Vijaya Sree

A quadratic equation has two solutions that may be or may not be distinct. The result may be real numbers or imaginary numbers. Learn the important formulas of quadratic equation, definition here. Let us learn about the introduction to the quadratic equations from this article. You can find examples of quadratic equations with step-by-step explanations.

What is a Quadratic Equation?

In the name, quadratic “quad” means square because the equation is square. A quadratic equation is an algebraic expression of the 2nd degree in variable x. The variable x has two answers real or complex numbers. The answers or solutions of x are called roots of the quadratic equations. They are specified as (α, β). The standard form of the quadratic equation is ax² + bx + c = 0. Where a, b is the coefficient of x² and c is the constant. a,b, c are not fractions nor decimals.

Quadratic Equation Formula

The formula for the quadratic equation is an easy method to find the roots of the equation. Without the formulas, the values are not factorized, and can find the roots in the easiest way. The roots of Q.E helps to find the sum of the roots and product of the roots of the quadratic equation.
Quadratic Equation (α, β) = [-b ± √(b² – 4ac)]/2a.

Important Formulas to Solve Quadratic Equations

• The standard form of the quadratic equation is ax² + bx + c = 0.
• The discriminant(D) of quadratic equation is D = b² – 4ac.
• For the case, D = 0 the roots are real and equal.
• For the case, D > 0 the roots are real and distinct.
• For the case, D < 0 the roots do not exist, or the roots are complex.
• The product of the Root of the quadratic equation is αβ = c/a = Constant term/ Coefficient of x²
• The roots of the quadratic equation is x = [-b ± √(b² – 4ac)]/2a.
• The sum of the roots of a Q.E is α + β = -b/a = – Coefficient of x/ Coefficient of x²
• Quadratic equation in the form of roots is x² – (α + β)x + (αβ) = 0
• If α, β, γ are roots of a cubic equation ax³ + bx² + cx + d = 0, then, α + β + γ = -b/a, αβ + βγ + λα = c/a, and αβγ = -d/a
• The roots (α + iβ), (α – iβ) are the conjugate pair of each other.
• For a > 0, the quadratic expression f(x) = ax² + bx + c has a minimum value at x = -b/2a
• For a < 0, the quadratic expression f(x) = ax² + bx + c has a maximum value at x = -b/2a
• For a > 0, the range of the quadratic equation ax² + bx + c = 0 is [b² – 4ac/4a, ∞).
• For a < 0, the range of the quadratic equation ax² + bx + c = 0 is (∞, -(b² – 4ac)/4a]

Methods for Solving Quadratic Equations

There are three methods for solving quadratic equations. They are as follows,
1. Factorization method
2. Completing the square method
3. Quadratic Equation formula

Quadratic Equation Question and Answers

Example 1.
Solve 5x² + 7x + 2 = 0
Solution:
Coefficients are: a = 5, b = 7, c = 2
x = [-b ± √(b² – 4ac)]/2a
x = [-7 ± √(7² – 4.5.2)]/2.5
x = [-7 ± √(49 – 40)]/10
x = [-7 ± √(9)]/10
x = [-7+3]/10 = -4/10 = -2/5
x = [-7 – 3]/10 = -10/10 = -1
Thus x = -2/5 or x = -1

Example 2.
Find the range of k for which 4 lies between the roots of the quadratic equation x² + 2(k – 4)x + 5 = 0.
Solution:
6 will lie between the roots of the quadratic expression f(x) = x² + 2(k – 4)x + 5 if,
f(4) < 0
= 16 + 2(k – 4)4 + 5 < 0
= 16 + (2k – 8)4 + 5 < 0
= 16 + 8k – 32 + 5 < 0
= 8k – 11 < 0
= k < 11/8

Example 3.
Find the factors of the quadratic equation x² + 7x + 12 = 0
Solution:
x² + 7x + 12 = 0
x² + 3x + 4x + 12 = 0
x(x + 3) + 4(x + 3) = 0
(x + 4) (x + 3) = 0
x + 4 = 0 or x + 3 = 0
x = -4 or x = -3

FAQs on Quadratic Equation

1. What is the purpose of quadratic equations?

Quadratic equations are actually used in our daily life, as when calculating areas, determining a product’s profit or formulating the speed of an object.

2. What is the standard form of the quadratic equation?

The standard form of the quadratic equation is ax² + bx + c = 0

3. How many roots does a quadratic equation have?

The quadratic equation has two roots. The Q.E with real or complex coefficients has two solutions that are called roots.

Most of us understand the value of planning and preparation. The calendar plays an important role in planning a specific day. From a student’s point of view, it is important to remember the exam dates, friends’ birthdays, special dates, etc. Learn how to make use of Calendar guides from here. Students of 4th grade can know the importance of a calendar with the help of this article.

