Coordinate Graph Definition | Plot Ordered Pair on Coordinate Graph

A coordinate graph is a two-dimensional plane having two perpendicular lines intersect at the midpoint called the origin. It is also called the cartesian plane or coordinate plane. You can represent ordered points in the coordinate graph. It is needed for the students who are studying the class 8 coordinate geometry concept. So, students are advised to go through this entire article to get clarity on what is meant by the coordinate graph, why it is used, solved examples, and how to represent an ordered pair in the coordinate graph.

Coordinate Graph Definition

A coordinate graph has two lines that are perpendicular to each other. Those lines are called axes. The horizontal line is known as the x-axis, and the vertical line is known as the y-axis. Those two axes meet at zero. The point o intersection of axes is called the origin. This coordinate graph is also called the coordinate plane or cartesian plane or cartesian coordinate system.

What is meant by Coordinates Axes, Quadrants?

On a coordinate graph two perpendicular straight lines having names X’OX, Y’OY are called the coordinate axes. The line having the name X’OX is called the x-axis, the other line having the name Y’OY is called the y-axis, and the point O is called the origin. Each graph paper has both coordinate axes (x-axis, y-axis) is known as the cartesian plane.

Coordinate Geometry

This image shows how exactly the coordinate graph looks like. Every coordinate graph has 4 quadrants namely quadrant 1, quadrant 2, quadrant 3, and quadrant 4. Some important points about the quadrants are mentioned here:

  • The first quadrant, is a quadrant having both x and y coordinates positive. The plane having xoy is called quadrant 1.
  • The second quadrant is available to the left side of the first quadrant, it has abscissa as negative and ordinate as positive. The plane enclosed by yox’ is quadrant 2.
  • The third quadrant is available below the second quadrant and here both abscissa, ordinates are negative. The plane enclosed by x’oy’ is called quadrant 3.
  • The last quadrant or fourth quadrant is available right-hand side to the third quadrant and below the first quadrant. So, the x coordinate value is positive and the y coordinate value is negative.

Steps to Plot Ordered Pairs on a Coordinate Graph

The ordered pair is a point in the coordinate plane having both x-coordinate and y-coordinate values. It has two real numbers enclosed by braces ‘(‘, ‘)’ and separated by a comma. The first value in the ordered pair is called the abscissa or x coordinate and the second value is called the y coordinate or ordinate. The format of an ordered pair is (x_coordinate_value, y_coordinate_value). In the following sections, we are giving the simple steps and instructions to mark ordered pairs on a coordinate graph effortlessly. Have a look at them and follow them whenever necessary.

  • Let us take any ordered pair having 2 real numbers as abscissa and ordinate.
  • Check out the signs of those real numbers to determine under which quadrant the given point lies.
  • Then take the mentioned number of units on the x-axis and y-axis.
  • Highlight that point and write the ordered pair there.
  • Similarly, you can plot several ordered pairs.
  • And join those ordered pairs to get a shape.

Example Questions

Question 1.

Plot point A (0, 2) on the graph?

Solution:

Given ordered pair is A (0, 2)

The coordinates of the point are positive. So, the point lies in the first quadrant.

As the x coordinate value is zero, the point lies on the y-axis.

As the y coordinate is 2, the point is 2 units away from the y-axis.

Coordinate graph

Question 2.

Plot point B (-5, 5) on the graph?

Solution:

Given ordered pair is B (-5, 5)

The x coordinate of B is negative and the y coordinate is positive.

According to the coordinate graph rules, the point belongs to the second quadrant.

Take -5 units on the x-axis and 5 units on the y-axis to get point B.

Coordinate Graph

Question 3.

Plot point C (2, -7) on the graph?

Solution:

Given ordered pair is C (2, -7)

As the abscissa of the point is positive and the ordinate is negative, the ordered pair lies in the fourth quadrant.

Measure 2 units on the x-axis and move to the ordinate part. Measure -7 units on the y-axis. Mark that point as the ordered pair C (2, -7).

Coordinate Graph

Frequently Asked Questions on Coordinate Plane

1. What are the 4 parts of a coordinate plane?

The major four different parts of a coordinate plane are the first quadrant, second quadrant, third quadrant, and fourth quadrant. These quadrants are obtained by drawing two straight and perpendicular lines on the coordinate graph. These points meet one other at the origin. All these quadrants are adjacent and the 1st, 2nd is above the 3rd and 4th quadrants. Based on the sign of coordinates of a point, we can say the point stays on which quadrant.

2. What is a coordinate plane example?

We have so many real-time examples of a cartesian plane. One of them is when you are planning to arrange furniture in a room divide the room into four parts and place the object at a place and know the area occupied by that particular object in the room. Later, decide based on your wishes and requirements.

3. What are the coordinates on a graph?

An ordered pair (a, b) gives the position of a point on a coordinate plane where a is the number on the x-axis and b is the number on the y-axis. The values of a, b in the ordered pair are called the coordinates on a graph.

Number Puzzles and Games

Best Number Puzzles and Games | Missing Number Puzzles with Answers

Number Puzzles and Games is a complicated topic in playing with numbers chapter. For solving each puzzle game, you must know the logic behind it and you should be smart enough. Playing with numbers and puzzle games will improve your skills and knowledge in math. So, check out the different types of puzzles and detailed steps to solve them.

