Construct Different Types of Quadrilaterals | How to Construct Quadrilaterals of Different Types?

Do you want to know How to construct different types of quadrilaterals? Different types of quadrilaterals are developed depending on the sides, diagonals, and also angles. Have a look at a step by step explanation to construct various types of quadrilaterals. We have given different problems on the construction of quadrilaterals along with steps for better understanding. Look at them and practice all the problems given below and enhance your conceptual knowledge.

How to Construct Quadrilaterals? | Steps of Construction

You can refer to the below available various questions on constructing quadrilaterals along with a detailed explanation. For the sake of your comfort, we even jotted Steps of Construction for each and every problem so that you can solve similar kinds of questions easily.

1. Construct a parallelogram PQRS in which PQ = 7 cm, QR = 5 cm and diagonal PR = 7.8 cm.

Steps of Construction:
Firstly, draw a rough figure of the quadrilateral with the given dimensions.
construction of quadrilateral 13

1. Draw a line segment of length 7 cm and mark the ends as P and Q.
2. Take the point P as a center and draw an arc by taking the radius 7.8 cm.
3. Next, take point Q as a center and draw an arc by taking the radius 5 cm. Mark the point as R where the two arcs cross each other. Join the points Q and R as well as P and R.
Note: A parallelogram is a simple quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure.
4. By taking the point P as a center, draw an arc with a radius of 5 cm.
5. By taking the point R as a center, draw an arc with a radius of 7 cm.
6. Mark the point as S where the two arcs cross each other. Join the points R and S as well as P and S.

PQRS is a required parallelogram.

construction of quadrilateral 14

2. Construct a parallelogram, one of whose sides is 7.2 cm and whose diagonals are 8 cm and 8.4 cm.

Steps of Construction:
Firstly, draw a rough figure of the quadrilateral with the given dimensions.
construction of quadrilateral 15

1. Draw a line segment of length 7.2 cm and mark the ends as P and Q.
2. Take the point P as a center and draw an arc by taking the radius 4.2 cm.
3. Next, take point Q as a center and draw an arc by taking the radius 4 cm. Mark the point as O where the two arcs cross each other. Join the points Q and O as well as P and O.
4. By taking the point O as a center, draw an arc with the required radius.
5. Produce PO to R such that OR = PO and produce QO to S such that OS = OQ.
6. Join PS, QR, and RS.

PQRS is a required parallelogram.

3. Construct a parallelogram whose diagonals are 5.6 cm and 6.4 cm and an angle between them is 70°.

Steps of Construction:
1. Draw a line segment of length 5.6 cm and mark the ends as P and R.
2. Take the point O as a center in between P and R.
3. Next, take point O as a center and make a point by taking 70º using a protector. Draw a line XO to Y.
4. Set off OQ = 1/2 (6.4) = 3.2 cm and OS = 1/2 (6.4) =3.2 cm as shown.
5. Join PQ, QR, RS, and SP.
construction of quadrilateral 16

PQRS is a required parallelogram.

4. Construct a rectangle PQRS in which side QR = 5.2 cm and diagonal QS = 6.4 cm.

Steps of Construction:
Firstly, draw a rough figure of the quadrilateral with the given dimensions.
construction of quadrilateral 17

1. Draw a line segment of length 5.2 cm and mark the ends as Q and R.
2. Take the point R as a center and draw a perpendicular line to QR.
3. Next, take point Q as a center and draw an arc by taking the radius 6.4 cm. Mark the point as S where the line and arc cross each other. Join the points Q and S as well as R and S.
4. By taking the point S as a center, draw an arc with the required radius of 5.2 cm.
5. Take the point Q as a center and draw a perpendicular line to QR. Mark the point as P where the point and arc cross each other. Join the points Q and P as well as P and S.

PQRS is a required rectangle.

construction of quadrilateral 18

5. Construct a square PQRS, each of whose diagonals is 5.4 cm.

Steps of Construction:
1. Draw a line segment of length 5.4 cm and mark the ends as P and R.
2. Draw the right bisector XY of PR, meeting PR at O.
3. From O set off OQ = 1/2 (5.4) = 2.7 cm along OQ and OS = 2.7 cm along OX.
4. Join PQ, QR, RS, and SP.

PQRS is a required square.
construction of quadrilateral 19

6. Construct a rhombus with a side of 4.4 cm and one of its angles equal to 67°.

Steps of Construction:
Given that a rhombus with a side of 4.4 cm and one of its angles equal to 67°.
The adjacent angle = (180° – 67°) = 113°.
1. Draw a line segment of length 4.4 cm and mark the ends as Q and R.
2. Make ∠RQX = 113° and ∠QRY = 67°.
3. Set off QP = 4.4 cm along with QX and RS = 4.4 cm along with RY.
4. Join PS.

PQRS is a required rhombus.

construction of quadrilateral 20

 

Quadrilateral- Definition, Types, Properties, Formulas, Notes

Different Geometry shapes and objects are named based on the number of sides. If an object has three sides, then it is classified as Triangle, An object with 4 sides classified as Quadrilateral, etc. Let us learn about the Quadrilateral definition, types, formula, properties, etc. in detail in this article. Every concept is explained separately on our website. Access every topic and easily get a grip on the Quadrilateral concept.

List of Quadrilateral Concepts

Find different concepts of Quadrilateral by checking out the below links. All you need is simply tap on them to have an idea of the related concept.

Quadrilateral Definition

A quadrilateral defined as a figure that has four sides or edges. Also, the quadrilateral consists of four vertices. rectangle, square, trapezoid, and kite, etc. are some of the examples of Quadrilateral.

Types of Quadrilaterals

There are various types of Quadrilaterals available. All the Quadrilaterals must have 4 sides. Also, the sum of the angles of the Quadrilateral is 360 degrees.

  1. Trapezium
  2. Kite
  3. Parallelogram
  4. Rectangle
  5. Squares
  6. Rhombus

Also, the quadrilaterals are classified differently. They are
Convex Quadrilaterals: It is defined as both diagonals of a quadrilateral are always present within a figure.
Concave Quadrilaterals: Concave Quadrilaterals one diagonals present outside of the figure.
Intersecting Quadrilaterals: The pair of non-adjacent sides intersect in Intersecting Quadrilaterals. These are also called self-intersecting or crossed quadrilaterals.

Quadrilateral Formula

Check out the below formula of a Quadrilateral.

Area of a Parallelogram = Base x Height
Area of a Square = Side x Side
Area of a Rectangle = Length x Width
Area of a Kite = 1/2 x Diagonal 1 x Diagonal 2
Area of a Rhombus = (1/2) x Diagonal 1 x Diagonal 2

Quadrilateral Properties

Know the different properties of a Quadrilateral PQRS.

  • Four sides: PQ, QR, RS, and SP
  • ∠P and ∠Q are adjacent angles
  • Four vertices: Points P, Q, R, and S.
  • PQ and QR are the adjacent sides
  • Four angles: ∠PQR, ∠QRS, ∠RSP, and ∠SPQ.
  • ∠P and ∠R are the opposite angles
  • PQ and RS are the opposite sides

Important Properties of Quadrilateral

  • Every quadrilateral consists of 4 sides, 4 angles, and 4 vertices.
  • Also, the total of interior angles = 360 degrees

Properties of a Square

  • The sides of a square are parallel to each other.
  • Also, all the sides are equal in measure.
  • The diagonals of a square perpendicular bisect each other.
  • All the interior angles of a square are at 90 degrees.

Rectangle Properties

  • The diagonals of a rectangle bisect each other.
  • The opposite sides consist of equal length in a rectangle.
  • All the interior angles of a rectangle are at 90 degrees.
  • The opposite sides are parallel to each other

Properties of a Rhombus

  • By adding two adjacent angles of a rhombus we get 180 degrees.
  • The opposite sides are parallel to each other in a rhombus.
  • The diagonals perpendicularly bisect each other
  • All four sides of a rhombus are of equal measure.
  • The opposite angles are of the same measure.

Parallelogram Properties

  • The opposite angles of a parallelogram are of equal measure.
  • The opposite side is of the same length in a parallelogram.
  • The sum of two adjacent angles of a parallelogram is equal to 180 degrees.
  • The opposite sides are parallel to each other in a Parallelogram.
  • The diagonals of a parallelogram bisect each other.

Trapezium Properties

  • The two adjacent sides are supplementary in a trapezium.
  • Only one pair of the opposite side is parallel to each other in a trapezium.
  • The diagonals of a trapezium bisect each other in the same ratio

Kite Properties

  • The large diagonal bisects the small diagonal of a kite.
  • The pair of adjacent sides have the same length in a kite.
  • Only one pair of opposite angles are of the same measure.