A calendar is a figure that shows days and months. The calendar is one of the most important topics for fixing the exam dates for all government entrance exams. The calendar shows days, months, and years. Let us see how the calendar guides us to know about the special dates, even and odd months, leap and non-leap years, etc.

Importance of Calendar | Benefits of Using a Calendar

i. Total number of Sundays in a month.
ii. Number of days in a month.
iii. Number of months in a year.
iv. To remember a special day or definite day.
v. Different number of days in a month like January, February, April, etc.
v. We can know what is a leap year is and which is not a leap year.
vi. It helps to fix registration dates, competitions, etc.
vii. You can schedule your work whether it is urgent or important tasks with the help of a calendar.
viii. Not urgent and important can be scheduled and added to ‘to-do list’ and urgent and unimportant tasks can be delayed with the help of the calendar.

Wanna become a master in maths? If yes, then learn the fundamentals of maths at the primary level itself. Get the worksheets, practice tests, examples, word problems on 4th Grade Math and get guidance.

Examples on Purpose of a Calendar

Example 1.
How many days are there in February in a leap year?
Solution:
There are 29 days in February in a leap year.

Example 2.
Which is the first day of the year in 2020?
Solution:
The first day of the year 2020 is Wednesday, January 1st.

Example 3.
How many days are there in the month of March?
Solution:
There are 31 days in the month of March.

Example 4.
How many weeks are there in a year?
Solution:
There are 52 weeks in a year.

Example 5.
How many Fridays are there in the year 2021?
Solution:
There are 53 Fridays in the year 2021.

Reciprocal of a Fraction – The reciprocal of a fraction is nothing interchanging or switching the numerator and denominator. That means numerator becomes denominator and denominator becomes the numerator. In the case of a mixed fraction, you have to convert the mixed fraction to the improper fraction and then switch the numerator and denominator (top number to the bottom number). Learn how to find the reciprocal of a fraction with the help of the below examples.

Example: Suppose the fraction is a/b then the reciprocal of the fraction is b/a. Here b becomes numerator and a becomes denominator.

Do Refer:

• Reducing the Equivalent Fractions
• Simplification of Fractions
• Conversion of Fractions into Fractions having Same Denominator

How to find the Reciprocal of a Fraction?

Go through the simple process listed below to determine the reciprocal of a fraction. They are as follows

• Initially, determine the numerator and denominator of a given fraction.
• Fractions Reciprocal can be obtained by swapping or interchanging the numerator and denominators.
• In the case of Mixed Fraction, you first need to change to improper fractions and then interchange the numerator and denominator of the improper fraction.

Reciprocal of a Fraction Examples

Example 1.
What is the opposite reciprocal of \(\frac{5}{6}\)?
Solution:
The opposite reciprocal of the fraction is nothing but changing the sign of the number. A positive number becomes a negative number.
So, the opposite reciprocal of the fraction \(\frac{5}{6}\) is –\(\frac{6}{5}\)

Example 2.
Find the reciprocal of the fraction \(\frac{2}{1}\)
Solution:
Given the fraction \(\frac{2}{1}\)
The reciprocal of a fraction is nothing interchanging or switching the numerator and denominator.
Thus the reciprocal of the fraction \(\frac{2}{1}\) is \(\frac{1}{2}\)

Example 3.
Find the reciprocal of the fraction \(\frac{17}{58}\)
Solution:
Given the fraction \(\frac{17}{58}\)
The reciprocal of a fraction is nothing interchanging or switching the numerator and denominator.
Thus the reciprocal of the fraction \(\frac{17}{58}\) is \(\frac{58}{17}\)

Example 4.
Find the reciprocal of the fraction \(\frac{16}{64}\)
Solution:
Given the fraction \(\frac{16}{64}\)
The reciprocal of a fraction is nothing interchanging or switching the numerator and denominator.
Thus the reciprocal of the fraction \(\frac{16}{64}\) is \(\frac{64}{16}\) or \(\frac{4}{1}\)

Example 5.
Find the reciprocal of the fraction \(\frac{2}{3}\)
Solution:
Given the fraction \(\frac{2}{3}\)
The reciprocal of a fraction is nothing interchanging or switching the numerator and denominator.
Thus the reciprocal of the fraction \(\frac{2}{3}\) is \(\frac{3}{2}\)

Example 6.
Find the negative reciprocal of the fraction \(\frac{7}{129}\)
Solution:
Given the fraction \(\frac{7}{129}\)
The opposite reciprocal of the fraction is nothing but changing the sign of the number. A positive number becomes a negative number.
Therefore the negative reciprocal of the fraction \(\frac{7}{129}\) is –\(\frac{129}{7}\)