Students can find the detailed steps, example questions on the games number puzzles concept that helps you to solve the questions easily. Have a look at the below sections and follow them.

Number Puzzles Definition

Number puzzle is a mathematical concept that has numbers from 1 to 9 and they must be placed in a grid of cells based on the condition given in the question. The best example of a number puzzle is Sudoku.

Types of Number Puzzles

The below-mentioned list are the types of number puzzles.

  • Cross Number: It is a puzzle similar to a crossword, but the entries in it are numbers, not words.
  • Sudoku: It is an excellent way to develop logical skills while having fun, and you can enjoy it while solving it. No math or no guess is needed to solve these questions.
  • Math Riddle: It will help kids to exercise the brain and improve creativity and logical reasoning.
  • Brain Teasers: Brain teasers are a type of puzzle, they come in various forms often represented as a question, activity riddle. It required little extra brain to solve.
  • Jigsaw Puzzle: These puzzles are simple, and helps you to build skills like visual reasoning, short-term memory, special awareness, and logical thinking.
  • Yohaku Puzzles: These focus on additive and multiplicative thinking and you can become more efficient by recalling certain facts as well as develop problem-solving skills.
  • Magic Triangle: You have to place the numbers on the magic triangle based on the conditions provided.

Step by Step Procedure to Solve Sudoku

The simple and easy steps to complete the sudoku game are listed here. Go through and follow the instructions carefully to solve difficult puzzles like sudoku in a fraction of seconds.

  • Let us take a 9 x 9 table.
  • You have to enter 1 to 9 numbers on each row and each column.
  • Each grid has only one number.
  • There should be no repetition of numbers on the row or column.

Examples of Number Puzzles and Games

Example 1:

Insert the eight four-digit numbers in the 4 × 4 grid, four reading across and four reading down.

{ 5 4 1 7, 8 6 2 1, 1 2 3 5 , 9 1 3 2, 6 1 9 3, 2 7 3 5, 3 7 5 1, 1 4 7 6, 6 5 2 8}

Solution:

Given set of numbers are { 5 4 1 7, 8 6 2 1, 1 2 3 5 , 9 1 3 2, 6 1 9 3, 2 7 3 5, 3 7 5 1, 1 4 7 6, 6 5 2 8}

At the first grid enter the numbers which have the common digit at the thousand’s place which are (6 1 9 3, 6 5 2 8).

Next enter the numbers wither from rows or columns.

After completing entering numbers in all the grids.

Check out once all numbers are numbers or not.

6 5 2 8
1 4 7 6
9 1 3 2
3 7 5 1

Example 2:

Find the value of A in the following image?

A

+ A

+ A

= B A

Solution:

In this case, A is a number its three times sum is a number and itself. Therefore, the sum of two A’s should be 0. Then the possibilities of A will be either 0 or 5.
If A = 0, then the total will be 0, so B = 0.
But is not possible. Because two different alphabets represent different numbers.
If A = 5, then
A + A + A = 5 + 5 + 5 = 1 5
So, B = 1, A = 5

Example 3:

Complete the magic square given below so that the sum of the numbers in each row or in each column or along each diagonal is fifteen.

Solution:

The total of each row or each column or diagonal is 15.

So, unfilled number in the second column is 15 – (1 + 5) = 15 – 6 = 9

Number at third row, third column is 15 – ( 6 + 5) = 15 – 11 = 4

Number at the first column, third row is 15 – (9 + 4) = 15 – 13 = 2

Number at first row, second column is 15 – (2 + 6) = 15 – 8 = 7

Number at first row, third coloumn is 15 – (6 + 1) = 15 – 7 = 8

Number at second row, third coloumn is 15 – (7 + 5) = 15 – 12 = 3

Divisibility Test Rules

Divisibility Test Rules | Division Rules in Maths

The divisibility test checks whether the number is divisible by another number or not. If the number is completely divisible by another number then the quotient will be a whole number and the remainder will be equal to zero. But every number is not exactly divisible by other numbers such numbers leave a remainder other than zero.

Like other games, the test of divisibility is also one important topic in the play with numbers chapter. Students can get a clear idea of what type of numbers are divisible by the whole numbers from 1 to 13 easily by checking out the divisibility rules provided below. Have a look at those division rules and say whether the given numbers are completely divisible or not in a fraction of seconds.

Test of Divisibility Definition

A nonzero integer “a” divides another integer “b” provided that the second integer b ≠ 0, and there is an integer c such that b = ac. We can say that a is the divisor of b and m is the factor of b and use the notation a | b.

In maths, division is the most basic concept that everyone should learn to score good marks in the examination. The basic terms of division are dividend, divisor, quotient, and remainder. It is opposite to the arithmetic operation multiplication.

Check:

Division Rules in Maths

Go through the following sections of this page to learn the shortcut methods to divide the numbers with fewer efforts. And get the divisibility rules for the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14 with the best examples.

Divisibility Rule of 1

Every number is divisible by 1. There are no divisibility rules for 1. Any number divided by 1 gives the original number itself.

Example:

31 divided by 1 and 359896472 divided by 1 is completely the same. It gives the original number as quotient.

Divisibility Rule of 2

Even numbers are divisible by 2. It means a number which is having any of these numbers 0, 2, 4, 6, 8 at the unit digit place is divisible by 2.