Notes on Quadrilateral

  • A quadrilateral is a parallelogram is 2 pairs of sides are parallel to each other.
  • Also, a quadrilateral is a trapezoid or a trapezium if two of its sides are parallel to each other.
  • A quadrilateral is a rhombus if all the sides are of equal length and the two pairs of sides are parallel to each other.
  • The quadrilateral becomes kite when 2 pairs of adjacent sides are equal to each other.
  • Quadrilateral becomes Square and Rectangle when all internal angles are right angles, all angles are right angles, and also the opposite sides of a rectangle are same.
  • Furthermore, the Quadrilateral becomes Square and Rectangle when Opposite sides of a rectangle and square are parallel. The sides of a square are of the same length.

What is Ratio and Proportion? – Definition, Formulas, Examples with Answers

Ratio and Proportion are mainly explained using fractions. If a fraction is expressed in the form of a:b it is called a ratio and when two ratios are equal it is said to be in proportion. Ratio and Proportion is the fundamental concept to understand various concepts in maths. We will come across this concept in our day to day lives while dealing with money or while cooking any dish. Check out Definitions, Formulas for Ratio and Proportion, and Example Questions belonging to the concept in the further modules.

Quick Links of Ratio and Proportion Topics

If you want to get a good hold of the concept Ratio and Proportion you can practice using the quick links available for various topics in it. You just need to tap on the direct links available and get a good grip on the concept.

What is Ratio and Proportion?

Ratio and Proportion is a crucial topic in mathematics. Find Definitions related to Ratio and Proportion along with examples here.

In Certain Situations comparison of two quantities by the division method is efficient. Comparison or Simplified form of two similar quantities is called ratio. The relation determines how many times one quantity is equal to the other quantity. In other words, the ratio is the number that can be used to express one quantity as a fraction of other ones.

Points to remember regarding Ratios

  • Ratio exists between quantities of a similar kind
  • During Comparison units of two things must be similar.
  • There should be significant order of terms
  • Comparison of two ratios is performed if the ratios are equivalent similar to fractions.

Proportion – Definition

Proportion is an equation that defines two given ratios are equivalent to each other. In Simple words, Proportion states the equality of two fractions or ratios. If two sets of given numbers are either increasing or decreasing in the same ratio then they are said to be directly proportional to each other.

Ex: For instance, a train travels at a speed of 100 km/hr and the other train travels at a speed of 500km/5 hrs the both are said to be in proportion since their ratios are equal

100 km/hr = 500 km/5 hrs

Continued Proportion

Consider two ratios a:b and c:d then in order to find the continued proportion of two given ratio terms we need to convert to a single term/number.

For the given ratio, the LCM of b & c will be bc.

Thus, multiplying the first ratio by c and the second ratio by b, we have

The first ratio becomes ca: bc

The second ratio becomes bc: bd

Thus, the continued proportion can be written in the form of ca: bc: bd

Ratio and Proportion Formulas

Ratio Formula

Let us consider, we have two quantities and we have to find the ratio of these two, then the formula for ratio is defined as

a: b ⇒ a/b

a, b be two quantities. In this a is called the first term or antecedent and b is called the second term or consequent.

Example: In the Ratio 5:6 5 is called the first term or antecedent and 6 is called the consequent.

If we multiply and divide each term of the ratio by the same number (non-zero), it doesn’t affect the ratio.

Proportion Formula

Consider two ratios are in proportion a:b&c:d the b, c are called means or mean terms and a, d are known as extremes or extreme terms.

a/b = c/d or a : b :: c : d

Example: 3 : 5 :: 4 : 8 in this 3, 8 are extremes and 5, 4 are means

Properties of Proportion

Check out the important list of properties regarding the Proportion Below. They are as follows

  • Addendo – If a : b = c : d, then a + c : b + d
  • Subtrahendo – If a : b = c : d, then a – c : b – d
  • Componendo – If a : b = c : d, then a + b : b = c+d : d
  • Dividendo – If a : b = c : d, then a – b : b = c – d : d
  • Invertendo – If a : b = c : d, then b : a = d : c
  • Alternendo – If a : b = c : d, then a : c = b: d
  • Componendo and dividendo – If a : b = c : d, then a + b : a – b = c + d : c – d

Difference Between Ratio and Proportion

S.No.RatioProportion
1The ratio is used to compare two similar quantities having the same unitsThe proportion is used to express the relation of two ratios
2It is expressed using a colon (:), slash (/)It is expressed using the double colon (::) or equal to the symbol (=)
3The keyword to identify ratio in a problem is “to every”The keyword to identify proportion in a problem is “out of”
4It is an expressionIt is an equation

Fourth, Third and Mean Proportional

If a : b = c : d, then:

d is called the fourth proportional to a, b, c.
c is called the third proportion to a and b.
Mean proportional between a and b is √(ab).
Comparison of Ratios
If (a:b)>(c:d) = (a/b>c/d)

The compounded ratio of the ratios: (a : b), (c : d), (e : f) is (ace : bdf).

Duplicate Ratios

If a:b is a ratio, then:

  • a2:b2 is a duplicate ratio
  • √a:√b is the sub-duplicate ratio
  • a3:b3 is a triplicate ratio

Ratio and Proportion Tricks

Check out the Tricks and Tips to Solve Problems related to Ratio and Proportion. They are as under

  • If u/v = x/y, then u/x = v/y
  • If u/v = x/y, then uy = vx
  • If u/v = x/y, then v/u = y/x
  • If u/v = x/y, then (u-v)/v = (x-y)/y
  • If u/v = x/y, then (u+v)/v = (x+y)/y
  • If u/v = x/y, then (u+v)/ (u-v) = (x+y)/(x-y), it is known as Componendo Dividendo Rule
  • If a/(b+c) = b/(c+a) = c/(a+b) and a+b+ c ≠0, then a =b = c

Solved Questions on Ratio and Proportion

1. Are the Ratios 4:5 and 5:10 said to be in Proportion?

Solution:

Expressing the given ratios 4:5 we have 4/5 = 0.8

5:10 = 5/10 = 0.2

Since both the ratios are not equal they are not in proportion.

2. Out of the total students in a class, if the number of boys is 4 and the number of girls being 5, then find the ratio between girls and boys?

Solution:

The ratio between girls and boys is 5:4. The ratio can be written in factor form as 5/4

3. Two numbers are in the ratio 3 : 4. If the sum of numbers is 42, find the numbers?

Solution:

Given 3/4 is the ratio of any two numbers

Let us consider the numbers be 3x and 4x

Given, 3x+4x = 42

7x = 42

x = 42/7

x = 6

finding the numbers we have 3x = 3*6 = 18

4x = 4*6 = 24

Therefore, two numbers are 18, 24

 

 

 

Place Value Chart Definition | All About Indian & International System of Place Value Charts with Examples

In maths, Numbers are used for common tasks like counting, measurements, and comparisons. A Place Value is a basic mathematical concept, important for every arithmetic math operations. A place value can be represented for both whole numbers and decimals. A place value chart can assist students in identifying and comparing the position of the digits in the given numbers through millions.

The place value of a digit rise by ten times as we move left on the place value chart and drops by ten times as we move right. While representing the number in general form, the place of each digit will be expanded. Let’s understand What is Place Value Chart, what are the systems of place value charts, how to solve the place values explicitly from this article.

Place Value Chart

Place value charts in mathematics support students and even learners to ensure that the digits are in the correct places. To recognize the positional values of numbers correctly, writing the digits in the place value chart is the best way and then address the numbers in the general and the standard form.

Here, we have presented the Indian system place value chart & Internal system place value chart for reference. Go through these two charts and identify the place values of the given number.

Indian Place Value Chart System

It is a chart that represents the value of each digit in a number on the basis of its position. As you noticed in the below Indian place value chart, the nine places are grouped into four periods: Ones, Thousands, Lakhs, and Crores. When reading the number, all digits in the same period are read together as well as with the period name, exclude the one’s period.

Note: If a period contains zero, we do not name that period in the word form.

The Indian System of Place value chart is given below.

Place Value Chart For Indian System
CroresLakhsThousandsOnes
Ten Crores (TC)

(10,00,00,000)

Crores (C) (1,00,00,000)Ten Lakhs (TL) (10,00,000)Lakhs (L) (1,00,000)Ten-Thousands (TTh) (10,000)Thousands (Th) (1000)Hundreds (H) (100)Tens (T) (10)Ones (O) (1)

Below is an example that shows the relationship between the place or position and the place value of the digits in the given number 13548.

In 13548, 1 is in ten thousand’s place and its place value is 10,000,
3 is in the thousands place and its place value is 3,000,
5 is in the hundreds place and its place value is 500,
4 is in the tens place and its place value is 40,
8 is in one place and its place value is 8.

example for indian place value chart of the given number

International Place Value Chart

In the international place value chart, the digits are classified into three periods in a big number. The number is read from left to right as billion, million, thousands, ones.

