truth in a given mathematical statement.

Math and reasoning go hand in hand, forming a very crucial part of the syllabus for JEE and other competitive exams. Let us take a look at what reasoning in math means and how we can solve important questions.

What are Mathematically Acceptable Statements?

Suppose you are given a statement that says:

“The square root of 1 is always 1.”

Thus this statement can either be true or be false, but cannot be both. This is the basic rule to follow when you solve problems in mathematics and reasoning. The statement above states that the square root of 1 is always 1. Therefore it is a true statement. Such a statement is mathematically acceptable. Other ambiguous statements, such as “the sum of prime numbers is even” makes no sense because we cannot be sure about the outcome. Therefore the rules governing a mathematically acceptable statement are:

1. A statement can be mathematically acceptable only if it is either true or false. It cannot be both.

2. Statements in mathematical logical reasoning can be none of these three things: exclamatory, interrogative or imperative.

3. A statement may have variables in it. Such a declarative statement is considered an open statement, only if it becomes a statement when these variables are replaced by some constants.

4. A mathematical statement that is a combination of two or multiple statements is known as a compound statement.

5. Compound statements are usually joined with the help of “and” or (^). Such a statement is denoted with p^q.

6. A compound statement containing “and” is true, if and only if all its components are found to be true.

The Following Table Illustrates the Truth Values of p^q and q^p:

Before you proceed further into compound statements, various fallacies and laws governing reasoning math, let us learn more about the various types of reasoning in maths.

Types of Maths Logical Reasoning

Logical and mathematical reasoning is key to knowing mathematics and sailing through the world of practical math. Doing, or applying mathematical principles in real life is a creative act, and reasoning is the basis of that act. It is a very useful way to make sense of the real world and nurture mathematical thinking. 

In mathematics, two kinds of reasoning demonstrate the logical validity of any statement. These are:

a. Inductive Reasoning

This involves looking for a pattern in a given set of problem statements and generalising. For instance, a student may use inductive reasoning when looking at a set of different shapes, such as rectangles, rhombuses and parallelograms. Further, he/she might try to spot the similarities and differences, and figure what is common. Therefore between a set of squares and circles, a student can use this type of reasoning to consider the shapes that are not squares. The following figure will demonstrate a similar situation.

You may also use inductive reasoning to spot patterns in a series of even numbers, multiplying numbers by ten, or while working with roots, indices and exponents. For instance, you may look at 20/100 and 30/120 and inductively reason that common multiples in a fraction can always be cancelled. Similarly, if all these fractions form part of a series, you should remember to continue evaluating it this way, before generalising. Also, remember that inductive reasoning is always non-rigorous and statements are usually generalised. 

Rack Your Brains: Work out the following statement and find out if it uses inductive reasoning.

Statement: The cost of materials used to make a bar of Dairy Milk is Rs. 20, and the cost of labour required to manufacture it is Rs.15. The selling price of the chocolate is Rs.60.

Reasoning: It is clear from the above statement, that stores and shops will make a hefty profit selling this bar of chocolate.

Before you generalise a statement and look for the truth in it, you must practise care to prove it through deductive reasoning including mathematical reasoning.

b. Deductive Reasoning

This involves drawing logical conclusions, stating a logical argument, and then generalising a specific situation. For instance, if you know what a parallelogram or a square is, you will be able to apply this generalisation to a new pattern of figures. You will be able to conclude if each of these shapes is a square or not.

In fact, deductive reasoning is the opposite of inductive reasoning. In deductive reasoning, we apply the general rules to a given statement and see if we can make the subsequent statements true.

This was all about the different types of reasoning in math. Check out our expertly-curated collection of mathematical reasoning NCERT questions, notes on different types of statements, and reference notes, available on the site. Watch free demo classes on the app and make math fun.

How To Find Mean Value

Before heading to mean, let’s get to know where it arises from and why it is so important to define mean in math. You might already be familiar that it is super complicated to decide and deem from anything unprocessed until the data is arranged properly. For a data arranged properly into a frequency distribution, the details contained in the data could be understood effortlessly. Now the point is where does the mean come from? Mean is one type of average that represents all the values of the distribution in some specific way.

Types of Averages

Most frequently used average are:-

1. Mean

2. Median

3. Mode

4. Range

When you have a huge set of data there are all sorts of easy ways to mathematically express the data. “Average” is used profusely with data sets. Mean, median, and mode, together with range are all different kinds of averages which help simplify the data for easy description.