Example:

938 is divisible by 2. But the next number 939 is not divisible 2.

The step by step procedure is as follows:

  • Given number 938 is having 8 at the unit place.
  • 8 is divisible by 2.
  • Simply, 938 is also divisible by 2.

Divisibility Rule of 3

Any number is divisible by 3 if the sum of digits in the number is completely divisible by 3.

Example:

Let us take one number 3975 and check whether it is divisible by 3 or not.

Find the sum of digit in the number 3 + 9 + 7 + 5 = 24

The sum 24 is divisible by 3 and gives the remainder 0, quotient 8.

So, the original number 3975 is also divisible by 3.

Divisibility Rule of 4

Divisibility rule for 4 states that the last two digits of the number i.e digits at the unit place, tens place are divisible by 4, then the whole number is a multiple of 4.

Example:

Take one number 4582 and check whether it is divisible 4 or not.

The last two digits of the number are ’84’

84 is completely divisible by 4

Finally, the original number 4582 is also divisible by 4.

Divisibility Rule of 5

If the number has a last digit either 0 or 5, then it is divisible by 5.

Example:

10, 5, 95, 1000, 1565, 18895, etc are multiples of 5.

Divisibility Rule of 6

A number is exactly divisible by 6 if it is divisible by 2 and 3 both. For checking the divisibility rule with the number 6 we have to apply both the rules of divisibility for 2 and 3. Because by multiplying 2 and 3 we will get 6.

Example:

Let us take one number 12,582

The last digit of 12,582 is 2. So it is divisible 2.

The sum of digits of the original number is 1 + 2 + 5 + 8 + 2 = 18.

18 is divisibly by 3.

So, 12,582 is divisible by 6.

Divisibility Rule of 7

The divisibility rule by 7 is a bit difficult which can be understood by the simple steps provided below:

  • Remove the last digit of the number and double it.
  • Subtract the double from the remaining number.
  • If the number is zero or the recognizable 2 digit multiple of 7, then the original number is divisible by 7.
  • Otherwise, repeat the process.

Example:

Take the number 2107.

Remove the last digit of 2107 i.e 7

Double the removed number i.e 14

Subtract 14 from 210

210 – 14 = 196

Remove the last digit of 196 i.e 6

The double of 6 is 12.

19 – 12 = 7

As 7 is divisible by 7, 1027 is also divisible by 7.

Divisibility Rule of 8

To check whether a number is divisible 8, you have to check that the last three digits of that particular number should be divisible 8.

Example:

Let us consider one number 2,87,256

The last three digits are 256

As, 256 is completely divisible by 8, so the original number 2,87,256 is also divisible by 8.

Divisibility Rule of 9

The rule for divisibility by 9 is similar to the divisibility by 3. Which is the sum of digits of the original number is divisible by 9, then the number is exactly divisible by 9.

Example:

Consider the number 3897.

The sum of digits are 3 + 8 + 9 + 7 = 27

27 is divisible by 9, so 3897 is also divisible by 9.

Divisibility Rule of 10

The divisibility rule for 10 states that any number whose last digit is 0, is divisible by 10.

Example:

10, 100, 20, 250, 89570, etc

Divisibility Rule of 11

If the difference of the sum of alternative digits of a number is divisible by 11, then that number is divisible by 11.

Example:

Let us take the number 2816.

Group the alternative digits i.e digits in odd places together and digits in even places together. 21 and 86 are two different groups.

Take sum of digits of each group i.e 2 + 1 = 3 and 8 + 6 = 14

Now, get the difference of the sums, 14 – 3 = 11

The difference number 11 is divisible by 11, so the original number 2716 is divisible by 11.

Divisibility Rule of 13

The divisibility rule for 13 says that add four times the last digit original number to the remaining number and repeat the process until you get a two-digit number. If the obtained two-digit number is divisible by 13, then the original given number is also exactly divisible by 13.

Example:

Check whether the number 99,867 is divisible by 13 or not?

The four times of last of the given number is 7 * 4 = 28

Add product to the remaining number 9986 + 28 = 10,014

Repeat the process,

1001 + (4 * 5) = 1001 + 20

= 1021

Repeat the process,

102 + (1 * 4) = 102 + 4 = 142

142 is not divisible by 13

Hence, 99,867 is also not divisible by 13.

Divisibility Rule of 14

If the number is divisible by both 7 and 2, then the original number is divisible by 14. This means the number should be an even number and subtract the double of the last digit from the remaining number. Repeat the process, until you left a two-digit number. If the resultant two-digit number is divisible by 7, then the original number is divisible by 14.

Example:

Is 266 is divisible by 14?

266 is divisible by 2. Because it is an even number.

26 – (6 * 2) = 26 – 12 = 14

14 is divisible by 7.

So, 266 is divisible by 14.

Solved Example Questions

Example 1:

Check whether 2848 is divisible by 11 or not?

Solution:

Given number is 2848

The alternative number groups are 24, 88

2 + 4 = 6, 8 + 8 = 16

16 – 6 = 10

10 is not divisible by 11

∴ 2848 is not divisible by 11.

Example 2:

Is 768 is divisible by 7?

Solution:

Given number is 768

The last digit of 768 is 8.

double of 8 is 16

76 – 16 = 60

60 is not divisible by 7

∴ 768 is not divisible by 7.