  • 100,000 = 100 thousand
  • 1,000,000 = 1 million
  • 10,000,000 = 10 millions
  • 100,000,000 = 100 millions

The place value chart of the International System is given below:

Place Value Chart For International System
MillionsThousandsOnes
Hundred- Millions (HM)
(100,000,000)
Ten-Millions (TM)
(10,000,000)
Millions (M)
(1,000,000)
Hundred -Thousands (HTh)
(100,000)
Ten- Thousands (TTh)
(10,000)
Thousands (Th)
(1000)
Hundreds (H)
(100)
Tens (T)
(10)
Ones (O) (1)

Comparison Between Indian and International System of Place Value

In this section, you will have a glance at the comparison between both the Indian and International place value system:

Indian Place Value ChartInternational Place Value Chart
Nine places are grouped into four periods: ones, thousands, lakhs, and crores.Nine places are grouped into three periods: ones, thousands, and millions.
Place Values: Ones, tens, hundreds, thousands, ten thousand, lakhs, ten lakhs, crores, and ten crores.Place Values: Ones, tens, hundreds, thousands, ten thousand, hundred thousand, millions, ten million, and hundred million
Lakhs and crores are Indians units in the Indian Place Value Chart.Millions and billions are international units in the International Place Value Chart.

Decimals Place Value

In decimals, place value represents the position of each digit after the decimal point and before the decimal point. A place-value chart tells you how many hundreds, tens, and ones to use. The place value of decimals is based on multiplying by 1/10.

  Hundred
Thousands
      Ten
Thousands
ThousandsHundredsTensOnes.OnesTenthsHundredths

place value chart for decimals

Place Value Table

NumberPlace ValueValue of digit
67,891,234Units / Ones4
67,891,234Tens30
67,891,234Hundreds200
67,891,234Thousands1,000
67,891,234Ten thousand90,000
67,891,234Hundred thousand800,000
67,891,234Millions7,000,000
67,891,234Ten million60,000,000

Solved Examples:

Example 1: Find the place value for the number 27349811 in the International place value system & address it with commas and in words.

Solution:

MILLIONTHOUSANDSONES
T.MMH.ThT.ThThHTO
27349811

The given number place values representation with commas is 27,349,811 and in words is Twenty-seven million three hundred forty-nine thousand eight hundred eleven.

Example 2: Identify the place value of digits for the given number 13548 using base-ten blocks?

Also, The place value of digits of the number can be positioned using base-ten blocks and aid learners write numbers in their expanded form. The below image has a solution for place value of digits for the given number 13548 using base-ten blocks:

example of place value chart using base-ten blocks

FAQs on Place Value Chart

1. What is Place Value with Example?

The position of each digit in a number is known as a place value. The place value of digits is determined as ones, tens, hundreds, thousands,ten-thousands, and so on, based on their place in the number. For instance, the place value of 7 in 1672 is tens, i.e. 70.

2. What is the place value chart for the number 50?

The place value chart for 50 number is

5 digit – Tens – 50

0 digit – ones – 0

3. Is place value different from face value?

Yes, The place value outlines the position of a digit in the given number but the face value represents the exact value of a digit. As an example here we are taking a number ie., 790, and identify both values of digit 9, the place value of 9 is Tens whereas the face value of 9 is 9.

Trigonometrical Ratios Table of All Angles | Tips & Tricks to Learn Trigonometric Functions Table

Trigonometry Ratios Table 0-360: Trigonometry is a branch of mathematics that deals with the study of the length and angles of a triangle. It is usually associated with a right-angle triangle in which one of the angles is 90 degrees. It has a vast number of applications in the field of mathematics. You can figure out many geometrical calculations much simpler if you are aware of the Trigonometric Functions and Table.

Trigonometric Ratios Table help you find the trigonometric standard angles such as 0°, 30°, 45°, 60°, and 90°. You can find Trigonometric Ratios such as sine, cosine, tangent, cosecant, secant, cotangent, etc. In short, you can write the Trigonometric Ratios as sin, cos, tan, cosec, sec, and cot. You can solve Trigonometry Problems easily if you know the standard values of the Trigonometric Ratios. Thus, remember the standard angle values to make your job easier.

Trigonometric Table has a wide range of applications and it was used ever since before the existence of calculators.  Another Important Application of the Trigonometric Table is in the Fast Fourier Transforms.

Trigonometric Ratios Table for Standard Angles

Trigonometrical Ratios Table in Degrees and Radians

Trig Values Table: 0 to 360 Degrees

Trigonometry Ratio Table for All Angles

Tricks to Remember Trigonometry Table

It is easy to remember the trigonometry table. If you are aware of the trigonometric formulas remembering the table is quite simple. The Trigonometric Ratios Table is dependant on the Trigonometric Formulas. Try to remember the trigonometric table easily by going through the simple formulas.

  • sin x = cos (90° – x)
  • cos x = sin (90° – x)
  • tan x = cot (90° – x)
  • cot x = tan (90° – x)
  • sec x = cosec (90° – x)
  • cosec x = sec (90° – x)
  • 1/sin x = cosec x
  • 1/cos x = sec x
  • 1/tan x = cot x

How to Create a Trigonometric Ratio Table?

Check out the simple guidelines listed below to create a Trigonometric Table having Values of Standard Angles. They are in the following fashion

Step 1:

Create a table having the top row and list out the angles 0°, 30°, 45°, 60°, 90° and also write trigonometric functions such as sin, cos, tan, cosec, sec, cot.

Step 2: Determine the Value of Sin

In the second step determine the value of sin, divide 0, 1, 2, 3, 4 by 4 under the root.

\(\sqrt{\frac{0}{4}}=0\)

 

Angles (In Degrees)30°45°60°90°180°270°360°
sin01/21/√2√3/210-10

Step 3: Determine the Value of Cos

Cos is opposite to sin and to find the value of cos divide by 4 in the opposite sequence of sin. For instance, divide 4 with 4 under the root to obtain the value of cos 0°

\(\sqrt{\frac{4}{4}}=1\)

 

Angles (In Degrees)30°45°60°90°180°270°360°
cos1√3/21/√21/20-101

Step 4: Determine the value of tan

Tan is obtained by dividing sin with cos. To find the value of tan 0° divide the Value of Sin 0° by the Value of Cos 0°.

Angles (In Degrees)30°45°60°90°180°270°360°
tan01/√31√300

Step 5: Determine the value of the cot

Value of cot is equal to reciprocal of tan. The value of cot at 0° is obtained by dividing 1 with the value of tan at 0°. In the same way, you can find the value of the cot for all the angles.

Angles (In Degrees)30°45°60°90°180°270°360°
cot√311/√300

Step 6: Determine the value of cosec

Cosec value at 0° is the reciprocal of sin at 0°. You can find all the angles of cosec as such

Angles (In Degrees)30°45°60°90°180°270°360°
cosec2√22/√31-1

Step 7: Determine the value of sec

sec values can be obtained by the reciprocal values of cos. Sec value at 0° is the opposite of cos on 0°. In the similar way entire table of values is given.

Angles (In Degrees)30°45°60°90°180°270°360°
sec12/√3√22-11

FAQs on Trigonometric Ratios Table

1. How to find the Trigonometric Functions Values?

All the Trigonometric Functions Values can be found easily using the formulas and they are given as such

  • Sin = Opposite/Hypotenuse
  • Cos = Adjacent/Hypotenuse
  • Tan = Opposite/Adjacent
  • Cot = 1/Tan = Adjacent/Opposite
  • Cosec = 1/Sin = Hypotenuse/Opposite
  • Sec = 1/Cos = Hypotenuse/Adjacent

2. What is Trigonometric Values Table?

Trigonometric Values table is made of trigonometric ratios that are interrelated to each other – sine, cosine, tangent, cosecant, secant, cotangent.

3. What are Trigonometric Ratios?

Trigonometric Ratios is a relationship between measurements of length and angles of a right angle triangle.

Whole Numbers – Definition, Symbol, Properties, Examples

Whole Numbers is a part of a number system that includes all the positive integers from 0 to infinity. These numbers are present on the number line and are usually called real numbers. Thus, we can say that Whole Numbers are Real Numbers but not all Real Numbers are Whole Numbers. Complete Set of Natural Numbers including “0” are called Whole Numbers.

Whole Numbers – Definition

Whole Numbers are numbers that don’t have fractions and is a collection of positive integers including zero. It is denoted by the symbol “W” and is given as {0, 1, 2, 3, 4, 5, ………}. Zero on a whole denotes null value or nothing.

  • Whole Numbers: W = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10……}
  • Natural Numbers: N = {1, 2, 3, 4, 5, 6, 7, 8, 9,…}
  • Integers: Z = {….-9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,…}
  • Counting Numbers: {1, 2, 3, 4, 5, 6, 7,….}

Whole numbers are positive integers along with zero and don’t have fractional or decimal parts. You can perform all the basic operations such as Addition, Subtraction, Multiplication, and Division.

