Common Core Math

Are you looking for ways on how to convert from decimals to fractions? Then, look no further as we have listed the easy ways to convert from decimal to fraction in a detailed way here.  Before doing so you need to understand firstly what is meant by a decimal and fractions. Refer to the entire article to know about the procedure on how to convert a decimal to a fraction along with solved examples.

What is meant by a Decimal?

A Decimal Number is a number that has a dot between the digits. In other words, we can say that decimals are nothing but fractions with denominators as 10 or multiples of 10.

Examples: 4. 35, 3.57, 6.28, etc.

What is meant by Fraction?

A fraction is a part of the whole number and is expressed as the ratio of two numbers. Let us consider two numbers a, b and the fraction is the ratio of these two numbers i.e. \(\frac { a }{ b } \) where b≠0. Here a is called the numerator and b is called the denominator. We can perform different arithmetic operations on fractions similar to numbers. There are different types of fractions namely Proper, Improper, Mixed Fractions, etc.

How to Convert Decimal to Fractions?

Follow the simple steps listed below to convert from decimal to fraction and they are as such

• Firstly, obtain the decimal and write down the decimal value divided by 1 i.e. \(\frac { decimal }{ 1 } \)
• Later, Multiply both the numerator and denominator with 10 to the power of the number of digits next to the decimal point. For else, if there are two digits after the decimal point then multiply the numerator and denominator with 100.
• Simplify or Reduce the Fraction to Lowest Terms possible.

Solved Examples on Decimal to Fraction Conversion

1. Convert 0.725 to Fraction?

Solution:

Step 1: Obtain the Decimal and write the down the decimal value divided by 1 i.e. \(\frac { 0.725 }{ 1 } \)

Step 2: Count the number of decimal places next to decimal point. Since there are three digits multiply both the numerator and denominator with 10³ i.e.  we get \(\frac { 725 }{ 1000 } \)

Step 3: Reduce the Fraction to the Lowest Terms by dividing with GCF. GCF(725, 1000) is 25. Dividing with GCF we have the equation as follows

= \(\frac { (725÷25) }{ 1000÷25 } \)

= \(\frac { 29 }{ 40 } \)

Therefore, 0.725 converted to fraction is \(\frac { 29 }{ 40 } \)

2. Convert 0.15 to fraction?

Solution:

Step 1: Obtain the decimal and write the decimal value divided by 1 i.e. \(\frac { 0.15 }{ 1 } \)

Step 2: Count the number of decimal places next to the decimal point. Since there are two digits multiply both the numerator and denominator with 10² i.e. we get \(\frac { 15 }{ 100 } \)

Step 3: Simplify the Fraction Obtained to Lowest Terms if possible by dividing with GCF.

We know GCF(15, 100) = 5

Dividing with GCF we get the equation as follows

= \(\frac { (15÷5) }{ 100÷5 } \)

= \(\frac { 3 }{ 20 } \)

Therefore, 0.15 converted to fraction is \(\frac { 3 }{ 20 } \)

FAQs on Converting Decimals to Fractions

1. What is a Decimal?

A Decimal Number is a Number whose Whole Number Part and Fractional Part are separated by a decimal point.

2. How to Convert a Decimal to Fraction?

We can convert Decimal to Fraction by placing the given number without considering decimal point as the numerator and denominator 1 followed by a number of zeros next to the decimal value. Later, reduce the fraction obtained to the lowest terms to obtain the resultant fraction form.

3. What is 0.62 as a Fraction?

0.62 in Fraction Form is obtained by dividing numerator with denominator 10 to the power of the number of zeros next to decimal point i.e. \(\frac { 62 }{ 100 } \). Simplify the fraction to the lowest terms. Thus, we get the fraction form as \(\frac { 31 }{ 50 } \)

The line of symmetry is the axis or imaginary line that passes through the center of an object and divides it into identical halves. If we cut an equilateral triangle into two halves, then it forms two right-angled triangles. Similarly, rectangle, square, circle are examples of a line of symmetry. The line of symmetry is also called an axis of symmetry. Also, it is named a mirror line where it forms two reflections of an image. A basic definition of a line of symmetry is it divides an object into two halves.

Types of Lines of Symmetry

There are mainly two types considered in Lines of Symmetry concepts. They are
1. Vertical Line of Symmetry
2. Horizontal Line of Symmetry

Vertical Line of Symmetry:  If the axis of the shape cuts it into two equal halves vertically then it is called a Vertical Line of Symmetry. The mirror image of the one half appears in a vertical or straight standing position. Examples for vertical Line of Symmetry are H, M, A, U, O, W, V, Y, T.
Horizontal Line of Symmetry: If the axis of the shape cuts it into two equal halves horizontally, then it is called as Horizontal Line of Symmetry. The mirror image of the one half appears as the other similar half. Examples of Horizontal Line of Symmetry are C, B, H, E.
Three Lines of Symmetry: An equilateral triangle is an example of three lines of symmetry. This is symmetrical along its three medians.
Four Lines of Symmetry: A square is an example of Four Lines of Symmetry. The symmetrical lines are two along the diagonals and two along with the midpoints of the opposite sides.
Five Lines of Symmetry: A regular pentagon is an example of Five Lines of Symmetry. The symmetrical lines are joining a vertex to the mid-point of the opposite side.
Six Lines of Symmetry: A regular hexagon is an example of Six Lines of Symmetry. The symmetrical lines are 3 joining the opposite vertices and 3 joining the mid-points of the opposite sides.
Infinite Lines of Symmetry: A circle is an example of Infinite Lines of Symmetry. It has infinite or no lines of symmetry. It is symmetrical along all its diameters.

Line of Symmetry Examples

Check out some of the examples of Line of Symmetry and learn completely with clear details.

1. Line segment:

From the figure, there is one line of symmetry. line of symmetry of a Line segment passes through its center. There may infinite line passes through the line segment and forms different angles. But we only consider a line as a line of symmetry that cuts the line segment into two equal halves. The Line segment AB is symmetric along the perpendicular bisector l.

2. An angle:

From the figure, there is one line of symmetry. An angle measures the amount of ‘turning’ between two straight lines that meet at a point. The figure is symmetric along the angle bisector OC.

3. An isosceles triangle:

From the figure, there is one line of symmetry. The isosceles triangle figure is symmetric along the bisector of the vertical angle. The Median PL. If the isosceles triangle is also an equilateral triangle, then it has three lines of symmetry. An isosceles triangle has exactly two sides of equal length. Therefore, it has only 1 line of symmetry that passes from the vertex between the two sides of equal length to the midpoint of the side opposite that vertex.

4. Semi-circle:

From the figure, there is one line of symmetry. The Semi-circle figure is symmetric along the perpendicular bisector l. of the diameter AB. A semi-circle does not have any rotational symmetry.

5. Kite:

From the figure, there is one line of symmetry. The Kite is symmetric along with the diagonal BS. A kite is a quadrilateral with two different pairs of adjacent sides that are equal in length and also have only one line of symmetry.

6. Isosceles trapezium:

From the figure, there is one line of symmetry. The Isosceles trapezium figure is symmetric along the line l joining the midpoints of two parallel sides AB and DC. The isosceles trapezium is a convex quadrilateral consists a pair of non-parallel sides that are equal and another pair of sides is parallel but not equal.

7. Rectangle:

From the figure, there are two lines of symmetry. The Rectangle figure is symmetric along the lines l and m joining the midpoints of opposite sides. There are 2 symmetry lines of a rectangle which are from its length and breadth. They cut the rectangle into two equal halves. They appear mirror to each other.

8. Rhombus:

From the figure, there are two lines of symmetry. The Rhombus figure is symmetric along the diagonals AC and BD of the figure. Both the lines of symmetry in a rhombus are from its diagonals. So, it can also say the rhombus lines of symmetry are both diagonals.

Lines of Symmetry in Alphabets

Have a look at the letters that have the line of symmetry.
One Line of symmetry: The letters consist of One line of symmetry are A B C D E K M T U V W Y.
Vertical Line of symmetry: The letters consist of a Vertical line of symmetry is A M T U V W Y.
Horizontal Line of symmetry: The letters consist of a Horizontal line of symmetry is B C D E K.
Two Lines of Symmetry: The letters consist of Two lines of symmetry are H I X. These are having both horizontal and vertical lines of symmetry.
No Lines of Symmetry: The letters consist of No lines of symmetry are F G J L N P Q R S Z. These have neither horizontal nor vertical lines of symmetry.
Infinite Lines of Symmetry: The letter having Infinite lines of symmetry is O.

Learn all the relations of Trigonometrical Ratios of (90° + θ). There are six different trigonometrical ratios depends on the (90° + θ). Let a rotating line OP rotates about O in the anti-clockwise direction, from starting position to ending position. It makes an angle ∠XOP = θ again the same rotating line rotates in the same direction and makes an angle ∠POQ =90°. Therefore, we see that ∠XOQ = 90° + θ.