Example 3:

Is 1440 is divisible by 15?

Solution:

Given number is 1440

The last digit is 0 so it is divisible by 5.

The sum of digits of 1440 is 1 + 4 + 4 + 0 = 9

9 is the multiple of 3.

∴ 1440 is the multiple of 15.

FAQs on Test of Divisibility

1.  What is meant by divisibility rules?

Divisibility test is a process of identifying the given dividend by a fixer divisor without performing the actual division process. If the dividend is completely divided by the divisor, then quotient should a whole number and the remainder is zero.

2. What is the divisibility rule for 18 and give an example?
A number which is divisible by 9 and 2 is the number divisible by 18. 1710 is an example for the divisibility of 18. It is an even number and sum of digits in it is 1 + 7 + 1 + 0 = 9. 9 is divisible by 9. so, 1710 is divisible by 18.

3. Prove that 35,16,48,792 is divisible by 396?

The factors of 396 = 4 * 9 * 11

If the number 35,16,48,792 is divisible by 4, 9, and 11 then it is divisible by 396.

The last two digits 92 is divisible by 4.

The sum digits is 3 + 5 + 1 + 6 + 4 + 8 + 7 + 9 + 2 = 45

45 is divisible by 9.

35,16,48,792 is divisible by 11.

Hence proved.

4. Explain the divisibility rule of 7?

Remove the last digit of the number and subtract the double of it from the remaining number. If the obtained number is multiple of 7, then it is divisible by 7.

How much Percentage One Quantity is of Another?

Wanna Express One Quantity as a Percentage of Another then continue reading as you will learn completely about it. Check the detailed procedure for determining how much percentage one quantity is of another. To help you understand the concept better we have provided solved example questions. Refer to them and learn the method used easily.

Expressing One Quantity as a Percentage of Another

Follow the below mentioned steps to know how much percentage one quantity is of another. They are as under

  • Take the numerator as the Quantity to be Compared.
  • Consider the Quantity with which it is to be Compared as the denominator.
  • Express the Quantities in Fraction Form and then multiply with 100.

While Expressing One Quantity as a Percentage of Another the two quantities must be of the same kind and should have the same units.

Let suppose we need to express m as a percentage of n then the formula is

Percentage = m/n*100 where m, n should have the same units.

Example Questions on What Percentage is One Number of the Another

1. Ram Scored 40 out of 60 in the exam. Calculate the Percentage of Marks gained by him?

Solution:

From the given data numerator is the Quantity to be Compared i.e. 40

The denominator is the quantity with which it is to be compared = 60

Express it as a fraction and multiply with 100 i.e. 40/60*100

= 66.66%

2. Express 1 Hour 20 Minutes as a Percentage of 2 Hour 40 Minutes?

Solution:

1 Hour 20 Minutes = 80 Minutes

2 Hour 40 Minutes = 160 Minutes

= 80/160*100

= 50%

3. Find the Number if 15% of it is 75?

Solution:

Let the number be m

15% of m = 75

15/100*m = 75

m = (75*100)/15

= 500

Therefore 15% of 500 is 75.

Word Problems for finding how much percentage one quantity is of another

1. What Percent of $12 is 60 Cents?

Solution:

We know 1 $ = 100 Cents

$12 = 1200 Cents

Percentage = (60/1200)*100

= 5%

Therefore, 60 cents is 5% of 12$.

2. What Percent of 45 Kg is 3 Kg?

Solution:

Required Percent = 3/45*100

= 100/15

= 6.66%

3. 250 is what percent of 3000?

Solution:

m% of 3000 = 250

m/100*3000 = 250

m = (250*100)/3000

= 8.33%

How To Find Coordinates of a Point on Graph With Examples

A set of values that shows the exact position of a point in a two-dimensional coordinate plane are called the coordinates. These represent the exact location of a point on a coordinate graph having both x, y axes. You can check the definition of coordinates, step by step detailed procedure to find the coordinates of a point with solved examples.

Coordinates Definition

A pair of numbers that describe the exact position of a point on a cartesian plane by using the horizontal and vertical lines called the coordinates. Usually represented by (x, y) the x value and y value of the point on a graph. Every point or an ordered pair contains two coordinates. The first one is the x coordinate or abscissa and the second is the y coordinate or ordinate. The values of the coordinates of a point can be any positive or negative real number.

The other types of coordinates are map coordinates (north/south, east/west), three-dimensional coordinates, polar coordinates (distance, angle), etc. Detailed information about x coordinate, y coordinate of a point follows:

x‐coordinate (Abscissa): The first number or the number which is located to the left side of a comma in the point is the x coordinate of the ordered pair. It represents the amount of movement along the x-axis from the origin. The movement is to the right side if the number is positive and to the left side of the origin if the number is negative.

y‐coordinate (Ordinate): The number which is located to the right side of the comma in the ordered pair or the second number is known as the y coordinate of the ordered pair. This ordinate indicates the amount of movement along the y-axis. If the number is positive, then the movement is above the origin and the movement is below the origin if the number is negative.

Point on x-axis: A point on the x-axis means its movement along the horizontal line is always zero and the y-coordinate of all points on the x-axis is zero. Therefore, the coordinates of a point on the x-axis are of the form (x, 0).