Symbol

The Symbol to denote the Whole Numbers is given by the alphabet W = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,…

  • All-natural numbers are whole numbers
  • All positive integers including zero are whole numbers
  • All whole numbers are real numbers
  • All counting numbers are whole numbers

Properties of Whole Numbers

Whole Numbers Properties depend on arithmetic operations such as Addition, Subtraction, Multiplication, Division. When you multiply or add two whole numbers the result will always be a Whole Number. If you Subtract Two Whole Numbers the result may not always be a Whole Number and it can be an Integer too. Division of Whole Numbers can result in a Fraction at times. Let us see few more Properties of Whole Numbers by referring below.

Closure Property: Whole Numbers can be closed under addition or multiplication. If a, b are two whole numbers then a.b and a+b is also a whole number.

Commutative Property of Addition and Multiplication: Sum and Product of Two Whole Numbers will be the same no matter the order in which they are added or multiplied. If a, b are two whole numbers then a+b = b+a, a.b = b.a

Additive Identity: If a Whole Number is added to 0 the result remains unchanged. If a is a whole number then a+0 = 0+a = a

Multiplicative Identity: Whenever you multiply a whole number with 1 the result remains unchanged. Let us consider a whole number “a” then a.1 = 1. = a

Associative Property: If you are grouping the whole numbers and adding or multiplying a set the result remains the same irrespective of the order. If a, b, c are whole numbers then a + (b + c) = (a + b) + c and a. (b.c)=(a.b).c

Distributive Property: If a, b, c are three whole numbers then the distributive property of multiplication over addition is given by a.(b+c) =(a.b)+(a.c), Similarly Distributive Propoerty of Multiplication over Subtraction is given by a.(b-c) = (a.b)-(a.c)

Multiplication by Zero: If you multiply a Whole Number with Zero the result is always zero. i.e. a.0=0.a=0

Division by Zero: If you divide a Whole Number with Zero the result is undefined, i.e. a divided by 0 is not defined.

Difference between Natural Numbers and Whole Numbers

Whole NumbersNatural Numbers
Whole Numbers: {0, 1, 2, 3, 4, 5, 6,…..}Natural Numbers: {1, 2, 3, 4, 5, 6,……}
All whole numbers are not natural numbersAll Natural numbers are whole numbers
Counting starts from 0Counting starts from 1

By referring to the below sections you will better understand the difference between Whole Numbers and Natural Numbers.

Difference Between Whole Numbers and Natural Numbers

 

Solved Examples on Whole Numbers

1. Are 101, 147, 193, 4028 whole numbers?

Yes, 101, 147, 193, 4028 are all whole numbers.

2. Solve 8 × (3 + 12) using the Distributive Property?

We know as per the Distributive Property a.(b+c) =(a.b)+(a.c)

Applying the Input Numbers in the formula we have the equation as such

8 × (3 + 12) = 8*3 +8*12

= 24+108

= 132

FAQs on Whole Numbers

1. Is 0 a Whole Number?

Yes, 0 is a Whole Number.

2. What is the Symbol of Whole Numbers?

The Letter W represents the Whole Numbers.

3. Are all Natural Numbers Whole Numbers?

Yes, all Natural Numbers are Whole Numbers but not all Whole Numbers are Natural Numbers. Natural Numbers begin from 0 and counts till infinity. Whole Numbers begin from 0 and end at infinity.

4. What is the set of whole numbers?

The whole numbers are the natural numbers together with 0. The set of whole numbers is a subset of the integers but does not include the negative integers.

Standard Form in Math | Definition, Formulas, How to Find Standard Form with Examples

Basically, Standard Form in Maths can be represented for numbers like decimal numbers, rational numbers, fractions, polynomials, equations, linear equations, etc. As we all know that the simplified form of the fractions called decimal numbers. Eventually, we can say that standard form is the representation of large numbers in small numbers irrespective of any form of numbers. In this article, we have explained what is the standard form of numbers, decimals, equations, etc. How to do it, Rules, and solved Standard form examples.

So, Let’s dive into it!

What is Standard Form?

A Standard Form is a process of formulating a given mathematical concept such as number, equation, polynomial, etc. in the standard form by following certain basic rules.

Standard Form of a Number

The definition of the standard form of a number is representing the very large expanded number in a small number. Here, we will see how to write a numerical in standard form with an example.

Now, we are taking a large number in the expanded form and convert the number into standard form.

Example: 

Write the given expanded form into standard form.

10,00,00,000 + 5,00,00,000 + 4,00,000 + 80,000 + 2,000 + 60 + 9.

Solution:

With the help of this below table, we will write the given expanded form of number into the standard form of a number.

TCCTLLTthThHTO
100000000
50000000
400000
80000
2000
60
9
150482069

Expanded Form                                                                                                   Standard Form

10,00,00,000 + 5,00,00,000 + 4,00,000 + 80,000 + 2,000 + 60 + 9       =             150482069

Standard Form of a Decimal Number

In Britain, the Scientific Notation’s other name is the Standard Form. And in other countries, it means not in the form of the large expanded number. We all know that it’s tough to read numbers such as 0.000000002345678 or 12345678900000. In order to help you all while reading these large numbers, we convert them into standard form.

Any number which we will address as a decimal number, among 1.0 and 10.0, multiplied by a power of 10, is stated to be in standard form. For example, 0.67X10¹³ is in the standard form.

Example:

Write down the standard form of the given decimal number 5326.6?

Solution:

The given decimal number 5326.6 can be written in this way in a scientific notation:

decimals standard form or scientific notation

Because 5326.6 = 53266 X 1000 =  5.3266 × 103.

Standard Form of a Linear Equation

The “Standard Form” for writing down a Linear Equation is Ax + By = C

A shouldn’t be negative, A and B shouldn’t both be zero, and A, B, and C should be integers.

Example:

Put this y = 7x + 4 in linear equation standard form?

Solution:

Given equation is Y= 7x + 4

Now, you have to change 7x from right to left like this, −7x + y = 4

Multiply all by −1:

7x − y = −4

Note: A = 7, B = −1, C = −4.

Standard Form of a Quadratic Equation

The Standard Form for writing down a Quadratic Equation is ax^2 + bx + c = 0, a is not equal to 0.

Example: 

Put this equation x(x−1) = 4 in the standard form?

Solution:

Given equation is x(x−1) = 4

In the first step, you have to expand x(x−1):

x(x−1) = x² − x

x² − x = 4

Next step is to move the 4 numerical to left side,

x² − x − 4 = 0.

Where A=1, B= -1, C= -4

The standard form of a quadratic equation x(x−1) = 4 is x² − x − 4 = 0. 

FAQs on Standard Form Rules in Maths

1. What is the standard form of an Equation?

An Equation in Standard Form looks like (some expression) = 0 ie., x + y = 0, where the left side of an equation are x and y terms and the Zero is on the right side.

2. What is meant by the standard form of a circle?

The graph of a circle is fully defined by its center and radius. The standard form for the equation of a circle is (x−h)²+(y−k)²=r². The center is (h,k) and the radius measures r units.

3. How to write 81 900 000 000 000 in standard form?

Steps on how to find or write the standard form of a given number ie., 81 900 000 000 000:

  • In step-1, write the first digit 8
  • Now, add a point then it becomes 8.
  • Count the remaining digits after the digit 8 and write the count in the power of 10.
  • For this example, there are 13 digits. So, the standard form of 81 900 000 000 000 is 8.19 × 10¹³.

Proof of De Morgan’s Law in Sets | Demorgan’s Law Definition, Statement and Proof

According to Demorgan’s Law Complement of Union of Two Sets is the Intersection of their Complements and the Complement of Intersection of Two Sets is the Union of Complements. The Law can be expressed as such ( A ∪ B) ‘ = A ‘ ∩ B ‘. By referring to the further modules you can find Demorgan’s Law Statement, Proof along with examples.

For any two finite sets A and B, we have

(i) (A U B)’ = A’ ∩ B’ (which is a De Morgan’s law of union).

(ii) (A ∩ B)’ = A’ U B’ (which is a De Morgan’s law of intersection).

De Morgan’s Laws Statement and Proof

A Set is a well-defined collection of objects or elements. You can perform various operations on sets such as Complement, Union, and Intersection. These Operations and usage can be further simplified by a set of simple laws called Demorgan’s Laws.

Any Set that includes all the elements related to a particular context is called a universal set. Let us assume a Universal Set U such that A and B are Subsets of it.

De Morgan’s Law Proof: (A∪B)’= A’∩ B’   

As per Demorgan’s First Law, the Complement of Union of Two Sets A and B is equal to the Intersection of Complements of Sets A and B.