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Define Mean In Math

When people say “average” they are generally talking about the mean. And it is commonly the arithmetic mean. Mean, typically in arithmetic math is the “average” number; established by summing up all data points and dividing by the number of data points.

Significance Of Mean In Mathematics And Statistics

Whether you are a math student, researcher, econometrician, statistician, you would invariably need to compute the average of multiple numbers from now and then.

A crucial characteristic of the mean is that it includes every value in your data series as a component of the calculation. More so, the mean is the only measure of central tendency where the total value of the deviations of each observation from the mean is always zero. The statistical mean also provides an effective scheme about simplifying the statistical data.

Calculating The Mean

You can calculate the arithmetic mean by totaling the value of all the numbers in the data and then dividing by the number of data points. To say, mean=sum of data# of data points

Formula to calculate mean written in mathematical expression: Mean = [frac{sum_{x_{i}}^{n}}{nsum x_{i}}]​​

Mean = Sum of all Data Points/Number of Data Points

Mean = Assumed Mean + (Sum of all Derivations/Number of Data Points)

Solved Example 

Example Problem Finding Mean

Find the mean of the following data set:

5, 11, 15, 17, 22

Finding the mean:

First add up all the numbers: 5 + 11 + 15 + 17 + 22 = 70

Then divide 70 by the total number of data points i.e. 5, and you get 14. Thus, the mean is 14.

Specific “Means” Commonly Used In Statistics

Statistical mean is a computation of central tendency and provides you a proposition about where the data appears to assemble around. Most common types of mean applied in stats for a number of different experimentation include;-

Example Problem Of Using Statistical Mean

The mean marks secured by a student Jessica in an examination are needed to accurately measure the performance of a student in that test. If the student obtains a low percentage, but is far ahead of the mean, then it states the exam paper is difficult and thus Jessica’s performance is satisfactory, something that simply a percentage will be unable to deduce.

Mean and its Types in Statistics

There are various other types of mean in statistics that can be used in various branches of math. Most have very narrow applications as well as meanings to domains like physics, finance.

Most common types of mean you’ll come across are as follows:-

1. Weighted Mean

Fairly frequent in statistical application, weighted mean are particularly used when studying populations. Rather of each data point bestowing equally to the final average, some data points impart more than others. If all the weights are equal, then this will make the arithmetic mean equivalent.

2. Geometric Mean

With ultra limited, and specific uses in finance, social sciences, stock analysis and technology., geometric mean can be used to determine the average rate of return. For example, you own mutual funds that earn 7% the first year, 25% the second year, and 15% the third year. If you seek to find out the average rate of return, you can’t use the arithmetic average. Wondering Why? It is due to the reason that when you are identifying rates of return you are multiplying, not adding.

3. Arithmetic-Geometric Mean

This is widely used in calculus and in machine computation or computer calculations. It has a relation to the perimeter of an ellipse as because when it was first invented by Gauss, it was used to compute planetary orbits. With arithmetic-geometric, you can find simple and clear explanations of complicated math concepts.

4. Root-Mean Square

With wide and specific application, this type of mean works quite fruitfully in fields that study electrical engineering like electromagnetism, sine waves. This specific type is also known as the quadratic average.

5. Harmonic Mean

Fairly used in the field of finance, the harmonic mean is a type of average used to identify the average for financial numerals such as price-to-earnings (P/E) ratio. To figure out the harmonic mean, use the harmonic formula which is dividing the number of observations in the data series by the sum of each value of reciprocals (1/x_i) in the data series.

6. Heronian Mean

This mean type is used in geometry to determine volume of a pyramidal frustrum. A pyramidal frustrum is a pyramid made by chopping off the top of the pyramid. 

Fun Facts

• There are some situations where weighted mean can provide inaccurate information, as illustrated by Simpson’s Paradox

• Harmonic mean provides better average than the arithmetic mean in certain circumstances involving rates and ratios

• “Median” contains just the same number of letters as “Middle”.

Mensuration is a division of mathematics that studies geometric figure calculation and its parameters such as area, length, volume, lateral surface area, surface area, etc. It outlines the principles of calculation and discusses all the essential equations and properties of various geometric shapes and figures.

 

What is Mensuration?

Mensuration is a subject of geometry. Mensuration deals with the size, region, and density of different forms both 2D and 3D. Now, in the introduction to Mensuration, let’s think about 2D and 3D forms and the distinction between them.

What is a 2D Shape?

A 2D diagram is a shape laid down on a plane by three or more straight lines or a closed segment. Such forms do not have width or height; they have two dimensions-length and breadth and are therefore called 2D shapes or figures. Of 2D forms, area (A) and perimeter (P) is to be determined.