                                                 
                                               

Take a point R on OP and draw RS perpendicular to OX or OX’. Again, take a point T on OQ such that OT = OR and draw TV perpendicular to OX or OX’. From the right-angled ∆ ORS and ∆ OTV, we get,
∠ROS = ∠OTV [since OQ ⊥ OP] and OR = OT.
Therefore, ∆ ORS ≅ ∆ OTV (congruent).
Therefore according to the definition of a trigonometric sign, OV = – SR, VT = OS and OT = OR
We observe that in diagrams 1 and 4 OV and SR are opposite signs and VT, OS are either both positive. Again we observe that in diagrams 2 and 3 OV and SR are opposite signs and TV, OS are both negative.

Check out Worksheet on Trigonometric Identities to learn more about Trig Identities and know the relation between all six quadrants.

Evaluate Trigonometrical Ratios of (90° + θ)

1. Evaluate sin (90° + θ)?

Solution:
To find sin (90° + θ), we need to consider the following important points.
(i) (90° + θ) will present in the IInd quadrant.
(ii) When we have 90°, “sin” will become “cos”.
(iii) In the IInd quadrant, the sign of “sin” is positive.
From the above points, we have sin (90° + θ) = cos θ
sin (90° + θ) = VT/OT
sin (90° + θ) = OS/OR, [VT = OS and OT = OR, since ∆ ORS ≅ ∆ OTV]

sin (90° + θ) = cos θ

2. Evaluate cos (90° + θ)?

Solution:
To find cos (90° + θ), we need to consider the following important points.
(i) (90° + θ) will present in the IInd quadrant.
(ii) When we have 90°, “cos” will become “sin”.
(iii) In the IInd quadrant, the sign of “cos” is negative.
From the above points, we have cos (90° + θ) = – sin θ
cos (90° + θ) = OV/OT
cos (90° + θ) = – SR/OR, [OV = -SR and OT = OR, since ∆ ORS ≅ ∆ OTV]

cos (90° + θ) = – sin θ.

3. Evaluate tan (90° + θ)?

Solution:
To find tan (90° + θ), we need to consider the following important points.
(i) (90° + θ) will present in the IInd quadrant.
(ii) When we have 90°, “tan” will become a “cot”.
(iii) In the IInd quadrant, the sign of “tan” is negative.
From the above points, we have a tan (90° + θ) = – cot θ
tan (90° + θ) = VT/OV
tan (90° + θ) = OS/-SR, [VT = OS and OV = – SR, since ∆ ORS ≅ ∆ OTV]

tan (90° + θ) = – cot θ.

4. Evaluate csc (90° + θ)?

Solution:
To find csc (90° + θ), we need to consider the following important points.
(i) (90° + θ) will present in the IInd quadrant.
(ii) When we have 90°, “csc” will become “sec”.
(iii) In the IInd quadrant, the sign of “csc” is positive.
From the above points, we have csc (90° + θ) = sec θ
csc (90° + θ) = 1/sin(90°+θ)
csc (90° + θ) = 1/cosθ

csc (90° + θ) = sec θ.

5. Evaluate sec (90° + θ)?

Solution:
To find sec (90° + θ), we need to consider the following important points.
(i) (90° + θ) will present in the IInd quadrant.
(ii) When we have 90°, “sec” will become “csc”.
(iii) In the IInd quadrant, the sign of “sec” is negative.
From the above points, we have sec (90° + θ) = – csc θ
sec (90° + θ) = 1/cos(90° + θ)
sec (90° + θ) = 1/-sinθ

sec (90° + θ) = – csc θ.

6. Evaluate cot (90° + θ)?

Solution :
To find cot (90° + θ), we need to consider the following important points.
(i) (90° + θ) will present in the IInd quadrant.
(ii) When we have 90°, “cot” will become “tan”
(iii) In the IInd quadrant, the sign of “cot” is negative.
From the above points, we have cot (90° + θ) = – tan θ
cot (90° + θ) = 1/tan(90° + θ)
cot (90° + θ) = 1/-cotθ

cot (90° + θ) = – tan θ.

Solved Examples on Trigonometrical Ratios of (90° + θ)

1. Find the value of sin 135°?

Solution:
Given value is sin 135°
sin 135° = sin (90 + 55)°
We know that sin (90° + θ) = cos θ
= cos 45°;
The value of cos 45° = 1/√2.

Therefore, the value of sin 135° = 1/√2.

2. Find the value of tan 150°?

Solution:
Given value is tan 150°
tan 150° = tan (90 + 60)°
We know that tan (90° + θ) = – cot θ
= – cot 60°;
The value of -cot 60° = 1/√3.

Therefore, the value of tan 150° = 1/√3.

Addition and Subtraction of Fractions tips and tricks are here. Know the various formulae, methods, and rules involved in adding or subtracting the fractions. Refer to steps on how to add and subtract fractions from each other. Find Solved Example Problems on fractions addition and subtraction of values. Check the below sections to know the complete details regarding fraction values addition and subtraction.

Addition and Subtraction of Fractions

Addition and Subtraction of Fractions are not that easy as adding or subtracting the whole numbers. It requires an extra procedure to get the desired results. There are certain steps to follow while adding or subtracting the fractions. Though you come to know various steps, you have to practice more problems to become perfect in this concept.

How to Add and Subtract Fractions?

Follow the below-listed procedure to know in detail about Adding and Subtracting Fractions. By following these simple steps you can solve Addition, Subtraction of Fractions Problems easily. They are as under

• In the first step, you have to verify if both the denominators are equal or not.
• In the case of different denominators, you have to convert the denominator to the same value to make them equivalent fractions. Equivalent fractions are those which has the same denominator value.
• Once the denominator is the same for both the fractions, then we go for further simplification.
• We add or subtract the numerator values in the simplification process.
• Write the answer to the numerator value with the common denominator.

Tips for Adding and Subtracting Fractions

•  Make sure, denominators are similar or equal before adding or subtracting the fractions.
• On multiplying the top and bottom of the fraction with the same number, the value of the fractions remains the same.
• Before starting the simplification of adding and subtracting fractions, practice converting fractions to common denominators beforehand.
• You have to simplify your answer once addition and subtraction are done. In some of the cases, we find that the result of the answers can be reduced even if the original fractions cannot be reduced. The same procedure is followed in both the cases of adding and subtracting fractions.
• In the case of mixed fractions, first, you have to convert them to improper fractions and then start the simplification of adding and subtracting fractions.

Methods of Adding and Subtracting Fractions

Fractions addition and subtraction involve different methods. They are

• Like fractions addition and subtraction
• Unlike fractions addition and subtraction
• Mixed Fractions addition and subtraction

Like Fractions Addition and Subtraction

The fraction values which possess the same denominators are called “like fractions”. Addition and subtraction of these like fractions are easy because the value of the denominator is the same for both the fractions.

Example 1:

Solve the equation 1/4 + 2/4

Solution:

Given equation is 1/4 + 2/4

In the above equation, the denominator has the same values. Therefore, we have to consider the common denominator and then add the numerator values.

Step 1: Verify if the denominators of the fractions are the same.

Step 2: As the denominator’s values are the same, then add the numerator values to get the final result.

Step 3: Simplification of the equation

1+\(\frac { 2 }{ 4 } \) = \(\frac { 3 }{ 4 } \)

Therefore, the final result is \(\frac { 3 }{ 4 } \)

Example 2:

Solve the equation \(\frac { 6 }{ 8 } \)– \(\frac { 2 }{ 8 } \)?

Solution:

Given equation is \(\frac { 6 }{ 8 } \)– \(\frac { 2 }{ 8 } \)

In the above equation, we have the same denominator values, so we can directly subtract the values to get the final result.

Step 1: Verify if the denominators of the fractions are the same.

Step 2: As the denominator’s values are the same, subtract the numerator values to get the final result.

Step 3: \(\frac { 6 }{ 8 } \)– \(\frac { 2 }{ 8 } \) = \(\frac { (6-2) }{ 8 } \)

= \(\frac { 4 }{ 8 } \)

Step 4: Check if further simplification is possible

As the fraction value is \(\frac { 4 }{ 8 } \), on further simplification it results as \(\frac { 1 }{ 2 } \)

Unlike Fractions Addition and Subtraction

The fractions with different denominators are called “unlike fractions”. It is mandatory that the denominator value should be the same to add or subtract the numerator values. So, for the unlike fractions, first, we have to convert them to equivalent fractions and then simplify the equation.

Example 1:

Solve the equation \(\frac { 4 }{ 6 } \) + \(\frac { 2 }{ 8 } \)?

Solution:

Given equation is \(\frac { 4 }{ 6 } \) + \(\frac { 2 }{ 8 } \)

In the above equation, the denominator values are different and hence the fractions should be made equivalent first and then simplify them.

Step 1: Verify if the denominators are the same. The denominators are not the same and hence to make it equivalent. First, take the LCM of denominator values ie., 6 and 8. The LCM is 24

Step 2: To get the same denominator value, we multiply 6 and 8 with the multiple that converts both into 24. Multiple the multiple to both numerator and denominator of the fraction.

For the fraction value \(\frac { 4 }{ 6 } \), \(\frac { 4 }{ 4 } \) is to be multiplied. Therefore,

\(\frac { 4 }{ 6 } \) . \(\frac { 4 }{ 4 } \)

= \(\frac { 16 }{ 24 } \)

For the fraction value, \(\frac { 2 }{ 8 } \), \(\frac { 3 }{ 3 } \) is to be multiplied. Therefore,

\(\frac { 2 }{ 8 } \) . \(\frac { 3 }{ 3 } \)

= \(\frac { 6 }{ 24 } \)

Step 3: After the simplification, now verify if the denominators are the same.