(A∪B)’= A’∩ B’

Let P = (A U B)’ and Q = A’ ∩ B’

Consider x to be an arbitrary element of P then x ∈ P ⇒ x ∈ (A U B)’

⇒ x ∉ (A U B)

⇒ x ∉ A and x ∉ B

⇒ x ∈ A’ and x ∈ B’

⇒ x ∈ A’ ∩ B’

⇒ x ∈ Q

Therefore, P ⊂ Q …………….. (i)

Let us consider y to be an arbitrary element of Q then y ∈ Q ⇒ y ∈ A’ ∩ B’

⇒ y ∈ A’ and y ∈ B’

⇒ y ∉ A and y ∉ B

⇒ y ∉ (A U B)

⇒ y ∈ (A U B)’

⇒ y ∈ P

Therefore, Q ⊂ P …………….. (ii)

Combining (i) and (ii) we get; P = Q i.e. (A U B)’ = A’ ∩ B’

De Morgan’s Law Proof: (A ∩ B)’ = A’ U B’

Let us consider Sets M = (A ∩ B)’ and N = A’ U B’

Let x be an arbitrary element of M then x ∈ M ⇒ x ∈ (A ∩ B)’

⇒ x ∉ (A ∩ B)

⇒ x ∉ A or x ∉ B

⇒ x ∈ A’ or x ∈ B’

⇒ x ∈ A’ U B’

⇒ x ∈ N

Thus, M ⊂ N …………….. (i)

Again, let y be an arbitrary element of N then y ∈ N ⇒ y ∈ A’ U B’

⇒ y ∈ A’ or y ∈ B’

⇒ y ∉ A or y ∉ B

⇒ y ∉ (A ∩ B)

⇒ y ∈ (A ∩ B)’

⇒ y ∈ M

Thus, N ⊂ M …………….. (ii)

Combining (i) and (ii) we get; M = N i.e. (A ∩ B)’ = A’ U B’

Examples of De Morgan’s Law

1. If U = {a,b,c, d, e}, X = {b, c, d} and Y = {b, d, e}. Prove that De Morgan’s law: (X ∩ Y)’ = X’ U Y’?

Solution:

Given Sets are U = {a,b,c, d, e}, X = {b, c, d} and Y = {b, d, e}

Firstly find the (X ∩ Y) = {b, c, d} ∩ {b, d, e}

= { b,d}

(X ∩ Y)’ = U – (X ∩ Y)

= {a,b,c, d, e} – { b,d}

= {a,c,e}

X’ =  U – X

= {a,b,c, d, e} – {b, c, d}

= {a,e}

Y’ = U – Y

= {a,b,c, d, e} – {b, d, e}

= {a,c}

X’ U Y’ = {a,e} U {a,c}

= {a,c,e}

Hence Prooved (X ∩ Y)’ = X’ U Y’

2. Let U = {1, 2, 3, 4, 5, 6, 7, 8}, A = {2, 3, 4} and B = {5, 6, 8}. Show that (A ∪ B)’ = A’ ∩ B’?

Solution:

Given Sets are U = {1, 2, 3, 4, 5, 6, 7, 8}, A = {4, 5, 6} and B = {5, 6, 8}
(A ∪ B) = {4, 5, 6} ∪ {5, 6, 8}

= {4, ,5, 6, 8}

(A ∪ B)’ = U – (A ∪ B)

= {1, 2, 3, 4, 5, 6, 7, 8} – {4, ,5, 6, 8}

= {1, 2, 3,7}

A’ = U – A

= {1, 2, 3, 4, 5, 6, 7, 8} – {4, 5, 6}

= {1, 2, 3, 7, 8}

B’ = U – B

= {1, 2, 3, 4, 5, 6, 7, 8} – {5, 6, 8}

= {1, 2, 3, 4, 7}

A’ ∩ B’ = {1, 2, 3, 7, 8} ∩ {1, 2, 3, 4, 7}

= {1, 2, 3, 7}

Hence Prooved (A ∪ B)’ = A’ ∩ B’

 

Binary Subtraction Definition, Table, Rules, Examples | How to Subtract Binary Numbers?

Binary Subtraction is one among the four binary operations in which we perform subtraction of binary numbers i.e. 0 or 1. It is similar to the Basic Arithmetic Operation of Decimals. When we subtract 1 from 0 we need to borrow 1 from the next digit to reduce the digit by 1 and the remainder left here is 1. Go through the entire article to know about Binary Subtraction Rules, Subtraction Table, Tricks and Procedure on How to Subtract Binary Numbers, etc.

What is meant by Binary Subtraction?

Binary Numbers Subtraction is similar to Subtraction of Decimals or Base 10 Numbers. For instance, 1+1+1 is 3 in base 10 whereas in a binary number system 1+1+1 is 11. While Performing Addition and Subtraction in Binary Numbers be careful with borrowing as you might need to do them quite often.

While performing subtraction of several columns of binary digits you need to consider the borrowing. If you subtract 1 from 0 the result will be 1 where 1 is borrowed from the next highest order digit.

Binary Subtraction Table

Binary NumberSubtraction Value
0 – 00
1 – 01
0 – 11 (Borrow 1 from next high order digit)
1 – 10

On Adding Two Binary Numbers 1 and 1 we get the result 10 in which we consider 0 and carry forward 1 to the next higher-order bit. On Subtracting 1 from 1, the result is 0 and nothing will be carry forwarded.

While subtracting 1 from 0 in the case of decimal numbers we borrow 1 from the preceding higher-order number and make it 10 and after subtracting result becomes 9. However, in the case of Binary Subtraction, the result is 0.

Rules for Binary Subtraction

Binary Subtraction is quite simple compared to Decimal Subtraction if you remember the following tips and tricks.

0 – 0 = 0
0 – 1 = 1 ( with a borrow of 1)
1 – 0 = 1
1 – 1 = 0

You can look at the binary subtraction examples provided below for better understanding.

How to Subtract Binary Numbers?

Follow the below-listed steps to perform Binary Subtraction. You will find the Subtraction of Binary Numbers much easier after going through the below steps. They are as follows

  • Align the numbers similar to an ordinary subtraction problem. Write the larger number up and the smaller number below it. If the Smaller Digit has few digits align them towards the right same as in decimal subtraction.
  • Begin from the right column and perform the subtraction operation of binary numbers. While doing so keep the binary subtraction rules in mind and do accordingly.
  • Solve column by column moving from right to left.

Binary Subtraction Examples

1. Find the Value of 1010011 – 001110?

Solution:

Write the given numbers as if you subtracting decimal numbers. Align them to the right and fill them with leading zeros so that both the numbers have the same digits.

1011011

(-)0001010

——————

1010001

Binary Notation

Decimal Notation

The decimal Equivalent of given numbers is

1011011 = 91

001010 = 10

91-10 = 81

2. Find the value of 1100010 – 001000?

Solution:

Write the given numbers as if you subtracting decimal numbers. Align them to the right and fill them with leading zeros so that both the numbers have the same digits.

1100010

(-)0001000

——————

1,011,010

Decimal Notation

The decimal Equivalent of given numbers is

1100010 = 98

1000 = 8

98-8 = 90

Binary Subtraction using 1’s Complement

Go through the below procedure and perform the Binary Subtraction easily. They are as follows

  • Firstly, write the 1’s complement of the subtrahend.
  • And then add the 1’s complement subtrahend with the minuend.
  • If the result has a carryover add the carryover in the least significant bit.
  • If it has no carryover take the 1’s complement of resultant and it is negative.

Questions on Binary Subtraction using 1’s Complement

1. (11001)2  – (1010)2

Solution:

(11001) = 25

(1010) = 10 – subtrahend

Fill with leading zeros till you have the same number of digits in both the numbers. Firstly, take the 1’s complement of subtrahend i.e. (01010)2.

1’s complement of the subtrahend is 10101. Add 1 to the 1’s complement of the second number

10101
+   1

10110

Now instead of subtracting add the 1’s complement of the second number to the first one

11001
+ 10110

——————
101111

Remove the leading 1 to obtain the result.

Then, remove the leading zeros as it will not alter the result and you can write the final result of subtraction as such

11001
– 1010

——————

1111

FAQs on Binary Subtraction

1. What are the Rules of Binary Subtraction?

Rules of Binary Subtraction are as follows

0 – 0 = 0
0 – 1 = 1 ( with a borrow of 1)
1 – 0 = 1
1 – 1 = 0

2. How many basic binary subtraction combinations are possible?

There are four possible binary subtraction combinations when subtracting binary digits.

3. What signs does the binary digit 0 and 1 represent?

0 represents the positive sign and 1 represents the negative sign.