 

What is a 3D Shape?

A 3D shape is a structure surrounded by a variety of surfaces or planes. These are also considered robust types. Unlike 2D shapes, these shapes have height or depth; they have three-dimensional length, breadth and height/depth and are thus called 3D figures. 3D shapes are made up of several 2D shapes. Often known as strong forms, volume (V), curved surface area (CSA), lateral surface area (LSA) and complete surface area (TSA) are measured for 3D shapes.

 

Difference between 2D and 3D shapes

 

 

Introduction to Menstruation: Important Terms

Until we switch to the list of important formulas for measurement, we need to clarify certain important terms that make these measurement formulas:

Area (A):

The area is called the surface occupied by a defined closed region. It is defined by the letter A and expressed in a square unit.

Perimeter (P): 

The total length of the boundary of a figure is called its perimeter. Perimeter is determined by only two-dimensional shapes or figures. It is the continuous line along the edge of the closed vessel. It is represented by P and measures are taken in a square unit.

Volume (V):

The width of the space contained in a three-diMensional closed shape or surface, such that, the area by a room or cylinder. Volume is denoted by the alphabet V and the SI unit of volume is the cubic meter.

Curved Surface Area (CSA):

The curved surface area is the area of the only curved surface, ignoring the base and the top such as a sphere or a circle. The abbreviation for the curved surface area is CSA.

Lateral Surface Area (LSA):

The total area of all of a given figure’s lateral surfaces is called the Lateral Surface Area. Lateral surfaces are the layers covering the artefact. The acronym for the lateral surface area is LSA.

Total Surface Area (TSA):

The calculation of the total area of all surfaces is called the Cumulative Surface Region in a closed shape. For example, we get its Total Surface Area in a cuboid by adding the area of all six surfaces. The acronym for the total surface area is TSA.

Square Unit (/):

One square unit is simply the one-unit square area. When we quantify some surface area, we relate to the sides of one block square to know how many of these units will fit in the figure given.

Cube Unit (/):

One cubic unit is the one-unit volume filled by a side cube. When we calculate the volume of any number, we refer to this cube of one unit and how many these component cubes will fit in the defined closed form. 

 

Tools Require for Mensuration

Calliper – A tool for measuring the diameter.

Try Square – A tool for determining the squareness and flatness of a surface.

Meter Stick – A measuring device with a one-meter length.

Compass – An arc and circle drawing tool.

 

List of Mensuration Formulas for 2D shapes:

As our introduction to Mensuration and the relevant words are over, let’s switch to the equations for Mensuration, as this is a discussion focused on an equation. The 2D figure has a list of formulas of measurement that define a relationship between the various parameters. Let’s look into detail about the estimation equations of some kinds.

Square:

Area  = [(side)^{2}] sq. units.

Perimeter = [(4 times sides)] units.

Diagonal = [sqrt{2 times side}]units.

 

Rectangle:

Area = [(length times breadth) ]sq. units.

Perimeter = 2(length + breadth) units.

Diagonal, D = [sqrt{length^{2} + breath^{2}}]units.

 

Scalene Triangle:

Area, [A = frac{height times base}{2} ]sq. units 

Perimeter = (side a + side b + side c) units.

 

Equilateral Triangle:

Area = [ frac{sqrt{3}}{4} times side^{2} ] sq. units.

Perimeter = [(3 times side)] units.

 

Isosceles Triangle:

Area = [A = frac{height times base}{2} ]sq. units 

Perimeter = [(2 times side ] + base) units.

 

Right Angled Triangle:

Area = [A = frac{leg_{a}times leg_{b}}{2} ]sq. units 

Perimeter = [leg_{a} + leg_{b} + sqrt {leg_{a}^{2} + leg_{b}^{2}}] units

Hypotenuse = [sqrt {leg_{a}^{2} + leg_{b}^{2}}] units

 

Circle:

Area = [ π times radius^{2} ] sq. units.

Circumference =[ 2π times radius] units.

Diameter, D = [2 times radius ] units.

 

List of Mensuration Formulas for 3D shapes:

The 3D figure has a list of formulas for measurement that define a relationship between the various parameters. Let’s look into the details about the estimation equations of some kinds.

 

Cube:

Volume =  [ side^{3} ] cubic units.

Lateral Surface Area =  [4 times side^{2}] sq. units.

Total Surface Area = [6 times side^{2}] sq. units.

Diagonal Length d = [sqrt {length^{2} + width^{2} + height^{2}}] units. 