The equations are \(\frac { 16 }{ 24 } \) and \(\frac { 6 }{ 24 } \). Therefore the denominators are the same.

Step 4: Now that the denominators are the same, now add the numerator values and give the common denominator

= \(\frac { 16 }{ 24 } \) + \(\frac { 6 }{ 24 } \)

= \(\frac { 22 }{ 24 } \)

Example 2:

Solve the equation \(\frac { 4 }{ 6 } \) – \(\frac { 2 }{ 8 } \)?

Solution:

The given equation is \(\frac { 4 }{ 6 } \) – \(\frac { 2 }{ 8 } \)

In the above-given equation, the denominator values are different and hence the fractions should be made equivalent first and then simplify them.

Step 1: Verify if the denominators are the same. The denominators are not the same and hence to make it equivalent. First, take the LCM of denominator values ie., 6 and 8. The LCM is 24

Step 2: To get the same denominator value, we multiply 6 and 8 with the multiple that converts both into 24. Multiple the multiple to both numerator and denominator of the fraction.

For the fraction value \(\frac { 4 }{ 6 } \), \(\frac { 4 }{ 4 } \) is to be multiplied. Therefore,

\(\frac { 4 }{ 6 } \) . \(\frac { 4 }{ 4 } \)

= \(\frac { 16 }{ 24 } \)

For the fraction value, \(\frac { 2 }{ 8 } \), \(\frac { 3 }{ 3 } \) is to be multiplied. Therefore,

\(\frac { 2 }{ 8 } \) . \(\frac { 3 }{ 3 } \)

= \(\frac { 6 }{ 24 } \)

Step 3: After the simplification, now verify if the denominators are the same.

The equations are \(\frac { 16 }{ 24 } \) and \(\frac { 6 }{ 24 } \). Therefore the denominators are the same.

Step 4: Now that the denominators are same, now subtract the numerator values and give the common denominator

= \(\frac { 16 }{ 24 } \) – \(\frac { 6 }{ 24 } \)

= \(\frac { 10 }{ 24 } \)

Mixed Fractions Addition and Subtraction

Example for addition of fractions

Solve the equation 3 \(\frac { 3 }{ 4 } \) + 2 \(\frac { 2 }{ 4 } \)?

Solution:

Given equation is 3 \(\frac { 3 }{ 4 } \) + 2 \(\frac { 2 }{ 4 } \)

The given fraction is a mixed fraction, hence we have to convert it to the whole number and simplify further.

Step 1: First of all, add the whole number to the mixed fractions

3 + 2 = 5

Step 2: Now, in the next step add the fractional part of the mixed number

\(\frac { 3 }{ 4 } \)  + \(\frac { 2 }{ 4 } \)

= \(\frac { 5 }{ 4 } \)

Step 3: As both the fractions are added separately, then convert an improper fraction into a proper fraction

\(\frac { 5 }{ 4 } \) = 1 \(\frac { 1 }{ 4 } \)

Step 4: Combining the equation

5 + 1 \(\frac { 1 }{ 4 } \) = 6 \(\frac { 1 }{ 4 } \)

Example for subtraction of mixed fractions

Solve the equation 3 \(\frac { 3 }{ 4 } \) – 2 \(\frac { 2 }{ 4 } \)?

Solution:

Step 1: In the first step, we subtract the whole numbers of both the fractions

3 – 2 =1

Step 2: In the next step, we subtract the fraction part of the mixed numbers

\(\frac { 3 }{ 4 } \)  – \(\frac { 2 }{ 4 } \)

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

Step 3: Now, that the result of the 2 parts are found, combine the equations

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

Unlike Mixed Fractions Addition and Subtraction

In unlike mixed fractions, the denominator value is different and it needs to be made the same by using the LCM method.

Adding Mixed Unlike Fractions

Solve the equation 3 \(\frac { 3 }{ 4 } \) + 2 \(\frac { 2 }{ 6 } \)?

Step 1: In the first step, add the whole numbers of the equation i.e.,

3 + 2 = 5

Step 2: As the denominators of both the equations are different, take the LCM of both the denominators.

Hence the LCM value of 4 and 6 is 12.

To get the same denominator value, we multiply 4 and 6 with the multiple that converts both into 12. Multiple the multiple to both numerator and denominator of the fraction.

For \(\frac { 3 }{ 4 } \), \(\frac { 3 }{ 3 } \) should be multiplied to get the denominator value as 12.

For \(\frac { 2 }{ 6 } \), \(\frac { 2 }{ 2 } \) should be multiplied to get the denominator value as 12.

Step 3: Now, that both the fractions have the same denominators

\(\frac { 9 }{ 12 } \) + \(\frac { 4 }{ 12 } \)

= \(\frac { 13 }{ 12 } \)

Step 4: Convert improper fraction to proper fraction

\(\frac { 13 }{ 12 } \) = 1 \(\frac { 1 }{ 12 } \)

Step 5: Now combine both the fractions

5 +1 \(\frac { 1 }{ 12 } \) = 6 \(\frac { 1 }{ 12 } \)

Subtracting unlike mixed fractions

Solve the equation 3 \(\frac { 3 }{ 4 } \) – 2 \(\frac { 2 }{ 6 } \)?

Solution:

Step 1: In the first step, add the whole numbers of the equation i.e.,

3 – 2 = 1

Step 2: As the denominators of both the equations are different, take the LCM of both the denominators.

Hence the LCM value of 4 and 6 is 12.

To get the same denominator value, we multiply 4 and 6 with the multiple that converts both into 12. Multiple the multiple to both numerator and denominator of the fraction.

For \(\frac { 3 }{ 4 } \), \(\frac { 3 }{ 3 } \) should be multiplied to get the denominator value as 12.

For \(\frac { 2 }{ 6 } \), \(\frac { 2 }{ 2 } \) should be multiplied to get the denominator value as 12.

Step 3: Now, that both the fractions have the same denominators

\(\frac { 9 }{ 12 } \) – \(\frac { 4 }{ 12 } \)

= \(\frac { 5 }{ 12 } \)

Step 4: Now combine both the fractions

1 + \(\frac { 5 }{ 12 } \) = 1 \(\frac { 5 }{ 12 } \)

Divisibility Rules or Tests are mentioned here to make the procedure simple and quick. Learning the Division Rules in Math helps you solve problems in an easy way.  Division Rules of Numbers 2, 3, 4, 5 can be understood easily. However, Divisibility Rules for 7, 11, 13 are a bit difficult to understand and refer to them in depth.

Solving Math Problems can be hectic for a few of us. At times, you need tricks and shortcuts to solve math problems faster and easier without lengthy calculations. Refer to the Solved Examples on Division Rules with Solutions to learn the approach for solving math problems by employing these basic rules.

Divisibility Test or Division Rules – Definition

From the name itself, we can understand that Divisibility Rules are used to test whether a number is divisible by another number or not without even performing the actual division operation. If a number is completely divisible by another number it will leave a remainder zero and quotient whole number.

However, not every number is exactly divisible and leaves a remainder other than zero. In such cases, these Division Rules will help you determine the actual divisor of a number by considering the digits of the number. Check out the Division Rules explained here in detail along with solved examples and learn the shortcuts to divide numbers easily.

List of Divisibility Rules for 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13

Below is the list of divisibility rules for numbers 1 to 13 explained clearly making it easy for you to do your divisions much simply. Understand the logic behind the Divisibility Test for 1 to 13 clearly and solve the math problems easily. They are as follows

Divisibility Rule of 1

Every Number is divisible by 1 and there is no prefined rule for that. Any number divided by 1 will give the number itself irrespective of how large the number is.

For Example, 4 is divisible by 1 and 3500 is also divisible by 1.

Divisibility Rule of 2

If a number is even or last digit of the number is even i.e. 0, 2, 4, 6, 8, 10… then the number is completely divisible by 2.

Example: 204

204 is an even number and thus it is divisible by 2. To check whether it divisible or not you can refer to the following process

• Check for the last digit of the number whether it is even or not
• Take the last digit 4 and divide with 2
• If the last digit 4 is divisible by 2 then the number 204 is also divisible by 2.

Divisibility Rule for 3

Divisibility Rule of 3 States that if the sum of digits of the number is divisible by 3 then the number is also divisible by 3.

Consider a number 507. to check 507 is divisible by 3 or not simply find the sum of digits i.e. (5+0+7) = 12. Check whether the sum of digits is divisible by 3 or not. If it is divisible by 3 or multiple of 3 then the number 507 is also divisible by 3.

Divisibility Rule of 4

If the last two digits of a number is a multiple of 4 then the number is exactly divisible by 4.

Example: 2124 here, last two digits are 24. since 24 is divisible by 4 the number 2124 is also exactly divisible by 4.

Divisibility Rule of 5

Numbers having 0 or 5 as the last digits are exactly divisible by 5.

Example: 20, 4560, 34570, etc.

Divisibility Rule of 6

Numbers that are divisible by both 2 and 3 are divisible by 6. It means the last digit of the given number is even and the sum of the digits is a multiple of 3 then the given number is a multiple of 6 and divisible by 6.