Point on y-axis: A point on the y-axis means the distance from the y-axis is zero and the x coordinate of every point on the y-axis is zero. Hence, the coordinates of a point on the y-axis are (0, y).

How to Find Coordinates of a Point?

Below given are the steps that are helpful to find the coordinates of a point. Go through them.

  • Go to the coordinate graph having lines X’OX, Y’OY.
  • Check out which quadrant of the graph has an ordered pair or a point.
  • To get the abscissa, measure the distance of the point from the x-axis.
  • Likewise, measure the distance of the point from the y-axis to obtain the ordinate value.

Solved Example Questions

Example 1.

In the adjoining figure, XOX’ and YOY’ are the co-ordinate axes. Find out the coordinates of point C?

Coordinate Graph

Solution:

To locate the position of point C draw straight and perpendicular lines from point C to the x-axis OX, y-axis OY’.

Measure the distance between the newly obtained point on the x-axis, origin, new point on the y-axis, and origin.

The value of that point on the x-axis is 2. And the value of the point on the y-axis is -7.

So, the x coordinate is 2, y coordinate is -7.

As the abscissa is positive and ordinate is negative, the point lies in the fourth quadrant.

∴ The ordered pair is C (2, -7)

Example 2.

Find the coordinates of three marked in the following figure?

 

Solution:

To locate the position of point P:

The ordered pair P is located in the first quadrant where both coordinates are positive.

The perpendicular distance of P from the x-axis is 5 units and the y-axis is 4 units.

So, the abscissa is 5, ordinate is 4

Therefore, the coordinates of P are (5, 4).

To locate the position of point Q:

The ordered pair Q is also located in quadrant 1.

The perpendicular of the point Q from the x-axis is 0.

So, the x coordinate is 0 and the point lies on the y-axis.

the distance of Q from the origin on the y-axis is 3 units.

So, y coordinate is 3.

The coordinates of point Q (0, 3)

To locate the position of point R:

The point R is located on the x-axis means its y coordinate is 0.

The distance of the point Q from the origin is -2 units.

So the x coordinate is -2.

The Coordinates of point R (0, -2).

Frequently Asked Questions on Coordinates of a Point

1. How many coordinates are there in a point?

For every point on a two-dimensional plane, we have two coordinates. One is x coordinate or abscissa and the second is y coordinate or ordinate.

2. What is meant by coordinates?

In a cartesian plane, the values of a point are called coordinates.

Plot Ordered Pairs | How to Plot an Ordered Pair on a Coordinate Plane?

Ordered pairs are used to locate a point on a cartesian plane. It has parenthesis, a comma, and two real numbers. The first real number is called the x coordinate or abscissa, and the second one is known as the y coordinate or ordinate. Those numbers can be either positive or negative. Depending on the sign of the coordinates, you can say that point belongs to which quadrant.

On this page, we are providing useful information like how do you define an ordered pair, steps of plotting ordered pairs on a coordinate plane, the detailed procedure to identify the quadrant of ordered pairs.

Ordered Pair Definition

The ordered pair is nothing but a point in the two-dimensional coordinate plane. It is used to locate a point on the graph. Every ordered pair has two coordinates namely abscissa and ordinate.

How do you plot an ordered pair of points on a Cartesian plane?

The following are simple and easy steps to locate a point in a cartesian plane. Check out the guidelines, terms, and conditions to plot ordered pairs.

  • Let us take one ordered pair.
  • Get the sign of x coordinate, y coordinate of the ordered pair.
  • Based on those signs, identify which quadrant the ordered pair belongs to.
  • Count x coordinate value on the x-axis starting from the origin.
  • Similarly, count the y coordinate value on the y-axis from the origin.
  • Mark the obtained point as the ordered pair.

Solved Example Questions

Example 1.

Plot P (4, 1) on the graph?

Solution:

Given point P (4, 1)

The x coordinate of the point is positive, 4. While the y coordinate is also positive, 1.

So, both the coordinates are positive, the point P belongs to the first quadrant.

To locate this point P, measure 4 units on the x-axis from the origin (towards the right). And count 1 unit on the y-axis from the origin (towards up). Draw a line from 4 and 1, the point those lines meet is called the ordered pair P(4, 1).

Example 2.

Plot the following ordered pairs in one coordinate plane.

a. A (5, 0) b. B (-2, -6) c. C (6, -3) d. D (7, -1)

Solution:

Given ordered pairs are A (5, 0), B (-2, -6), C (6, -3), D (-7, 1)

The abscissa for A is 5, the ordinate is 0, both are positive. Therefore this ordered pair lies in the first quadrant. Take 5 units on the x-axis (towards the right) and 0 units on the y-axis and mark the point with strict or any other symbol.

The x coordinate is -2, the y coordinate is -6, both are negative. So, point B lies in the third quadrant. Measure 2 units from the origin on the x-axis (towards left), 6 units on the y-axis (downwards). Mark that particular point as B (-2, -6).

For the ordered pair C (6, -3), the x coordinate is positive and the y coordinate is negative. Then, the point lies in the fourth quadrant. To represent the ordered pair on the coordinate graph, you need to draw horizontal lines naming XOX’, YOY”. Count 6 units on the x-axis towards the right from the origin and count 3 units on the y-axis towards from the origin. Plot that point on the graph as C.