Example 7.
Find the negative reciprocal of the fraction \(\frac{3}{5}\)
Solution:
Given the fraction \(\frac{3}{5}\)
The opposite reciprocal of the fraction is nothing but changing the sign of the number. A positive number becomes a negative number.
Therefore the negative reciprocal of the fraction \(\frac{3}{5}\) is –\(\frac{5}{3}\)

Example 8.
Write the reciprocal of the fraction \(\frac{3}{9}\)
Solution:
Given the fraction \(\frac{3}{9}\)
The reciprocal of a fraction is nothing interchanging or switching the numerator and denominator.
Therefore the reciprocal of the fraction \(\frac{3}{9}\) is \(\frac{9}{3}\) or \(\frac{3}{1}\) or 3.

FAQs on Reciprocal of a Fraction

1. What is a reciprocal of the fraction?

The reciprocal of a fraction will be obtained by interchanging the numerator and denominator.

2. Is 1 the reciprocal of 1?

Yes, 1 is the reciprocal of 1 itself. Since 1 can be written as \(\frac{1}{1}\)

3. What is the reciprocal of \(\frac{1}{3}\) as a fraction?

The reciprocal of the given fraction is \(\frac{3}{1}\) which means 3.

For Estimating Sums and Differences we use the concept of Rounding Off Numbers. Estimation is nothing but taking the values that is closer to the exact answer. Estimating Sums and Differences means writing answers that are approximately equal to the exact answer. Estimating the Values helps your child to improve mental math. Refer to the Solved Examples on Estimating the Sums and Differences explained step by step in the later modules.

Do Read:

• Estimating a Sum
• Estimating Products
• Estimating the Quotient

How to Estimate the Sums and Differences of Whole Numbers?

Estimation means finding the answer closer to the accurate solution. The concept which is used for estimating addition and subtraction is round-off numbers. We can round the number nearest to ten, hundred, thousand, etc to estimate the answer. Bullet points to keep in mind is

• If the number is less than 5, round down (means 0)
• If the number is greater than 5, round up (means 1)

Advantages of Estimating Sum and Difference

There are many benefits of estimating sums and differences. Some of the advantages are shown below.

• Estimating Addition and subtraction helps to improve mental math.
• Your fluency in calculation will be improved.
• You can understand the concept of rounding off numbers in the number system by learning the concept of estimation.

Estimating Sums and Differences Examples

Example 1.
Estimate the sum of 79, 89, 58.
Solution: 
9 is greater than 5, so you can add 1 to the tens place value and 0 to the unit place value.
The number 79 nearest to ten is 80
The number 89 nearest to ten is 90
The number 58 nearest to ten is 60
Now add three numbers 80 + 90 + 60 = 230
Now check whether the estimated answer is closer to the actual answer.
79 + 89 + 58 = 226
6 is greater than 5, so you can add 1 to the tens place value and 0 to the unit place value.
226 nearest to 10 is 230.

Example 2.
Estimate the difference between 219 and 17.
Solution:
9 is greater than 5, so you can add 1 to the tens place value and 0 to the unit place value.
7 is greater than 5, so you can add 1 to the tens place value and 0 to the unit place value.
219 nearest to ten is 220.
17 nearest to ten is 20.
Estimated difference is 220 – 20 = 200
Now check whether the estimated answer is closer to the actual answer.
219 – 17 = 202
202 nearest to ten is 200.

Example 3.
Estimate the sum and difference of 311 and 92.
Solution:
1 is less than 5, so you can round down to 0 to the unit place value.
2 is less than 5, so you can round down to 0 to the unit place value.
311 nearest to ten is 310
92 nearest to ten is 90
Estimated Sum: 310 + 90 = 400
Estimated Difference: 310 – 90 = 220
Now check if the estimated answer is closer to the actual answer.
311 + 92 = 403
403 is closer to 400.
311 – 92 = 219
219 is closer to 220.
So, the solution is correct.