Example: 660

Here the last digit is 0 and is even and is divisible by 2

The Sum of digits is 6+6+0 = 12 is also divisible by 3

Thus, 660 is divisible by 6

Divisibility Rule for 7

Divisibility Rule of 7 can be a bit tedious to understand. Remove the last digit of the number and double it. Subtract the remaining number and check whether the number is zero or multiple of 7 then divide it with 7. Or else repeat the process again i.e. double the last digit of a number and then subtract from the remaining number.

Example: Is 1074 divisible by 7?

Since the last digit of the number is 4 double it and subtract from the remaining number.

107-8 = 99

Double the last digit number i.e. 9 and we get 18

Remaining Number 9-18 = -9

Since it is not divisible by 7 the given number 1074 is not divisible by 7.

Divisibility Rule for 8

If the last three digits of the number are divisible by 8 then the number is completely divisible by 8.

Example: 24328 In this last three digits are 328 since the last three digits are divisible by 8 then the given number is divisible by 8.

Divisibility Rule for 9

The Divisibility Rule of 9 is similar to the Divisibility Rule of 3. If the sum of digits is divisible by 9 then the number is completely divisible by 9.

Example: 88565 here sum of digits = 8+8+5+6+5 = 32. Since 32 is not divisible by 9 the number 88565 is not divisible by 9.

Divisibility Rule of 10

Divisibility Rule for 10 States that any Number whose last digit is 0 is divisible by 10.

Example: 100, 200, 330, 450, 670,….

Divisibility Rule of 11

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

To check whether a number 2134 is divisible by 11 go through the below process

• Group the alternative digits i.e. digits in odd places and digits in eve places together. Here 23 and 14 are two different groups.
• Find the sum of each group i.e. 2+3 = 5 and 1+4 =5
• Find the Difference of Sums i.e. 5-5 =0
• Here 2 is the difference and if the difference is divisible by 11 then the number is also divisible by 11. Here the difference is 0 and is divisible by 11 so the number 2134 is also divisible by 11.

Divisibility Rule of 12

Divisibility Rule of 12 states that if the number is divisible by both 3 and 4 then the number is exactly divisible by 12.

Example: 4654

The sum of digits 4+6+5+4 = 19(Not a Multiple of 3)

Last two digits = 54(Not divisible by 4)

Given number is neither divisible by 3 nor 4 so it is not divisible by 12

Divisibility Rule of 13

To check whether a given number is divisible by 13 or not simply add four times the last digit of the number to the remaining number and repeat the process until you get a two-digit number. Check if the two-digit number is divisible by 13 or not and if it is divisible then the number is exactly divisible by 13.

Example: 2045

Here the last digit is 5

Add four times the last digit of the number to the remaining number

= 204+4(5)

= 204+20

= 224

224 = 22+4(4)

= 22+16

= 38

Since 38 is not divisible by 13 the given number is not divisible by 13.

Solved Examples on Divisibility Rules

1. Check if 234 is divisible by 2?

Solution:

last digit = 4

4 is divisible by 2 so the given number is also divisible by 2.

2. Check if 164 is divisible by 4 or not?

Solution:

Last 2 digits = 64

Since the last two digits are divisible by 4 given number is also divisible by 4.

FAQs on Divisibility Rules

1. What is meant by Divisibility Rules?

Divisibility Rules are used to test whether a number is divisible by another number or not without even performing the actual division operation.

2. Write down the Divisibility Rule of 3?

If sum of the digits of the number is divisible by 3 then the given number is divisible by 3.

3. What is the divisibility rule of 10?

Any number whose last digit is 0 is divisible by 10.

Equation of a Line Parallel to Y-Axis: As we all aware of the infinite points in the coordinate plane so take an arbitrary point P(x,y) on the XY Plane and a line L. Now, finding the point that lies on the line is a very essential task for bringing an equation of straight lines into the picture in 2-D geometry.

In an equation of a straight line, terms involved in x and y. So, in case, the point P(x,y) meets the equation of the line, then the point P lies on the Line L. Now, you all will come to know about the Equation of a Line Parallel to Y-Axis, how to find it for the given point, and much more like different forms of equations of a straight line in the below modules.

Find the Equation of a Line Parallel to Y-Axis

Now, we will explain how to find the equation of Y-axis and the equation of a line parallel to Y-axis. By following this explanation, you will understand how easy to calculate and solve the equation of a straight line parallel to Y-axis. So, let’s start with the process of finding an Equation of a Line Parallel to the y-axis.

Let AB be a straight line parallel to the y-axis at some distance assume ‘a’ units from the Y-axis. From the below figure, it is clear that line L is parallel to y-axis and passing through the value ‘a’ on the x-axis. So, the equation of a line parallel to y-axis is X=a.

The equation of the y-axis is x = 0, as, the y-axis is a parallel to itself at a distance of 0 from it.

Or

If a straight line is parallel and to the left of the x-axis at a distance a, then its equation is x = -a.

Different Forms of Equations of a Straight Line

In addition to the equation of a line parallel to the y-axis, let’s have a glance at some various forms of the equation of a straight line. Here is the list of different forms of the equation of a straight line:

• Slope intercept form
• Point slope form
• Two-point form
• Intercept form
• Normal form
• Point-slope form

Worked-out Examples on Equation of y-axis and Equation of a line parallel to the y-axis

1. Write the equation of a line parallel to y-axis and passing through the point (−2,−4).

Solution:

As we know that the Equation of line parallel to y-axis is x=a.

The given point (−2,−4) lies on our required line, so that

⟹ x = -2

Therefore, the equation of the required line is x=−2. 

2. Calculate the equation of a straight line parallel to y-axis at a distance of 4 units on the left-hand side of the y-axis.

Solution: 

According to the statements that we know about the equation of a straight line is parallel and to the left of the x-axis at a distance a, then its equation is x = -a.

Hence, the equation of a straight line parallel to y-axis at a distance of 4 units on the left-hand side of the y-axis is x = -4, 

3. Find the equation of a line parallel to the y-axis and passing through the point (5,10)?

Solution:

A line parallel to the y-axis will be of form x=a

Given the line passes through (5,10)

So, x=5

Hence, The equation of a line is x – 5 = 0.

FAQs on Line Parallel to Y-Axis

1. What is the equation representing Y-axis?

The equation of a line which is representing the y-axis is x=0.

2. How to calculate the equation of a line?

Typically, the equation of a line is addressed as y=mx+b where m is the slope and b is the y-intercept.

3. What is the formula for point-slope form?

The formula for Point-Slope of the line by the definition is, m = \(\frac { y − y₁ }{ x − x₁ } \)

y − y₁ = m(x − x₁).

Odds and Probability: In mathematical concepts, we use odds and probability calculations in many ways like while solving the Playing Cards Probability and calculating the problems like the trains may be late, it may take an hour, to reach home and so forth. Here we will be discussing Odds & Probability Topic. The definitions for both are given in this article.

However, Probability is not similar to odds, as it describes the probability that the event will occur, upon the probability that the event will not occur. So, have a look at the difference between odds and probability provided below. Also, Go through the given solved examples based on Odds and Probability to learn the concept better.

What is the Definition of Odds?

The definition of Odds in the probability of a particular event is the ratio between the number of favorable outcomes of an event to the number of unfavorable outcomes.

In short, odds are defined as the probability that a particular event will occur or not. The range of Odds is from zero to infinity, if the odds is 0, the event is not likely to occur, but if it is ∞, then it is more likely to occur.

What is the Definition of Probability?

In mathematics, the probability is the likelihood of an event or more than one event happening. It denotes the chances of obtaining certain outcomes and can be calculated with the help of simple formula. Also, you can calculate the probability with multiple events by breaking down each probability into separate single considerations and then multiplying each output together to achieve a single likely result. Probabilities constantly range between 0 and 1.

In case, odds are declared as an A to B, the chance of winning then the winning probability can be P_W = A / (A + B) while the losing probability is P_L = B / (A + B).

Comparison Chart of Odds and Probability

Here is the table of comparison chart to learn about odds and probability basics:

Basis for Comparison	Odds	Probability
Meaning	Odds refers to the possibilities in favor of the event to the chances against it.	Probability refers to the likelihood of occurrence of an event.
Expressed in	Ratio	Percent or decimal
Lies between	0 to ∞	0 to 1
Formula	Occurrence/Non-occurrence	Occurrence/Whole

Key Difference Between Odds & Probabilities

The key difference between odds and probability are explained here in a simple manner to understand and learn the concepts easily and quickly:

• The term Odds is utilized to outline that if there are any possibilities of the occurrence of an event or not. Whereas the term ‘probability’ is defined as the possibility of the happening of an event, ie., how frequently the event will happen.
• Commonly, Odds range from zero to infinity, where zero represents the impossibility of happening of an event, and infinity signifies the possibility of the event. In contrast, probability lies between zero to one. Therefore, the closer the probability to zero, the more are the possibilities of its non-occurrence, and the closer it is to one, the higher are the possibilities of the event.
• Odds are the ratio of positive events to negative events. However, the probability can be measured by dividing the favorable event by the overall number of events.