The abscissa for point D is -7 and negative and ordinate is positive and 1. You can say that the point lies in the second quadrant. At first, draw one horizontal line (x-axis), vertical line (y-axis) meet at origin O. Write down the numbers as a coordinate graph. Now, measure 7 units on the axis towards the left from the origin, 1 unit on the y-axis towards up.

Example 3.

Plot the below given three points on the graph?

a. P (5, 4) b. Q (0, 3) c. R (-2, 0)

Solution:

Given ordered pairs are P (5, 4), Q (0, 3), R (-2, 0)

First point P (5, 4) is 5 units away from the origin on the x-axis, 4 units on the y-axis. As coordinates of the point are positive, it lies in the first quadrant.

As the second point has the x coordinate is zero, the point available on the y-axis and 3 units away from the origin. It also lies in quadrant 1.

In the same way, the third point also contains the y coordinate is zero, it is present on the x-axis. And its x coordinate is negative, so it lies in the third quadrant.

Playing with Numbers

Playing With Numbers for Class 8 | Number in General Form, Examples

Are you studying in 8th grade? If yes, then you must learn the Fun With Numbers unit. One of the most important concepts in that unit is Playing With Numbers. The name itself says that the topic is totally about the numbers. There are 4 different types of numbers which are natural numbers, whole numbers, integers, and rational numbers.

Natural numbers sometimes called the counting numbers which do not include 0 and include from 1 to n. Whole numbers are similar to the natural numbers which include zero. In integers, we have positive integers and negative integers and zero. And rational numbers are nothing but fractions in the form of a/b and b does not equal to zero.

Read further sections of this page to know completely about how to play games with numbers. You will see the definition of play with numbers, General form of numbers, Games with numbers, tests of divisibility, solved examples in the below sections.

Play with Numbers Definition

The name itself says that this chapter contains more information about numbers. There are various games to play with numbers. The name of the game is reversing the digits of a number, general form of the number, letters for digits, and tests of divisibility. The detailed process of all these games is mentioned below.

Number in General Form

You can represent any number in a general form easily by checking out the simple procedure. The only thing is to identify the position of each digit in the given number. And multiply the position of digit i.e tens digit 10 * number and add remaining digits. For example, take a two-digit number AB. Its general form is 10 * A + B. The simple thing is to perform the addition, multiplication operations and the result will be the original number.

Games with Numbers

Reversing the Digits:

Reverse the digits of the number, add the original number, and reversed number. Divide the sum by 11, the remainder will be automatically zero. You can play this game with either a two-digit number or a three-digit number. But for three-digit numbers, the procedure will be different.

Letters for Digits:

In this method, we are going two have two or more numbers and expressions in between them. Perform those arithmetical operations and by trial and error method get which number suits in expression that satisfies the equation. It is one of the tricky and easy games to play with numbers.

Tests of divisibility:

The test of divisibility means a list of numbers divisible by a particular number and leaves the remainder zero or not.

Number Puzzles:

Playing with number puzzles and games will increase your skills and knowledge in math numbers. You have to fill the given numbers in the grid or magic triangle or any other by satisfying the conditions provided.

Solved Examples

Example 1:

Write the following list of numbers in the generalized form.

a. 125 b. 76 c. 59 d. 895

Solution:

Given numbers are 125, 76, 59, 895

General form of 125 is as follows:

125 can be expressed as 100 + 20 + 5

1 * 100 + 2 * 10 + 5 is 125.

76 = 70 + 6

= 7 * 10 + 6

59 = 50 + 9

= 5 * 10 + 9

895 = 800 + 90 + 5

= 8 * 100 + 9 * 10 + 5

Example 2:

Prove that the following numbers satisfy the reversing the digits method?

a. 12 b. 756

Solution:

Given two-digit number is 12.

Reverse of 12 is 21.

12 + 21 = 33

11) 3 3 (3

–   3 3

=   0

Three digit number is 756

Reverse of 756 is 657

756 – 657 = 99

99) 9 9 (1

–   9 9

=   0

Example 3:

In a two-digit number, the digit in the units place is 4 times the digit in the tens place and the sum of the digits is equal to 10. What is the number?

Solution:

Let us take two digit number as ‘ab’

b = 4 * a

a + b = 10

Substitute b = 4a in the above equation.

a + 4 * a = 10

5a = 10

a = 2

Substitute a = 2 in the equation to get b value.

2 + b = 10

b = 10 – 2 = 8

∴ The number is 28.

Example 4:

Check is 195 divisible by 4 or not?

Solution:

Given divisor is 195 and dividend is 4

The last digit of 195 is 5, which is not divisible by 4.

Hence, 195 is not divisible by 4.

Example 5:

In the given 3 × 3 grid, arrange the digits from 1 to 9 (without repetition) such that the sum of the numbers surrounding any cell is a multiple of the number in the cell.

Solution:

The only way is provided here.

4 8 9
1 3 7
5 6 2

Check out the games and learn hose to be perfect in mathematics.

Ordered Pair of a Coordinate System| How to Find Ordered Pairs?

The ordered pair is nothing but a point on the coordinate graph. It is used to determine the position of a point in the cartesian plane. The more useful details regarding the ordered pair are provided below. Get an idea regarding the Ordered Pair Definition, Its Applications, and Solved Examples in the forthcoming modules.