Decimal Place Value Chart: Definition, How to Write, and Examples

Mathematics is the study of numbers, shapes, and patterns. It includes various complex and simple arithmetic topics that help people in their daily life routine. In Maths, Numbers play a major role and they can be of different types like Real Numbers, Whole Numbers, Natural Numbers, Decimal Numbers, Rational Numbers, etc. Today, we are going to discuss one of the main topics of Decimal Numbers. In Decimals, identifying the Decimal Place Values is a fundamental topic and everyone should know the techniques clearly. So, here we will be discussing elaborately the topic of Decimal Place Values Chart.

Let’s get into it.

What is a Decimal in Math?

In algebra, a decimal number can be represented as a number whose whole number part and the fractional part is divided by a decimal point. The dot in a decimal number is called a decimal point. The digits following the decimal point show a value smaller than one.

What is the Place Value of Decimals?

Place value is a positional notation system where the position of a digit in a number, determines its value. The place value for decimal numbers is arranged exactly the identical form of treating whole numbers, but in this case, it is reverse. On the basis of the preceding exponential of 10, the place value in decimals can be decided.

Decimal Place Value Chart

Decimal Place Value Chart table image

On the place value chart, the numbers on the left of the decimal point are multiplied with increasing positive powers of 10, whereas the digits on the right of the decimal point are multiplied with increasing negative powers of 10 from left to right.

  • The first digit after the decimal represents the tenths place.
  • The second digit after the decimal represents the hundredths place.
  • The third digit after the decimal represents the thousands place.
  • The rest of the digits proceed to fill in the place values until there are no digits left.

How to write the place value of decimals for the number 132.76?

  • The place of 6 in the decimal 132.76 is 6/100
  • The place of 7 in the decimal 132.76 is 7/10
  • The place of 2 in the decimal 132.76 is 2
  • The place of 3 in the decimal 132.76 is 30
  • The place of 1 in the decimal 132.76 is 100.

Examples:

1.  Write the place value of digit 7 in the following decimal number: 5.47?

The number 7 is in the place of hundredths, and its place value is 7 x 10 -2 = 7/100 = 0.07.

2. Identify the place value of the 6 in the given number: 689.87?

Given number is 689.87

The place of 6 in the decimal 689.87 is 600 or 6 hundreds. 

3. Write the following numbers in the decimal place value chart.

(i) 4532.079

(ii) 490.7042

Solutions: 

(i) 4532.079

4532.079 in the decimal place value chart.

example of decimal place value for the given number

(ii) 490.7042

490.7042 in the decimal place value chart.

decimal place value chart examples

Math Conversion Chart | Conversion of Units of Measurement | Metric Conversion Chart

In your day-to-day life, you might need to convert from one unit to another. To Perform the Required Calculations you might need to learn about Mathematical Conversions. Thus, it is necessary to learn about Unit Conversions to change from one unit to another unit.  Before understanding the Units of Measurement you need to be aware of the relationship between them.

Go through the entire article to learn about units of length conversion chart, unit of mass and weight conversion chart, unit of time conversion chart, units of capacity and volume conversion chart, unit of a temperature conversion chart, etc. Math Conversion Charts are quite important and interesting to learn.

What is Conversion of Units?

Depending on the Situation each unit differs. For instance, the area of the room is expressed in Meters whereas the thickness of the pencil is expressed in mm. Thus, it is necessary to convert from one unit to another.

Importance of Mathematical Conversions

In Order to have accuracy and avoid confusion in measurement, we need to change from one unit to another. For example, we will measure the length of the pencil in cm but no in km. Likewise, we need to convert from one unit to another. While converting from one unit to another of the same quantity we use multiplicative conversion factors. Let’s see how to convert different units of length and mass in the later modules.

Length Conversion Chart

Length Conversion Chart

Mass Conversion Chart

Mass Conversion Chart

Unit Conversion Table

Length Conversion Table

1 km = 10 hm

1 hm = 10 dam

1 dam = 10 m

1 m = 10 dm

1 dm = 10 cm

1 cm = 10 mm

1mm = \(\frac { 1 }{ 10 } \) cm

1 cm = \(\frac { 1 }{ 10 } \) dm

1 dm = \(\frac { 1 }{ 10 } \) m

1m = \(\frac { 1 }{ 10 } \) dam

1 dam = \(\frac { 1 }{ 10 } \) hm

1hm = \(\frac { 1 }{ 10 } \) km

Mass Conversion Table

1 kg = 10 hg

1 hg = 10 dag

1 dag = 10 g

1 g = 10 dg

1 dg = 10 cg

1 cg = 10 mg

1 mg = \(\frac { 1 }{ 10 } \) cg

1 cg = \(\frac { 1 }{ 10 } \) dg

1 dg = \(\frac { 1 }{ 10 } \) g

1 g = \(\frac { 1 }{ 10 } \) dag

1 dag = \(\frac { 1 }{ 10 } \) hg

1 hg = \(\frac { 1 }{ 10 } \) kg

Capacity Conversion Table

1 kℓ = 10 hℓ

1 hℓ = 10 daℓ

1 daℓ = 10 ℓ

1 ℓ = 10 dℓ

1 dℓ = 10 cℓ

1 cℓ = 10 mℓ

1 mℓ =\(\frac { 1 }{ 10 } \) cℓ

1 cℓ = \(\frac { 1 }{ 10 } \) dℓ

1dℓ = \(\frac { 1 }{ 10 } \) ℓ

1 ℓ = \(\frac { 1 }{ 10 } \) dℓ

1 daℓ = \(\frac { 1 }{ 10 } \) hℓ

1 hℓ = \(\frac { 1 }{ 10 } \) kℓ

Points to remember:

  • To convert bigger units to smaller units multiply
  • To convert smaller units to bigger units divide

Solved Examples on Conversion of Units

1. Convert 4 cm to km?

Solution:

Step 1:

Draw a line or box as below

kmhmdammdmcmmm

Step 2:

Put 1 at the larger unit to be converted.

Here we want to convert cm to km,

Since km is the larger unit, thus put 1 under the corresponding column (km).

km

1

hm

 

dam

 

m

 

dm

 

cm

 

mm

 

Step 3:

Now place 0 till the smaller unit

km

1

hm

0

dam

0

m

0

dm

0

cm

0

mm

 

Since the conversion is from a smaller unit to larger we need to divide the given length by 100000 i.e. 105

4 cm = 4/105

= 0.00004 km

2. Convert 7 gms to milligrams?

Solution:

We need to convert grams to milligrams

We know 1 gram = 1000 mg

Thus 7 g = 7*1000

= 7000 mg

 

 

 

What is 90 Degree Clockwise Rotation Rule? | Check How to Rotate 90° Point in Clockwise Direction with Examples?

In Geometry Topics, the most commonly solved topic is Rotations. A Rotation is a circular motion of any figure or object around an axis or a center. If we talk about the real-life examples, then the known example of rotation for every person is the Earth, it rotates on its own axis. However, Rotations can work in both directions ie., Clockwise and Anticlockwise or Counterclockwise. 90° and 180° are the most common rotation angles whereas 270° turns about the origin occasionally.

Here, in this article, we are going to discuss the 90 Degree Clockwise Rotation like definition, rule, how it works, and some solved examples. So, Let’s get into this article!

90 Degree Clockwise Rotation

If a point is rotating 90 degrees clockwise about the origin our point M(x,y) becomes M'(y,-x). In short, switch x and y and make x negative.

image for understanding 90 degree rotations rules

Before RotationAfter Rotation
(x, y)(y,-x)

Rule of 90 Degree Rotation about the Origin

  • When the object is rotating towards 90° clockwise then the given point will change from (x,y) to (y,-x).
  • When the object is rotating towards 90° anticlockwise then the given point will change from (x,y) to (-y,x).

Solved Examples:

Example 1:

Solve the given coordinates of the points obtained on rotating the point through a 90° clockwise direction?

(i) A (4, 7)

(ii) B (-8, -9)

(iii) C (-2, 8)

Solution:

When the point rotated through 90º about the origin in the clockwise direction, then the new place of the above coordinates are as follows:

(i) The current position of point A (4, 7) will change into A’ (7, -4)

(ii) The current position of point B (-8, -9) will change into B’ (-9, 8)

(iii) The current position of point C (-2, 8) will change into C’ (8, 2)

Example 2: 

Let P (-6, 3), Q (9, 6), R (2, 7) S (3, 8) be the vertices of a closed figure. If this figure is rotated 90° about the origin in a clockwise direction, find the vertices of the rotated figure.

Solution:

Given vertices are P (-6, 3), Q (9, 6), R (2, 7) S (3, 8)

Now, we will solve this closed figure when it rotates in a 90° clockwise direction,

In step 1, we have to apply the rule of 90 Degree Clockwise Rotation about the Origin

(x, y) → (-y, x)

Next, find the new position of the points of the rotated figure by using the rule in step 1.