Solved Examples on Odds & Probability

1. What is the difference between odds and probability?

Solution:

The difference between odds and probability is as illustrated below:

‘Odds’ of an event are the ratio of success to failure.

Hence, Odds = \(\frac { Success}{ Failures} \)

The ratio of the success to the amount of success and failures is known as the ‘Probability’ of an event.

Therefore, Probability = \(\frac { Success}{ (Success + Failures) } \)

2. A coin is thrown 3 times. What is the probability that at least one tail is taken?

Solution:

Let’s consider the sample space for a better understanding of the possibilities

Sample space = [HHH, HHT, HTH, THH, TTH, THT, HTT, TTT]

Total number of ways = 2 × 2 × 2 = 8.

Possible Cases for Tail = 7

P (A) = \(\frac { 7 }{ 8 } \)

OR

Probability (of getting at least one tail) = 1 – P (no tail)⇒ 1 – (\(\frac { 1 }{ 8 } \)) = \(\frac { 7 }{ 8 } \). 

FAQs on Odds Vs Probability

1. How do you convert odds to probability?

For converting odds to probability, we have to divide the odds by 1 + odds. For instance, let’s convert odds of 1/9 to a probability. Now, divide 1/9  by 10/9 to get the probability of 0.10.

2. What are the odds in favor?

The Odds in favor of an event equal to the number of favorable outcomes by the number of unfavorable outcomes.

P(A) = \(\frac { Number of favorable outcomes }{  Number of unfavorable outcomes } \)

3. What is the formula for odds against?

Odds against the probability formula is,

P(A) = \(\frac { Number of unfavorable outcomes }{ Number of favorable outcomes } \)

In the earlier classes of Math, you might have heard of the concept of Common Multiples. It is an important topic and is essential to find patterns in numbers. In this article, we will help you learn all about the Definition of Common Multiples, Facts, and Examples. The Solved Examples on Common Multiples make it easy for you to understand the entire concept quickly and easily.

What are Multiples?

Multiples are the results obtained by multiplying a number with an integer.  Multiples of Whole Numbers are obtained by considering the product of counting numbers and whole numbers.

For example, we can obtain the multiples of 5, 7 by multiplying them with the numbers 1, 2, 3, 4, 5, 6, …..

Multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40, 45……

Multiples of 7 are 7, 14, 21, 28, 35, 42, 49, 56…..

Common Multiples – Definition

Common Multiple is a whole number that is a shared multiple of two or more numbers. The Multiples that are common to two or more numbers are called the Common Multiples of those Particular Numbers.

Example:

List the Common Multiples of 8, 12

Multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96,…

Multiples of 12 are 12, 24, 36, 48, 60, 72, 84, 96, 108, 120,  132, 144….

Common Multiples of 8, 12 are 24, 72, 96

How to find Common Multiples of Two or More Numbers?

Go through the below listed simple and easy steps to find out the Common Multiples of Two or More Numbers. They are in the following way

• The first and foremost step is to assess your numbers.
• Make a list of multiples for the given numbers.
• Continue preparing the list until you find two or more common multiples.
• Then, Identify the common multiples from the multiples list prepared.

Solved Examples on Common Multiples

1. List out the Common Multiples of 5 and 15?

Solution:

Given Numbers are 5 and 15

Multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, ….

Multiples of 15 are 15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180,….

Common Multiples of 5 and 15 are 15, 30, 45, 60,75, 90

2. Find the Common Multiples of 6 and 9?

Solution:

Multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, 54, 60…..

Multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, 81, 90…..

Common Multiples of 6 and 9 are 18, 36, 54

Time is written in terms of Hours, Minutes, Seconds. Time can be converted from one unit to another with the help of conversion formulas. Among all Time conversions, Conversion of Seconds into Hours is the trickest one and highly seek help by students to solve their homework or assignments. To aid your preparation and homework, we have curated the details about seconds to hours conversion like definitions, conversion formula, the procedure to convert sec to hr, and some solved examples on second to hour conversion.

Go ahead and dig deep regarding Conversion of Seconds(sec) into Hours(hr) also look at the Math Conversion Chart to learn more about length, mass, capacity conversions along Time Conversions.

Definitions of Seconds & Hours

Seconds(sec)

A Second is a Unit of Time and its symbol is ‘s or sec’. As per the SI definition, the definition of second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the unperturbed ground state of the caesium 133 atoms at zero kelvins. According to the conversion base, 1 sec = 0.000277778 hr.

Hour(hr)

An Hour is a Unit of Time measurement, determined as 1/24th of a day or 60 Minutes. The Hour is the SI unit of time taken for the Metric System. The abbreviation of Hours is ‘hr’. As per the conversion base, 1 hr = 60 Minutes or 3600 Seconds.

Formula for Converting Second to Hour

Have a glance at the provided box and learn the mathematical formula which is used for converting seconds to hours:

Hours = Seconds ÷ 3,600

Second to Hour Conversion Table | Sec to Hr Conversion Chart

Here is the list of seconds converted to hours. Check out the conversion chart of sec to hrs and get some idea of How much is particular second in Hours.

Seconds	Hours	Hours, Minutes & Seconds
1	0.000278	0 hr 0 min 1 sec
100	0.027778	0 hr 1 min 40 sec
200	0.055556	0 hr 3 min 20 sec
300	0.083333	0 hr 5 min
400	0.111111	0 hr 6 min 40 sec
500	0.138889	0 hr 8 min 20 sec
600	0.166667	0 hr 10 min
700	0.194444	0 hr 11 min 40 sec
800	0.222222	0 hr 13 min 20 sec
900	0.25	0 hr 15 min
1,000	0.277778	0 hr 16 min 40 sec
2,000	0.555556	0 hr 33 min 20 sec
3,000	0.833333	0 hr 50 min
3,600	1	1 hr
4,000	1.1111	1 hr 6 min 40 sec
5,000	1.3889	1 hr 23 min 20 sec
6,000	1.6667	1 hr 40 min
7,000	1.9444	1 hr 56 min 40 sec
8,000	2.2222	2 hr 13 min 20 sec
9,000	2.5	2 hr 30 min
10,000	2.7778	2 hr 46 min 40 sec
20,000	5.5556	5 hr 33 min 20 sec
30,000	8.3333	8 hr 20 min
40,000	11.11	11 hr 6 min 40 sec
50,000	13.89	13 hr 53 min 20 sec
60,000	16.67	16 hr 40 min
70,000	19.44	19 hr 26 min 40 sec
80,000	22.22	22 hr 13 min 20 sec
86,400	24	24 hr
90,000	25	25 hr
100,000	27.78	27 hr 46 min 40 sec

Conversion of Seconds into Hours Process | How to Convert Seconds(sec) to Hours(hr)?

One of the easiest methods to convert seconds to hours is to divide the number of seconds by 3,600. In order to explain the reason for this conversion, first set up the conversion tables where first convert the number of seconds to minutes, and then the number of minutes to hours. These can be helpful to convert the sec to hr easily.

If not, you may also make use of the simple conversion formula to convert seconds to hours. The formula for second to hour unit of time conversion is given here.

We know 1 second is equal to 1/3600 hour

Hence, the Seconds to Hours Conversion Formula is given as

Hours = Seconds ÷ 3,600

Once, you have the formula for conversion between sec to hr just apply the given inputs in the formula and find out the hours within less time. For more learning knowledge, please go with the Conversion Of Seconds Into Hours solved examples explained by us and learn completely how to convert Sec to Hr easily within no time.

View Related Articles

• Conversion of Hours into Seconds
• Conversion of Minutes into Seconds
• Conversion of Minutes into Hours
• Conversion of Hours into Minutes

Worked-out Examples on Conversion of Seconds(sec) into Hours(hr)

1. Convert 1500 seconds into hours?

Solution:

First, we start converting seconds into minutes and then minutes into hours.

As we know the formula for conversion of sec into min, so apply it in step 1:

1500 seconds = (1500 ÷ 60) minutes (1 second = 1/60 minute)

= 25 minutes

Now, convert the minutes into hours by using the conversion formula:

= (25 ÷ 60) hours (1 minute = 1/60 hour)

= 0.416667 hours or 0 hr 25 min.

Therefore, 1500 seconds(sec) = 0 hr 25 min.

2. Convert 76 Seconds into Hours using conversion formula?

Solution:

We have 76 seconds and the conversion formula for Seconds(sec) to Hours(hr) conversion is as follows:

Hours = Seconds ÷ 3600

Now, apply the given time values in the formula and covert the sec to hr,

Hours = 76 ÷ 3600

= 0.021111 hr or 0 hr 1 min 16 sec

Hence, 76 seconds in hours is 0.021111 hr or 0 hr 1 min 16 sec. 

FAQs on Converting Seconds to Hours

1. What is the formula for converting Seconds to Hours?

The conversion formula for seconds to hours is Hours = Seconds ÷ 3600

2. How much is 15 seconds in hours? 

15 sec = 0.004167 hr or 15 sec = 0 hr 0 min 15 sec.

3. How to convert Sec to Hr?

As there are 3600 seconds in an hour to convert from Seconds to hours divide the time value by 3600.