Ordered Pair Definition

An ordered pair is the combination of coordinates which are x coordinate (abscissa) and y coordinate (ordinate). Those coordinates are real values enclosed by parenthesis and separated by a comma. The ordered pair is helpful to locate a point on the two-dimensional coordinate graph for better visual comprehension. The values in it can be integers or fractions.

Ordered Pair of a Coordinate System

Two numbers written in a certain order with parenthesis is also called an ordered pair. The usual representation of an ordered pair is (x, y). Where x is the horizontal value and y is the vertical value, and x, y are called the coordinates. The ordered pair (x, y) is never equal to the ordered pair (y, x). Whenever we write the coordinates of a point, first we write the x coordinate and then y coordinate value. As the ordered pair suggests the order in which values are written in a pair is very important.

Let us take a point P (2, 5)

The first number in the ordered pair shows the distance from the x-axis which is 2

The second number in the ordered pair shows the distance from the y-axis which is 5

To plot that point on the coordinate graph, count 2 steps towards the x-axis (towards the right) and start counting from the origin. And then 5 steps on the y-axis (upwards direction).

Applications:

  • This concept is useful in data comprehension and statistics.
  • The coordinate geometry uses ordered pairs to represent the geometric figures in an open space for virtual comprehension.
  • Geometric shapes can be circles, triangles, squares, rectangle,s, and polygons use the ordered pairs to represent the center, vertices, and the length of the sides with coordinates.

Solved Example Questions

Example 1.

What are the coordinates of an ordered pair A (9, -7)?

Solution:

Given ordered pair is A (9, -7)

Its coordinates are x coordinate is 9 and the y coordinate is -7.

As the x coordinate is positive and the y coordinate is negative, the point belongs to the fourth quadrant.

Example 2.

Find out the abscissa and ordinates of a point P (-9, 7)?

Solution:

Given point is P (-9, 7)

The abscissa of the point is -9.

The ordinate of the point is 7.

As the point abscissa is a negative integer and ordinate is a positive integer the point lies in the third quadrant.

Example 3.

Is ordered pair A (5, 8) and B (8, 5) are equal?

Solution:

Given that,

A (5, 8) and B (8, 5)

The real numbers in both ordered pairs are the same. But both A and B are not equal.

Why because the first ordered pair A is having 5 as the x coordinate, 8 as the y coordinate, and the second ordered pair B (8, 5) is having 8 as the x coordinate and 5 as the y coordinate. So A ≠ B.

∴ Given ordered pairs, A and B are not equal.

FAQs on Ordered Pair of a Coordinate System

1. What is an example of an ordered pair?

Few examples of an ordered pair are (8, 6), (-8, 6), (8, -6), (-8, -6). All these ordered pairs are not equal. Because they belong to different quadrants.

2. What comes first in an ordered pair?

For every ordered pair, the x coordinate comes first followed by the y coordinate. It is important to write both coordinates in ordered pairs.

3. What is the order of an ordered pair?

An ordered pair is a pair of numbers present in a specific order and contains two numbers. The order in which you write those numbers is very important. Those numbers are called coordinates of a point. The x coordinate is the first number and y coordinate is the second number. The ordered pair is useful to locate a point in the coordinate plane.

Quadrants of a Coordinate Plane | Quadrant I, II, III, and IV

When you draw the x-axis, y-axis on a coordinate graph quadrants is formed. In every two dimensional graph, we will get 4 quadrants. In this page, we are discussing the All Four Quadrants concept. It is one of the important topics in coordinate geometry. This article is very useful for students who are preparing for their exams.

We giving the complete details regarding quadrants, its definition, signs of coordinates on each quadrant, and its examples in the following sections. Have a look at them and know about quadrant 1, quadrant 2, quadrant 3, and quadrant 4.

Quadrant Definition

Take a graph and draw two perpendicular lines horizontal line (x-axis), vertical line (y-axis). These two axes divide the plane into 4 equal parts. Each part is called a quadrant. Those two perpendicular lines meet at a point called the origin. By convention, quadrants are named in an anticlockwise direction. The names of all four quadrants are quadrant I, quadrant II, quadrant III, quadrant IV. All these quadrants are differentiated by the sign of coordinates of the points.

Signs of Quadrants on the x-axis, y-axis

Depending upon the sign of coordinates, quadrants are divided into 4 types. You have to know the signs of abscissa and ordinate of an ordered pair to check under which quadrant the point lies in.

Quadrant I: In the first quadrant, both coordinates are positive values. The region is XOY.

Quadrant II: In this quadrant, the x value is negative and the y value is positive. The region of X’OY.

Quadrant III: In this quadrant, both x and y coordinates are negative. The region is X’OY’.

Quadrant IV: In the fourth quadrant, the x coordinate is positive and the y coordinate is negative. the region is XOY’.

All Four Quadrants

Example Coordinates of Each Quadrant

  • Point P (8, 5) lies in the first quadrant.
  • Point Q (-8 5) lies in the second quadrant.
  • Point R (-8, -5) lies in the third quadrant.
  • Point S(8, -5) lies in the fourth quadrant.

The below-listed points does not lie in any of the quadrants. They lie on the x-axis, y-axis.