(x, y) → (y, -x)

P (-6, 3) → P'(3, 6)

Q (9, 6) → Q’ (6, -9)

R (2, 7) → R'(7, -2)

S (3, 8) → S'(8, -3)

Finally, the Vertices of the rotated figure are P'(3, 6), Q’ (6, -9), R'(7, -2), S'(8, -3).

Example 3:

Find the new position of the given coordinates A(-5,6), B(3,7), and C(2,1) after rotating 90 degrees clockwise about the origin?

Solution:

90 degree clockwise rotation example graph

Given Coordinates are A(-5,6), B(3,7), and C(2,1)

The rule/formula for 90 degree clockwise rotation is (x, y) —> (y, -x). 

After applying this rule for all coordinates, it changes into new coordinates and the result is as follows:

A(-5,6) –> A'(6,5)

B(3,7) –> B'(7,-3)

C(2,1) –> C'(1,-2)

I believe that the above graph clears all your doubts regarding the 90 degrees rotation about the origin in a clockwise direction. At last, the result of the coordinates is A'(6,5), B'(7,-3), C'(1,-2). 

FAQs on 90 Degree Clockwise Rotation

1. Is a 90 Degree rotation clockwise or counterclockwise?

Considering that the rotation is 90 Degree, you should rotate the point in a clockwise direction.

2. What are the types of rotation?

You can see the rotation in two ways ie., clockwise or counterclockwise. In case, there is an object which is rotating that can rotate in different ways as shown below:

  • 90 degrees counterclockwise
  • 90 degrees clockwise
  • 180 degrees counterclockwise
  • 180 degrees clockwise

3. What is the rule of Rotation by 90° about the origin?

The rule for a rotation by 90° Counterclockwise about the origin is (x,y)→(−y,x)

The rule for a rotation by 90° Clockwise about the origin is (x,y)→(y,−x)

Practice Test on Profit and Loss | Profit and Loss Practice Questions

Solving the Profit and Loss Questions can give you an idea of how to solve related problems. Know different methods and formulae involved to calculate the Profit and Loss. Try to solve the Profit and Loss Questions on your own and then verify your solution with ours to know where you went wrong. By solving them regularly you can increase your speed and accuracy thereby attempt the exam well and score better grades in the exam.

Question 1:

If the manufacturer gains 15%, the wholesale dealer gets 20% and the retailer gets 30%, then find the cost of production of a blackboard, the retail price of which is $ 1360?

Solution:

Cost of production of blackboard be ‘X’

115% of 120% of 130% of cost price is

i.e. \(\frac { 115 }{ 120 } \)*\(\frac { 120 }{ 100 } \)*\(\frac { 130}{ 100} \)*1360

Cost of production of a blackboard $ 758.08

Question 2:

A man bought a dog and a kennel for $ 4500. He sold the dog at a gain of 25% and the carriage at a loss of 15%, thereby gaining 3% on the whole. Find the cost of the dog.

Solution:

Cost price (C.P.) of dog ‘X’

Cost price (C.P.) of kennel ‘4500 – X’

25% of x – 15% of (4500 – X) = 3% of 4500

Cost of a dog is $ 1373.68.

Question 3:

Profit earned by selling television for $ 6000 is 25% more than the loss incurred by selling the article for $ 4500. At what price should the article be sold to earn 25% profit?

Solution:

let cost price (C.P.) be ‘X’

By equalizing,

(6000 – X) = \(\frac { 125 }{ 100 } \)*(X-4500)

Desired selling price of the television is $ 6458.3.

Question 4:

A manufacturer undertakes to supply 2200 pieces of a particular component at $ 30 per piece. According to his estimates, even if 6% fail to pass the quality tests, then he will make a profit of 30%. However, as it turned out, 60% of the components were rejected. What is the loss to the manufacturer?

Solution:

Incurred cost = $(\(\frac { 100 }{ 130 } \)*30*(94% of 2200))

Loss = C.P. – S.P.

The loss to the manufacturer is $ 8123.07.

Question 5:

John bought a lorry for a certain sum of money. He spent 15% of the cost on repairs and sold the lorry for a profit of $ 1600. How did he spend on repairs if he made a profit of 30%?

Solution:

Let the Cost price (C.P.) of the lorry be ‘X’

Profit (P) = $16,00

Profit (%) = 30

Profit (%) = \(\frac { 100 }{ CP } \)*P

30 = \(\frac { 100 }{ CP} \)*1600

CP = \(\frac { 100 }{ CP} \)*1600

CP = \(\frac { 100 }{ 30} \)*1600

= $5333.3

The C.P Includes both original price and repairs cost

C.P +0.15C.P = $5333.3

1.15 C.P = $5333.3

C.P = $4637.68

Repairs Cost = Total Cost Price – Cost Price for which john bought the lorry

= $5333.3 – $4637.68

= $ 695.61

Expenditure spend on repairs = $ 695.61.

Question 6:

A boy bought 25 litres of milk at the rate of $ 12 per litre. He got it churned after spending $ 15 and 7kg of cream and 25 lit of toned milk were obtained. If he sold the cream at $ 40 per kg and toned milk at $ 6 per lit, what is his profit % in the transaction?

Solution:

Cost price (C.P.) = $ ((25*12) + 15)

Selling price (S.P.) = $ ((7*40) + (25*6))

Gained profit % on the above transaction is 36.50%.

Question 7:

Arun purchased 150 reams of paper at $ 90 per ream. He spent $ 300 on transportation, paid local tax at the rate of 50 paise per ream, and paid $ 76 to coolie. If he wants to have a gain of 10%, what must be the selling price per ream?

Solution:

Total investment = $ (((150*90) + 300) + ((50/100 * 150) + 76))

Find selling price per ream

Selling price per ream = $ 102.3

Question 8:

A retailer mixes three varieties of dal costing $ 50, $ 25, and $ 30 per kg in the ratio 3: 5: 2 in terms of weight, and sells the mixture at $ 35 per kg. What percentage of profit does he make?

Solution:

Cost price (C.P.) of dal for 10 kgs = $ ((3*50) + (5*25) + (2*30))

Selling price for 10 kgs = $ (10*35)

Profit percentage on transactions is 4.74%.

Question 9:

A man buys chocolates at 3 for $ 2 and an equal number at 5 for $ 3 and sells the whole at 6 for $ 4. His gain or loss percent is?

Solution:

Suppose he buys 7 eggs of each kind

Find C.P. and S.P. for 14 eggs

C.P. = $ ((\(\frac { 2 }{ 3 } \) * 7) + (\(\frac { 3 }{ 5 } \) * 7)) = $ 8.86

Similarly, find S.P.

The gain % obtain is 5.34%.

Question 10:

The manufacturer of a certain item can sell all he can produce at the selling price of $ 65 each. It costs him $ 45 in material and labor to produce each item and he has overhead expenses of $ 3500 per week in order to operate the plant. The number of units he should produce and sell in order to make a profit of at least $ 1200 per week is?

Solution:

Consider he produces ‘x’ items

Cost price (C.P.) = $ (45x + 3500)

Selling price (S.P.) = $ 65x

The number of units produced per week to gain profit is 235.

Absolute Value of an Integer Definition, Examples | How to find the Absolute Value of an Integer?

Wanna learn all the basics of Integers concepts? Then start with the absolute value of an integer. Integer Absolute Value is an important and basic concept to learn other concepts of Integers. Refer to all the rules, definitions, solved examples, types, etc to understand the concept more clearly. Follow the below sections to get more idea on how to find the Absolute Value of an Integer in detail and Solved Problems for finding the Integer Absolute Value using different approaches.

Absolute Value of an Integer – Introduction

Integers Absolute Value is the distance of that integer value from zero irrespective of (positive or negative) direction. While considering the absolute value, its numerical value is taken without taking the sign into consideration.

An integer is any positive or negative whole number. Therefore, the positive integer has a negative sign and vice versa.

A positive number is a number that is greater than zero. It is represented by a “+” symbol and can be written with or without a symbol in front of it. The gain in some value or something is written with a positive number. A negative number is a number that is less than zero. It is represented by a “-” symbol and can be written with a “-” symbol in front of it. The loss in some value or something is written with a negative number.

A number line is a kind of diagram on which the numbers are marked at intervals. These are used to illustrate simple and easy numerical operations. Using the number line allows seeing a number is in relationship to other numbers and from zero. Zero, is present in the middle and separates positive and negative numbers. On the right side of zero, we can find numbers that are greater than zero (positive numbers) and on the left side, we can find numbers that are less than zero (negative numbers). The absolute value of the integer is the same as the distance from zero to a specific number.

Representation of Absolute Value of an Integer

Integer Absolute Value is represented with two vertical lines. i.e., | |, one on either side of the integer.

|a| = a, when a is the positive integer

|a| = -a, when a is the negative integer

Examples:

  1. Absolute integer value of -15 is written as |-15| = 15 {here mod of -15 = 15}
  2. Absolute integer value of 8 is written as |8| = 8 {here mod of 8 = 8}

Adding and Subtracting Absolute Value of Integers

To add 2 integers with the same (positive or negative) sign, add absolute values and assign the sum with the same sign as same both values.