 

Cuboid:

Volume = (length + width + height) cubic units.

Lateral Surface Area =[ 2 times height (length + width) ]sq. units.

Total Surface Area = [2(length times width + length times height + height times width)] sq. units.

Diagonal Length = [ length^{2} + breadth^{2} + height^{2}] units.

 

Sphere:

Volume =  [ frac{4}{3} prod times side^{2} ] cubic units.

Surface Area = [4prod times radius^{2} ]sq. Units.

 

Hemisphere:

Volume = [ frac{2}{3} prod times side^{2} ] cubic units.

Total Surface Area =[3prod times radius^{2} ]sq. Units.

 

Cylinder:

Volume = [prod times radius^{2} times height ] cubic units.

Curved Surface Area (excluding the areas of the top and bottom circular regions) = [(2 times prod times R times h)] sq. units.

Where, R = radius

Total Surface Area =  [(2 times prod times R times h) + (2 times prod times R^{2}) ] sq. units 

 

Cone:

Volume =  [ frac{1}{3} prod times radius^{2} times height ] cubic units.

Curved Surface Area =[ 𝜋 times radius times height] sq. units.

Total Surface Area = 𝜋 x radius(length + height) sq. units 

Slant Height of Cone =[ sqrt{height^{2} + base radius^{2}}] sq. units.

Using these above formulas for the Mensuration, most of the Mensuration problems can be solved.

Fraction, Latin for ‘broken’ symbolises a part of a whole number, or in a more general way, equal parts of any number. In regular spoken English, the word ‘fraction’ is spoken as to describe the number of parts there are of a certain size, like one-third, eighty-fifth, etc.

A fraction has two parts, a numerator and a denominator. Numerator comes above the line, and a denominator (that is not a zero) comes below the line separating these two. In Mathematics, there are three types of fractions, proper, improper, and mixed fraction. In this following discussion, students get to learn about mixed fraction, an important topic of Mathematics. This sub-type of fraction helps to simplify the representation of fraction numbers.

Definition of Mixed Fraction

A mixed fraction is a combination of a proper fraction and a whole number. Usually, it represents a number between any two. For instance, 1(1/3) is a mixed fraction, where 1 is a whole number and 1/3 is a fraction, and together they form this subtype of fractions.

This concept is straightforward and easy to grasp. However, students must also know about proper and improper fractions to ensure that they comprehend this topic properly.

1. Proper Fraction

A proper fraction is a vital sub-type that students must know about, and to understand this concept of mixed fraction thoroughly. In this type, the numerator is smaller than its denominator. For example, in a fraction of 3/5, three the numerator is smaller than five.

2. Improper Fraction

In case of an improper fraction, the rules are reversed. Here the numerator is equal or larger than its denominator. For instance, in a fraction of 7/4, 7 the numerator is larger than its denominator, i.e. 4.

A point to note here is that the idea of mixed fractions helps to define improper fraction is a simpler way. In smaller fractions like the one mentioned above, it is not difficult to read them, but in case of a bigger one, it is not easy to read them.

Hence, this idea of a mixed fraction is of great help in this regard.

How to Convert Improper Fractions to Mixed Fractions?

Here are the steps of changing improper fractions to mixed numbers –

Step 1: 

Dividing the numerator with denominator. For instance 15/7

Step2: 

The integer of this division will be the integer of the following mixed fraction, i.e. 2

Step 3: 

The denominator will remain the same with 7

Step 4: 

Therefore, the improper fraction of 15/7 is now a mixed fraction of 2(1/7)

How to Convert Mixed Fraction to Improper Fraction?

Here are the step for convert to improper fraction from a mixed one –

Step 1: 

Taking a cue from the previous one, the first step here is to multiply the denominator with the whole number. Hence, 2X7 = 14

Step 2: 

After that, add the numerator to this result, i.e. 14+1= 15

Step 3: 

Retain the same denominator.

Step 4: 

The conversion of a mixed number to an improper fraction is now complete with 15/7 as the outcome

A mixed fraction is a vital part of Mathematics, and it is essential for students to know about it. It helps them learn how to calculate and represent fractions is a way that does not include decimal placing.

Moreover, using mixed fractions is easier and offer more clarity while reading and representing any complicated fraction numbers. Students can learn more about this concept from the website of India’s leading e-learning provider . At the website, they will find study materials, exam suggestions, practice sets for various topics that can help students to improve their performance. Furthermore, the live online classes and doubt clearing sessions also aid in exam preparations.