A (0, 5), B (-5, 0), C (0, -5), D (5 0), etc of this type.

Practice Test on Exponents

Practice Test on Exponents | Exponents Practice Problems & Solutions

Practice Test on Exponents prevailing will test your conceptual knowledge on Chapter Exponents. Assess your strengths and weaknesses and concentrate on the topics you are lagging accordingly. Try solving the problems on Exponents on your own and verify with the solutions provided. The Questions cover various exponent laws and you can apply them to make your calculations much simple.

1. The Value of (4/5)-2 is

(a) 16/25

(b) 25/2

(c) 25/16

(d) -2/5

Solution:

(a) 16/25

Given Expression is (4/5)-2

= 1/(4/5)2

= (5/4)2

= 25/16

2. The value of (-2)-3 is

(a) -8

(b) 8

(c) -1/16

(d) -1/8

Solution:

(d) -1/8

Given (-2)-3

= (1/-2)3

= (-1/2)3

= (-1/8)

= -1/8

3. What is the value of (-2/5)-2 × (4/3)-3?

(a) -24

(b) – 8192/2187

(c) 675/256

(d)  1/256

Solution:

(c) 675/256

Given (-2/5)-2 × (4/3)-3

= (5/2)2 × (3/4)3

= 25/4*27/64

= (25*27)/(4*64)

= 675/256

4. Find the value of {(2/-3)-2}3 is

(a) -3/2

(b) 4/3

(c) 729/64

(d) 1024/243

Solution:

(c) 729/64

Given {(2/-3)-2}3

= {(-3/2)2}3

= {-3/2}6

= 729/64

5. What is the value of (2-1 + 3-1)-1 ÷ 4-1 

(a) 4/10

(b) 10/3

(c) 3/5

(d) 4/15

Solution:

(b) 10/3

Given (2-1 + 3-1)-1 ÷ 4-1

= (1/2+1/3)÷1/4

= ((3+2)/6)÷1/4

= 5/6÷1/4

= 5/6*4/1

= 20/6

= 10/3

6. Simplify: (1/3)-2 + (2/4)-2 + (3/2)-2?

(a) 89/36

(b) 141/36

(c) 21/4

(d) 121/9

Solution:

(d) 121/9

Given (1/3)-2 + (2/4)-2 + (3/2)-2

= (3/1)2+ (4/2)2+(2/3)2

= 9+4+4/9

= (81+36+4)/9

= 121/9

7. The value of x for which (5/12)-4 * (5/12)3x = (5/12)5?

(a) -2

(b) -1

(c) 1

(d) 3

Solution:

(d) 3

(5/12)-4 * (5/12)3x = (5/12)5

Since bases are equal we need to add the powers.

(5/12)-4+3x = (5/12)5

Both the sides are equal and equating the powers we have the value of x as such

-4+3x = 5

3x = 5+4

3x = 9

x = 3

8. If 33x – 1 = 9, then x is equal to

(a) -2

(b) 0

(c) 1

(d) 2

Solution:

(c) 1

33x – 1 = 9

33x/3 = 9

33x = 27

33x = 33

3x = 3

x= 3/3

= 1

9.  [{(-1/3)3}-2]-1

(a) 1/15

(b) 23

(c) 729

(d) -15

Solution:

Given [{(-1/3)3}-2]-1

= (-1/3)6

= (3/1)6

= 729

10. {(1/2)-3 – (1/4)-3} ÷ (1/3)-3 = ?

(a) 17/64

(b) 27/16

(c) -56/27

(d) 15/24

Solution:

(c) -56/27

Given {(1/2)-3 – (1/4)-3} ÷ (1/3)-3

= {(2/1)3 -(4/1)3} ÷ (3/1)3

= ((8)-64)÷27

= -56/27

Signs of Coordinates of a Point

In an ordered pair, the sign of coordinates decides that particular point goes into which quadrant of the coordinate plane. If you are not aware of the importance of coordinate signs, then you may face issues while plotting points on the coordinate graph. So, check out the step by step process to find the quadrant of a point. Have a glance at the solved examples and understand the concept better.

Steps to Find Signs of Coordinates of a Point in All Quadrants

These are the simple steps to identify the signs of coordinates of a point easily. On a coordinate graph the lines X’OX, Y’OY represent the coordinate axes. In that line, OX is considered as the positive x-axis, and OX’ is taken as the negative x-axis. In the same way, OY is a positive y-axis and OY’ is a negative y-axis.

As we all know that in a point (x, y) x represents the abscissa or x coordinate value, and y represents the ordinate or y coordinate. You must observe the sign of these x and y coordinates to choose which part of the graph the point goes into.

If both x, y > 0, then the point is in quadrant I

If x < 0, y >0, then the point is in quadrant II

If x < 0, y < 0, then the point lies in quadrant III

If x > 0, y <0= 0, then point goes to quadrant IV

The below-mentioned graph will give brief information.

All Four Quadrants

The main important points about the point coordinates sign are listed below.

  • Any point having abscissa, ordinate as positive lies in the first quadrant.
  • The coordinates of a point lie in the second quadrant when its x coordinate is negative and the y coordinate is positive.
  • When a point is in the third quadrant, then its abscissa and ordinates are having negative signs.
  • The points in the fourth quadrant have the x coordinate positive sign and the y coordinate negative sign.