Percentages topic is widely utilized by the people in various fields like shopping for deals, buying things at veggie or fruit markets, etc. Also, it is commonly used in accounting and finance scenarios like Profits, Interest Rates, Sales, and Taxation. Moreover, the percentage is helpful for grading the student’s annual marks. So, finding percentages can be tricky but an easy mathematical process. If you have to calculate the ratio or portion of a quantity then you need help with percentages. Hence, check out this article properly and learn what is percentage, how to find the percent of a given number or quantity along with worked-out examples.

What is Percentage?

A percentage is a number or ratio as a fraction of 100. In other words, the word percent indicated one part in a hundred. Always, the number of a percentage is represented by a percent symbol (%) or simply “percent”. Here is the percentage illustration:

5 %, 10 %, 33 \(\frac { 1 }{ 6} \) %, 75 %

For example, 60 percent (or 60%) means 60 out of 100.

However, the percentage is the outcome when a particular number is multiplied by a percent. So, learn how to calculate the percentages for a given number or quantity in the below modules with solved word problems.

How to Find the Percent of a Given Number?

To calculate the % of a given number so easily, please follow the below steps:

• Take the number, say x.
• Let the percent as p%.
• To find the formula is P% of x
• Now, write these as a proportion as \(\frac { P }{ 100 } \) = \(\frac { ? }{ x } \)
• Finally, do cross multiplication and calculate the value of the “?” mark.

Solved Examples on Percentages

1. What is 3 ⅓ of 60 km?

Solution :

Given expression is 3 ⅓ of 60 km

\(\frac { 3 ⅓ }{ 100 } \) = \(\frac { x }{ 60 } \)

Now, Convert mixed fraction to improper fraction

\(\frac { 10 }{ 3 } \)/100 = \(\frac { x }{ 60 } \)

\(\frac { 10 }{ 3 } \) x100 = \(\frac { x }{ 60 } \)

\(\frac { 10 }{ 300 } \) = \(\frac { x }{ 60 } \)

Cross multiply and find the x value

300x = 10 . 60

300x = 600

x = \(\frac { 600 }{ 300 } \)

x = 2 km.

2. Find 41% of 400.

Solution:

Given is 41% of 400

Now find the % of a given number

41% of 400 = 41 × \(\frac { 1 }{ 100 } \) × 400

= \(\frac { 41 }{ 100 } \) × 400

= \(\frac { 41 × 400 }{ 100 } \)

= \(\frac { 16400 }{ 100 } \) (Finally divide 16400 by 100 and get the result)

= 164.

3. What is the sum of the money of which 5 % of $750?

Solution:

Let the required sum of money be $m.

5 % of $m = $750

⇒ \(\frac { 5 }{ 100 } \) × m = 750

⇒ m = \(\frac { 750 × 100 }{ 5 } \)

⇒ m = 15000

Hence, sum of the money = $15000.

4. Find 17% of $4500?

Solution:

Given expression is 17% of $4500

Now, convert 17% into decimal form

Then write it as 0.17 x $4500

Multiply  17 × $4,500 = $76,500

Finally, keep the decimal point

Therefore the result for 17% of $4500 is $765.00

5. The price of a TV was reduced by 40% to $500. What was the original price?

Solution:

To find the original price,

First, determine the percentage of the actual price by subtracting 40% from 100.

Later, Product the final price by 100 ie., 500 x 100 = 50000.

Now, divide the result by the percentage computed in step 1 above.

Then, \(\frac { 50000 }{ 60 } \) = $ 833.33

The actual price of a TV is $ 833.33.

Ascending order is arranging or placing numbers from smallest to largest. In ascending order, the numbers are in increasing order. For example, 1, 2, 3, etc are in ascending order. The reverse process of ascending order is called descending order. The decreasing order is arranging or placing numbers from largest to smallest. Examples for decreasing order are 3, 2, 1. The ascending order is represented by the less than symbol ‘<‘ where descending order is represented by the greater than symbol ‘>‘.

Ascending Order – Definition & Symbol

Ascending order is the arrangement of numbers from the lowest to the highest. In the case of Ascending Order, the Smallest Number will be on the top of the list when sorted. To represent the order of numbers we use the symbol “<”

Examples of Ascending Order

• For numbers or amounts, the ascending order is 5, 8, 11, 18, 23, 31.
• For words and letters, the ascending order is A, B, C, D, E……Y, Z.
• Also, for dates, the ascending order will be from the oldest dates to recent dates.

Problems on Ascending Order

1. Arrange the below numbers in an Ascending Order

(i) 2, 14, 3, 59, 46
(ii) 25, 8, 97, 47, 3
(iii) 5, 6, 82, 31, 24
(iv) 6, 7, 35, 14, 4
(v) 24, 8, 15, 94, 119

Solution:
(i) Given numbers are 2, 14, 3, 59, 46.
Compare the values and write down the smallest number.
Write down the smallest number first, and then compare with all the remaining numbers with the same number of digits.
2, 3, 14, 46, 59.

The Ascending Order of the numbers is 2, 3, 14, 46, 59.

(ii) Given numbers are 25, 8, 97, 47, 3.
Compare the values and write down the smallest number.
Write down the smallest number first, and then compare with all the remaining numbers with the same number of digits.
3, 8, 25, 47, 97.

The Ascending Order of the numbers is 3, 8, 25, 47, 97.

(iii) Given numbers are 5, 6, 82, 31, 24.
Compare the values and write down the smallest number.
Write down the smallest number first, and then compare with all the remaining numbers with the same number of digits.
5, 6, 24, 31, 82.

The Ascending Order of the numbers is 5, 6, 24, 31, 82.

(iv) Given numbers are 6, 7, 35, 14, 4.
Compare the values and write down the smallest number.
Write down the smallest number first, and then compare with all the remaining numbers with the same number of digits.
4, 6, 7, 14, 35.

The Ascending Order of the numbers is 4, 6, 7, 14, 35.

(v) Given numbers are 24, 8, 15, 94, 119.
Compare the values and write down the smallest number.
Write down the smallest number first, and then compare with all the remaining numbers with the same number of digits.
8, 15, 24, 94, 119.

The Ascending Order of the numbers is 8, 15, 24, 94, 119.

Descending Order – Definition & Symbol

Descending Order is the arrangement of numbers from the highest to the lowest. Descending Order is the Contradictory of Ascending Order. Here Numbers are arranged from bigger to smaller. The number that is largest among the sorted list will fall at the top of the Descending Order List and the Smallest Number will be at the Last. It is denoted by the symbol  ‘>’.

Examples of Descending Order

• For numbers or amounts, the descending order is 31, 23, 18, 11, 8, 5.
• For words and letters, the descending order is Z, Y, X, W……B, A.
• Also, for dates, the descending order will be from the recent dates to the oldest dates.

Problems on Descending Order

1. Arrange the below numbers in an Descending Order

(i) 2, 14, 3, 59, 46
(ii) 25, 8, 97, 47, 3
(iii) 5, 6, 82, 31, 24
(iv) 6, 7, 35, 14, 4
(v) 24, 8, 15, 94, 119

Solution:
(i) Given numbers are 2, 14, 3, 59, 46.
Compare the values and write down the highest number.
Write down the highest number first, and then compare with all the remaining numbers with the same number of digits.
59, 46, 14, 3, 2.

The Descending Order of the numbers is 59, 46, 14, 3, 2.

(ii) Given numbers are 25, 8, 97, 47, 3.
Compare the values and write down the highest number.
Write down the highest number first, and then compare with all the remaining numbers with the same number of digits.
97, 47, 25, 8, 3.

The Descending Order of the numbers is 97, 47, 25, 8, 3.

(iii) Given numbers are 5, 6, 82, 31, 24.
Compare the values and write down the highest number.
Write down the highest number first, and then compare with all the remaining numbers with the same number of digits.
82, 31, 24, 6, 5.

The Descending Order of the numbers is 82, 31, 24, 6, 5.

(iv) Given numbers are 6, 7, 35, 14, 4.
Compare the values and write down the highest number.
Write down the highest number first, and then compare with all the remaining numbers with the same number of digits.
35, 14, 7, 6, 4.

The Descending Order of the numbers is 35, 14, 7, 6, 4.

(v) Given numbers are 24, 8, 15, 94, 119.
Compare the values and write down the highest number.
Write down the highest number first, and then compare with all the remaining numbers with the same number of digits.
119, 94, 24, 15, 8.

The Descending Order of the numbers is 119, 94, 24, 15, 8.

FAQs on Ascending or Descending Order

1. What does Ascending Order mean?

When the Numbers are arranged in increasing order i.e. from smallest to largest then they are said to be in Ascending Order.

2. What does Descending Order mean?

Descending Order or Decreasing Order is the way of arranging numbers from biggest to smallest.

3. What are the signs used to represent Ascending Order and Descending Order?

Ascending order is denoted by the ‘<‘ (less than) symbol, whereas descending order is denoted using the ‘>’ (greater than) symbol.

Learn Regular and Irregular Polygon Concepts and properties along with examples. Know the difference between Regular Polygons and Irregular Polygons and find out the given shape is a regular or irregular polygon. Check out the step-by-step process to find the given shape is a polygon or not? Also, we have given clear details about Regular and Irregular Polygons. Go through the Solved Examples on Regular and Irregular Polygons and learn how to solve problems on them quickly and easily.