Example:

(-7) + (-4) = -(7 + 4)= – 11.

To add 2 integers with different signs, find the difference in absolute integer values and give that product the same sign of the largest absolute value.

Example:

(-7)+(2)= -5

How to find the absolute value?

  1. First of all, find the absolute values of (7 and 2)
  2. Find the difference of numbers between 7 and 2 (7-2=5)
  3. Find the sign of the largest absolute value i.e., negative, as 7 is of (-)sign
  4. Add the sign to the difference we got in Step 2.
  5. Hence the final solution is -5

When an integer is subtracted or added by another integer, the result will be an integer.

Multiplication of Absolute Value of Integers

To multiply the integer values, we have to multiply the absolute values. If the integers that are to be multiplied have the same sign, then the result will be positive. If both the integers have different signs, then the result will be negative.

When an integer is multiplied by another integer, then the result will be an integer itself.

Absolute Value of Integers Examples

Question 1:

Write the absolute value of each of the following?

(i) 15

(ii) -24

(iii) -375

(iv) 0

(v) +7

(vi) +123

Solution: 

(i) 15

(ii) 24

(iii) 375

(iv) 0

(v) 7

(vi) 123

Question 2: 

Evaluate the following integers

(i) |-7| + |+5| + |0|

(ii) |10| – |-15| + |+12|

(iii) – |+3| – |-3| + |-6|

(iv) |-8| – |17| + |-12|

Solution: 

(i) 12

(ii) 7

(iii) 0

(iv) 3

Question 3:

State whether the statements are true or false?

(i) The absolute value of -3 is 3.

(ii) The absolute value of an integer is always greater than the integer.

(iii) |+5| = +5

(iv) |-5| = -5

(v) – |+5| = 5

(vi) – |-5| = -5

Solution:

(i) True

(ii) False

(iii) True

(iv) False

(v) False

(vi) True

Methods to Compute the Absolute Value of Integer

There are 3 approaches to find the Integers Absolute value. These approaches are helpful while solving problems. Finding the absolute value is the first step in solving the problems.

Method 1: 

As the absolute value of the integer is always positive. For the positive integer, the absolute number is the number itself. For the negative number, the absolute number is multiplied by other negative numbers.

Method 2:

Negative numbers are saved in 2’s complement form. To get the absolute number, toggle bits of the number, then 1 to the result.

Method 3:

The built-in function library finds the absolute value of an integer.

To solve most of the problems, knowing the absolute value of the integer is important. This method is used in the real world and it has all sorts of math clues. Absolute values are often used in distance problems and also sometimes used for inequalities. Integers Absolute values are really helpful to get clarity on many things.

We have mentioned the important concept of Integers Absolute Value in the above article. Hope you got a clear idea of definitions, rules, and how to solve the problems. If you have any doubts, you can contact us from the below comment box. You can also directly ping us to know other information. Stay tuned to our page to get the latest and important information about all mathematical concepts.

Examples on Division of Integers | Dividing Integers Problems with Solutions

Get the complete practice test questions and worksheet here. Follow the step by step procedure to solve examples on the division of integers all the problems. Know the shortcuts, tricks, and steps involved in solving Division of Integers problems. Also, find the definitions, formulae before going to start the practice sessions. Go through the below sections to know the detailed information regarding formulas, definitions, and problems.

Integers Division Rules

Rule 1: The quotient value of a positive integer number and a negative integer number is negative.

Rule 2: The quotient value of two positive integer numbers is a positive number.

Rule 3: The quotient value of two negative integer numbers is a positive number.

Division of Integers Rules and Examples

Question 1: Find the value of ||-17|+17| / ||-25|-42|

Solution:

||-17|+17| / ||-25|-42|

= |17+17| / |25–42|

= |34| / |-17|

= 34 / 17

= 2

Question 2: Simplify: {36 / (-9)} / {(-24) / 6}

Solution:

{36 / (-9)} / {(-24) / 6}

= {36/-9} / {-24/6}

= – (36/9) / – (24/6)

= -4/-4

= 4/4

=1

Question 3: Find the value of [32 + 2 x 17 + (-6)] ÷ 15

Solution:

[32+2 x 17+(-6)] / 15

= [32+34+(-6)] / 15

= (66-6) / 15

= 60 / 15

= 4

Question 4: Divide the absolute values of the two given integers?

Solution:

The quotient of the absolute value of integer +24 and the absolute value of integer -8

From the rules given above, dividing the integers with different signs gives the final result as negative.

When positive(+) 24 is divided by negative(-) 8 results negative(-) 3, which can be defined as +24/(-8) = -3

Question 5: Prove that [((-8) / (-4)] ≠ =-8 / [(-4) / (-2)]

Solution:

From the given question

[((-8)/(-4)]/(-2)] ≠ -8/[(-4)/(-2)]

First of all, we will consider the LHS part i.e., [((-8) / (-4)] / (-2)]

To solve the equation, first, we divide 8 by 4, we get the result as 2

As both numbers have a negative sign, it will be positive.

Then we divide the result (2) by -2, then the final result will be -1.

Therefore, the result of the LHS part is -1.

Now, we consider the RHS part i.e., -8 / [(-4) / (-2)]

First of all, we divide -4 by -2, the result will be 2.

As both the numbers have a negative sign, the result will be positive.

Now, divide -8 by the above result 2.

Hence, the result will be -4.

Therefore, the result of RHS is -4.

Hence, LHS ≠ RHS

The above given equation [((-8) / (-4)] / (-2)] ≠ -8 / [(-4) / (-2)] is thus satisfied.

Question 6: In a maths test containing 10 questions, 2 marks are given for every correct answer and (-1) marks are given for every wrong answer. Rohith attempts all the questions and 8 questions answers are correct. What is Rohith’s total score?

Solution:

From the given question,

The marks given for every correct answer = 2 marks

Marks given for 8 correct answers = 2 * 8= 16 marks

Marks given for 1 incorrect answer = -1 marks

Marks given for 2 incorrect answers = -1 * 2 = -2 marks

Rohit’s total score = 16-2 = 14 marks

Therefore, the answer is 14 marks.

Question 7: Priya sells 20 pens and some pencils losing 2 Rs in all. If Priya gains 2 Rs on each pen and loses 1 Rs on each pencil. How many pencils does Priya sell?

Solution:

Suppose that Priya sells x pencils.

Total gain on pens = 2*20 = 40 Rs

Total loss on pencils = 1x = x

Total loss on selling pens and pencils = (-2)

40-x = (-2)

40+2 = x

x = 42

Therefore, Priya sells 42 pencils.

Thus, the answer is 42 pencils.

Question 8: To make ice cream, the room temperature must be decreased from 45degree C at the rate of 5 degrees C per hour. What will be the room temperature 12 hours after the freezing point of the icecream?

Solution:

As given in the question,

Temperature after 12 hours = 12*5 = 60degree C

Room temperature = 45degree C

Hence, the room temperature after freezing process of 12 hours = (45-60)degree C

= -15degree C

Question 9:

A car runs at a rate of 50km/hr. If the car starts at 5 km above the starting point, how long will it take to reach 2505km?

Solution:

As per the question,

Total distance covered by car = (2505-5) km = 2500 km

Rate of car = 50km/hr

From the above values, Distance = 2500 km, speed = 50 km/hr

Therefore, the time required by the car = distance/speed

=2500/50 = 50 hours

Therefore, the car will take 50 hours to travel 2505 km.

Hence, The final solution is 50 hours.

Question 10: Jason borrowed $5 a day to buy launch. She now owes $65 dollars. How many days did Jane borrow $5?

Solution:

As per the question,

Jason borrowed a launch at =  $5

Now she owes = $65

No of days = (-65) / (-5)

= 13 days

Therefore, Jane borrows $5 for 13 days.

Hence, The final solution is 13 days.

Division of Integers Word Problems

Question 11: Allen’s score in a video game was changed by -120 points because he missed some target points. He got -15 points for each of the missed targets. How many targets did he miss?

Solution: 

As per the given question,

Allen scored points in a video game = -120

He got points for missed targets = -15

No of targets he missed = -120/-15 = 8 targets.

Therefore, he missed 8 targets.

Question 12: Karthik made five of his truck payments late and was fined five late fees. The total change in his savings of late fees was -$30. What integer represents the one late fee?

Solution:

As given in the question, Karthik has made five of his truck payments. Therefore, it is positive.

He was fined -$30 as the late fees.

To find one late fee, we have to divide the fine by no of payments he did.

Therefore, One Late fee = -$30/5

=-$6

Thus, He paid $6 for each payment as late fees.

Hence, the final answer is -$6.