Regular Polygon – Definition

A polygon that consists of equal sides with equal length and also by having equal angles called a regular polygon.

Examples of a Regular Polygon

Let us have a look at the different Examples of a Regular Polygon below.

Equilateral Triangle:
An equilateral triangle is a triangle in which all three sides have the same length and all angles are equal.
In the below figure of an equilateral triangle PQR there are three sides i.e., PQ, QR, and RP are equal and there are three angles i.e., ∠PQR, ∠QRP, and ∠RPQ are equal.
Therefore, an equilateral triangle is a regular polygon.

Square:
A Square is a quadrilateral that has four equal sides and four equal angles.
In the below figure of a square PQRS there are four sides i.e., PQ, QR, RS and SP are equal and there are four angles i.e., ∠PQR, ∠QRS, ∠RSP, and ∠SPQ are equal.

Therefore, a square is a regular polygon.

Regular Pentagon:
A polygon is regular when all angles are equal and all sides are equal.
In the below figure of a regular pentagon PQRST there are five sides i.e., PQ, QR, RS, ST, and TP are equal and there are five angles i.e., ∠PQR, ∠QRS, ∠RST, ∠STP, and ∠TPQ are equal.

Therefore, a regular pentagon is a regular polygon.

Irregular Polygon – Definition

A polygon that consists of unequal sides with unequal lengths and also having unequal angles called an irregular polygon.

Examples of Irregular Polygon

Let us have a look at the different Examples of a Regular Polygon below.

Scalene Triangle:
A scalene triangle is a triangle that has all three sides are in different lengths, and all three angles are of different measures.
In the adjoining figure of a scalene triangle PQR there are three sides i.e., PQ, QR, and RS are unequal and there are three angles i.e., ∠PQR, ∠RSP, and ∠SPQ are unequal.

Therefore, a scalene triangle is an irregular polygon.

Rectangle:
A Rectangle is a four sided-polygon, having all the internal angles equal to 90 degrees. Each angle is at right angles. The opposite sides of the rectangle are equal in length which makes it different from a square.
In the adjoining figure of a rectangle PQRS there are four sides i.e., PQ, QR, RS, and SP where the opposite sides are equal i.e., PQ = RS, and QR = PS. So, all the sides are not equal to each other.
Similarly, among the four angles i.e., ∠PQR, ∠QRS, ∠RSP, and ∠SPQ where the opposite angles are equal i.e., ∠PQR = ∠RSP and ∠QRS = ∠SPQ. So, all the angles are not equal to each other.

Therefore, a rectangle is an irregular polygon.

Irregular Hexagon:
An irregular hexagon is a six-sided shape whose sides are not equal.
In the adjoining figure of an irregular hexagon PQRSTU there are six sides i.e., PQ, QR, RS, ST, TU, and UP are equal and there are six angles i.e., ∠PQR, ∠QRS, ∠RST, ∠STU, ∠TUP, and ∠UPQ are equal.

Therefore, an irregular hexagon is an irregular polygon.

Rounding Decimals to the Nearest Whole Number guide is available here. Check the steps to find out the process of rounding the decimals worksheet. Make use of number lines and visual models to round decimal numbers. Know the definition, rules, and methods to round the decimal values. Follow the below sections to gather the information regarding rounding decimal values to the nearest whole numbers. Also, check the solved problems and solution procedure.

Rounding Decimals to the Nearest Whole Number – Definition

The rounding of decimal values is a process to estimate a particular number in context. The rounding of decimals refers to the gaining of accuracy to a certain degree. We can easily round the decimals to the nearest whole numbers i.e., tenths or hundreds. With the help of rounding decimals to the nearest whole numbers, you can easily estimate the solution quickly and easily. It is also used to get the average score of the pupil in the class. There are various procedures to round off the whole numbers.

Basic Rules of Rounding to Whole Numbers

While rounding to the nearest whole numbers, you have to follow a few basic rules.

• Identify the value in the place you are rounding to. The smaller value you consider at the place value, the most accurate will be the final result.
• Look at the next smallest number in place value is the number to the right of the place value which you are rounding to. For suppose, if you are looking to round to the nearest ten, then you would look at the one’s place.
• If the number in the succeeding smallest place value is less than five (0, 1, 2, 3, 4 or 5), you have to leave the digit you want to round as a whole number to as in. Any digits after the tens place value (include the smallest place value you looked at) drop-off it or become zeros which are located after the decimal point. It is called rounding down.
• If the number in the succeeding smallest place value is greater than or equal to five (5, 6, 7, 8, or 9), you have to increase the digit value you are rounding to +1 (one). As seen in the previous step, the remaining digits before the decimal point become zero, and any values after the decimal point are dropped. It is called rounding up.

Also, Read: Round off to Nearest 100

How to Round to the Nearest Whole Number?

There are a few steps to round numbers to the nearest whole number.

Step 1: First, look at the number that you want to round.

Step 2: As you have to round the number to the nearest whole number, we mark the number in the one’s place.

Step 3: Now look at the digit which is at right to the decimal point, i.e., tenth’s place

Step 4: (i) If the number in the tenth column has 0,1,2,3,4, we have to round down the digit at the ones place to the nearest whole number.

(ii) If the number in the tenths columns has 5,6,7,8 or 9, we have to round up the digit at one’s place to the nearest whole number.

Step 5: You have to remove all the numbers after the decimal point. The desired answer will be the left-out number.

Rounding to Nearest Tenths

We have to follow the given steps in rounding numbers to the nearest tenths.

Step 1: First, look at the number that you want to round.

Step 2: As you have to round the number to the nearest tenths, we mark the number in the tenths place.

Step 3: Now look at the digit which is at right to the tenth’s point, i.e., hundredth’s place

Step 4: (i) If the number in the hundredth’s column has 0,1,2,3,4, we have to round down the digit at the tenths place to the nearest tenths.

(ii) If the number in the hundredth’s column has 5,6,7,8 or 9, we have to round up the digit at the tenth’s place to the nearest tenths.

Step 5: You have to remove all the numbers after the tenths column. The desired answer will be the left out number.

Rounding to Nearest Hundredths

We have to follow the given steps in rounding numbers to the nearest hundredths

Step 1: First, look at the number that you want to round.

Step 2: As you have to round the number to the nearest hundredths, we mark the number in the hundredth’s place.

Step 3: Now look at the digit which is at right to the hundredth’s point, i.e., thousandth’s place

Step 4: (i) If the number in the thousandth’s column has 0,1,2,3,4, we have to round down the digit at the hundredth’s place to the nearest hundredth’s.

(ii) If the number in the thousandth’s column has 5,6,7,8 or 9, we have to round up the digit at the hundredth’s place to the nearest hundredth’s.

Step 5: You have to remove all the numbers after the hundredth’s column. The desired answer will be the left out number.

How to Round Decimals Using a Calculator?

Follow the below procedure to round decimals to the whole numbers.

Step 1: Enter the decimal or the number in the given input field.

Step 2: Find the button “round” and click on it to get the final result.

Step 3: You will notice the value that is displayed in the output field.

Examples on Rounding Numbers to Nearest Whole Number

Problem 1:

Consider the number 46.8. Round the digit to the nearest whole number?

Solution:

As the given number = 46.8

The number in the tenths place = 8 which is greater than 5

As the value is greater than 5, then the whole part of the number increases with 1 and the number to the right of the decimal value means the tenth’s place becomes zero.

Hence, the final value of the rounded-up = 47

Problem 2:

Consider another number 45.379. Round the digit to the nearest whole number?

Solution:

As the given number is 45.379

The ones digit in the number = 5 and tenths digit is 3

Since the value in the tenths digit is 3, then the one’s digit will remain unchanged, i.e., 45.379

Rewrite the digit by dropping the decimal point and all the digits after it.

Therefore, 45 is the final answer.

Problem 3:

Round the following digits to the nearest tenths: 0.437

Solution:

As the given number is 0.437

The hundredth’s digit is 3 and therefore the tenth digit will remain unchanged

As the hundredth’s digit is less than 5, we have to drop off the remaining digits.

The final answer is 0.4

Rounding is making a number simple but keeping its value nearest to what it was. The result of round-off is less accurate and easier to use especially while doing arithmetical operations. Likewise, rounding a number to the nearest hundred means making the units and tens place zeros and either increasing or decreasing the remaining part of the number by 1. Students can get the solved examples questions, definitions, and round-off rules in the below-mentioned sections of this page.

Round off to Nearest 100

Round off to the Nearest 100 is a process where you need to convert the given number into an easy form for various reasons. The obtained easy number is not the actual value but is an approximate value of the original number. Rounding to the nearest hundred means, you need to convert the given number to the nearest 100.

The purpose of rounding the numbers is it makes the numbers easier to understand and remember, the calculations become easier. The application of rounding is when you want to estimate an answer or try to find the most sensible guess, then rounding is used.

Rules for Rounding Numbers to Nearest 100

• Rule 1: While round off to nearest 100, if the digit in the tens place is between 0 to 4 or <5, then the tens place is replaced by 0.
• Rule 2: If the digit in the unit’s place is equal to 5 or greater than 5, then the tens place is replaced by 0 and the hundreds place is increased by 1.

How to Round Numbers to Nearest Hundred?

Get the detailed steps to rounding to the nearest 100. They are along the lines

• Get the number we want to round.
• Identify the digit in the tens place.
• If the digit in the tens place is less than 5, then place zero’s in the tens, units place of the number.
• If the digit in the tens place is more than or equal to 5, then place zero’s in the tens and units place, increase the hundredth place digit by 1.
• Now, write the obtained number.

Also, Read: Rounding Decimals to Nearest Whole Number

Examples on Rounding Numbers to Nearest 100

Example 1:

Round the following numbers to the nearest 100.

(i) 148

(ii) 5520

(iii) 95

Solution:

(i) 148

Given number is 148

We see the digit in the tens place is 4 means that is less than 5. So, we round to the nearest multiple of a hundred which is less than the number. Keep zeros in the units and tens place.

Therefore, rounding of 148 to the nearest 100 is 100.

(ii) 5520

Given number is 5520

We can see that the digit in tens place is 2 which is less than 2. So, keep zero’s in the tens, units place, and write the remaining digits as it is.

Therefore, rounding of 5520 to the nearest 100 is 5500.

(iii) 95

Given number is 95

The tens digit of the given number is 9 that is more than 5. So, we need to place zeros in the tens, units place and increase the hundreds digit by 1. So, the obtained number is 100.

Example 2:

Round the following numbers to the nearest hundred.

(i) 696

(ii) 1,00,678

(iii) 12,05,896

Solution:

(i)696

Given number is 696

The digit in the tens place of the original number is 9. So, increase the hundreds digit by 1 and place zeros in the units, tens place.

Therefore, the obtained number is 700.

(ii) 1,00,678

Given number is 1,00,678

We can identify that the digit in the tens place of the number is 7 i.e > 5. Therefore, increase the digit in the hundreds place of the number by 1 and put zeros in the units, tens places.

Hence, the rounding off 1,00,678 to the nearest hundred is 1,00,700.

(iii) 12,05,896

Given number is 12,05,896

We see the digit in the tens place is 8, we round to the nearest multiple of hundred which is greater than the number. Hence, 12,05,896 is nearer to 12,05,900 than 12,05,800.

Example 3:

Round of To Nearest 100.

(i) 50

(ii) 255

(iii) 510

Solution:

(i) 50

Given number is 50

The digit in the tens position of the given number is 5. So, increase the hundreds position of the number by 1, place zeros in the tens, units place.

So, the round-off 50 to the nearest 100 is 100.

(ii) 255

Given number is 255

We choose the two multiple of 100 just greater than and just less than 255.

The nearest hundreds of 255 are 200, 300

255 – 200 = 55

300 – 255 = 45

As 300 has the lowest difference value.

So, 300 is the nearest 100 for 255.

(iii) 510

Given number is 510

We see the digit in the tens place is 1, we round to the nearest multiple of hundred which is less than the number. Hence, 510 is nearer to 500 than 600.

Angles are formed when two lines intersect each other at a point. The pair of angles are nothing but two angles. The angle pairs can relate to each other in various ways. Those are complementary angles, supplementary angles, vertical angles, adjacent angles, alternate interior angles, alternate exterior angles, and corresponding angles.

Pairs of Angles – Definition

The region between two infinitely long lines pointing a certain direction from a common vertex is called an angle. Which is the amount of turn is measured by an angle. The pairs of angles mean two angles. If there is one common line for two angles, it is known as angle pairs. Get the definitions and examples of all pairs of angles in the following section.

1. Complementary Angles

Two angles whose sum is 90° are called complementary angles and one angle is the complement of another angle.

Here, ∠AOB = 20°, ∠BOC = 70°

So, ∠AOB + ∠BOC = 20° + 70° = 90°

Therefore, ∠AOB and ∠BOC are called complementary angles.

∠AOB is a complement of ∠BOC and ∠BOC is the complement of ∠AOB.

Example:

(i) Angles of measure 50° and 40° are complementary angles because 50° + 40° = 90°.

Thus, the complementary angle of 50° is the angle measure 40°. The complementary angle of 40° is the angle measure 50°.

(ii) Complementary of 60° is 90° – 60° = 30°

(iii) Complementary of 45° is 90° – 45°= 45°

(vi) Complementary of 25° is 90 – 25° = 65°

Working rule: To find the complementary angle of a given angle subtracts the measure of an angle from 90°.

So, the complementary angle = 90° – given angle

Also, Read:

• Complementary and Supplementary Angles
• Linear Pair of Angles
• Classification of Angles

2. Supplementary Angles

The pair of angles whose sum is 180° is called the supplementary angles and one angle is called the supplement of the other angle.

Here, ∠AOC = 120°, ∠COB = 60°

∠AOC + ∠COB = 120° + 60° = 180°

Therefore, ∠AOC, ∠COB are called supplementary angles.

∠AOC is the supplement of ∠COB, ∠COB is a supplement of ∠AOC.

Example:

(i) Angles of measure 90° and 90° are supplementary angles because 90°+ 90° = 180°

Thus, the supplementary angle of 90° is the angle of measure 90°.

(ii) Supplement of 100° is 180° – 100° = 80°

(iii) Supplement of 50° is 180° – 50° = 130°

(iv) Supplement of 95° is 180° – 95°= 85°

(v) Supplement of 140° is 180°- 140° = 40°

Working rule: To find the supplementary angle of the given angle, subtract the measure of angle from 180°.

So, the supplementary angle = 180° – given angle

3. Adjacent Angles

Two non-overlapping angles are said to be adjacent angles if they have a common vertex, common arm, and other two arms lying on the opposite side of this common arm so that their interiors do not overlap.

In the above figure, ∠DBC and ∠CBA are non-overlapping, have BC as the common arm and B as the common vertex. The other arms BD, AB of the angles ∠DBC and ∠CBA are opposite sides of the common arm BC.

Hence, the arm ∠DBC and ∠CBA form a pair of adjacent angles.

4. Linear Pair of Angles

The angles are called liner pairs of angles when they are adjacent to each other after the intersection of two lines. Two adjacent angles are said to form a linear pair if their sum is 180°. The types of linear pairs of angles are alternate exterior angles, alternate interior angles, and corresponding angles.

Alternate interior angles

Two angles in the interior of the parallel lines and on opposite sides of the transversal. Alternate interior angles are non-adjacent and congruent.

Alternate exterior angles

Two angles in the exterior of the parallel lines, and on the opposite sides of the transversal. Alternate exterior angles are non-adjacent and congruent.

Corresponding angles

The pair of angles, one in the interior and another in the exterior that is on the same side of the transversal. Corresponding angles are non-adjacent and congruent.

5. Vertical Angles

Two angles formed by two intersecting lines having no common arm are called the vertically opposite angles.

When two lines intersect, then vertically opposite angles are always equal.

∠1 = ∠3

∠2 = ∠4

Pair of Angles Examples

Example 1:

Suppose two angles ∠AOC and ∠ BOC form a linear pair at point O in a line segment AB. If the difference between the two angles is 40°. Then find both the angles.

Solution:

Given that,

∠AOC and ∠BOC form a linear pair

so, ∠AOC + ∠BOC = 180° —- (i)

∠AOC – ∠BOC = 40° —- (ii)

Add both equations

∠AOC + ∠BOC + ∠AOC – ∠BOC = 180° + 40°

2∠AOC = 220°

∠AOC = 220° / 2

∠AOC = 110°

Now, substitute ∠AOC in (i)

110° + ∠ BOC = 180°

∠BOC = 180° – 110°

∠BOC = 70°

Therefore, two angles are 70°, 110°.

Example 2:

Find the values of the angles x, y, and z in the following figure.

Solution:

From the given figure,

lines AD and EC intersect each other and ∠DOC and ∠AOE are vertically opposite angles

When two lines intersect, then vertically opposite angles are always equal.

So, ∠DOC = ∠AOE

Therefore, z = 40°

AD is a line

∠DOE and ∠AOE are adjacent angles. The sum of adjacent angles are 180°

So, ∠DOE + ∠AOE = 180°

y + 40° = 180°

y = 180°- 40°

y = 140°

And, lines AD and CE intersect

∠DOE, ∠COA are vertically opposite.

When two lines intersect, then vertically opposite angles are always equal.

So, ∠DOE = ∠COA

y = ∠COB + ∠BOAA

140° = x + 25°

140° – 25° = x

x = 115°

Hence, x = 115°, y = 140°, z = 40°

Example 3:

Identify the five pairs of adjacent angles in the following figure.

Solution:

Adjacent angles are the angles that have a common side, vertex, and no overlap.

So, (i) ∠AOD, ∠AOE are the adjacent angles

The common side is AO, the common vertex is O and OE, OD is not overlapping.

(ii) ∠AOD, ∠DOB is the adjacent angles

The common vertex is O, the common side is OD and OA, OB is not overlapping.

(iii) ∠DOB, ∠BOC is the adjacent angles

The common side is OB, the common vertex is O, and OD, OC are not overlapping.

(iv) ∠COE, ∠BOC are adjacent angles

The common side is OC, the common vertex is O, and OE, OB are not overlapping.

(v) ∠COE, ∠EOA are adjacent angles

The common vertex is O, the common side is OE, and OC, OA is not overlapping.