Example 4.
Estimate the following additions and subtractions to the nearest ten, hundred and thousand.
i. 27 – 19
ii. 126 + 112
iii. 1002 + 996
iv. 2009 – 122
v. 39 – 12
Solution:
i. 27 – 19
9 is greater than 5, so you can add 1 to the tens place value and 0 to the unit place value.
7 is greater than 5, so you can add 1 to the tens place value and 0 to the unit place value.
27 to the nearest ten is 30.
19 to the nearest ten is 20.
Estimated Difference:
30 – 20 = 10
27 – 19 = 8
8 is closer to 10.
ii. 126 + 112
If the tens place is greater than 50, round up to the next hundred.
If the tens place is less than 50, round up to the previous hundred.
126 rounded to the nearest hundred is 100
112 rounded to the nearest hundred is 100
100 + 100 = 200
126 + 112 = 236
236 is closer to 200.
iii. 1002 + 996
If that digit is less than 5, you will round down to the previous thousand.
If that digit is greater than 5, you will round up to the next digit.
1002 to the nearest thousand is 1000.
996 to the nearest thousand is 1000.
1000 + 1000 = 2000
1002 + 996 = 1998
1998 is closer to 2000.
iv. 2009 – 122
If that digit is less than 5, you will round down to the previous thousand.
If that digit is less than 5, you will round down to the previous hundred.
2009 to the nearest thousand is 2000.
122 to the nearest hundred is 100.
2000 – 100 = 1900
2009 – 122 = 1880
1880 is closer to 1900.
v. 39 – 12
9 is greater than 5, so you can add 1 to the tens place value and 0 to the unit place value.
2 is less than 5, so you round down to 0 to the unit place value.
39 rounded to nearest ten is 40.
12 rounded to the nearest ten is 10.
40 – 10 = 30
39 – 12 = 27
27 is closer to 30.

Example 5.
Estimate the sum 711 and 625 to the nearest hundred.
Solution:
If that digit is less than 5, you will round down to the previous hundred.
If that digit is greater than 5, you will round up to the next hundred.
The unit place value is less than 5 so you have to round down to 0
711 number nearest to the hundred is 700.
The unit place value is equal to 5 so you have to round down to 0
625 number nearest to the hundred is 600.
700 + 600 = 1300
Now check if the estimated answer is closer to the actual answer.
711 + 625 = 1336
1336 is closer to 1300.

FAQs on Estimation of Addition and Subtraction

1. What is the actual difference and estimated difference?

If the exact difference is obtained, then it is called the actual difference. The estimated difference means the difference is obtained from the rounding off the given numbers.

2. How do you estimate the sum?

We estimate the addition by rounding off to the nearest numbers.

3. How do you estimate the difference?

We estimate the subtraction by rounding off to the nearest place values.

The Indian Numbering System is used in India to express large numbers. The terms like hundreds, thousands, lakhs, crores are the most commonly used terms to express the large numbers in Indian English. The zeroth power of 10 is 1, 10 power 1 is 10, 10 power of 2 is 100, the next powers of ten are called thousand, ten thousand, lakh, ten lakhs, crore.

Whereas in the Western System the next powers of ten are called one hundred thousand, one million, ten million, one hundred million, and so on. Get the solutions to the problems to calculate as per the Indian System or Indian Standard Number System.

Also, refer:

• International Place-value Chart
• Worksheet on International Numbering System
• International Numbering System

What is Indian Numbering System?

The Indian Numeral System is also known as Hindu-Arabic Numeral System. Indian Number System is a mathematical notation for expressing numbers, symbols, and digits or place values. In Arabic Number System or Hindu Number System ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are used to write the numbers called digits. The Indian Numbering System groups the rightmost three digits together until hundreds place and then groups them by sets of two digits.

Indian Numbering System Chart

The Number System is a way to express or represent numbers. There are different place values like one’s place, tens place, hundreds place, thousands place, ten thousand places, lakhs place, and so on. In the Indian Number System separators or commas are used to identify the place values in the numbers. Know the way to express the large numbers in detail from this page.

Period	Place Values	Unit Digits	Number of Digits
Tens	Ones	1	1
	Tens	10	2
	Hundreds	100	3
Thousands	Thousands	1000	4
	Ten Thousands	10000	5
Lakhs	Lakhs	100000	6
	Ten Lakhs	1000000	7
Crores	Crores	10000000	8
	Ten Crores	100000000	9

Indian Number System Vs International Number System

The main difference between the Indian and International numeral systems is the placement of the separator. In the Indian System of Numeration, lakhs are written after thousands whereas in the International Numeral System millions are written after thousands.

100 thousand = 1 lakh 1 million = 10 lakhs 10 millions = 1 crore 100 millions = 10 crores

Placement of Commas in Indian Number System

In the Indian Number System, we use commas or separators to mark different periods. Commas help using reading and writing large numbers. According to the Indian Numbering System, the first comma is placed after the hundreds place, the second comma is placed after thousands period, the third comma is placed after the lakhs period, continue to place the commas after every two digits.

Example: Using Hindu-Arabic Numeral System read the number 1053671.
Solution: We can read the number 1053671 by placing the commas 10,53,671.
The word form is Ten Lakhs Fifty Three Thousand Six Hundred and Seventy One.

Indian Number System in Words | How to Write Number Names in Indian Number System?

Indian Number System in Words helps to read large numbers easily. Read the values according to the place values and commas. Neglect zero while read the number in word form. Go through the examples given below to get an idea about how to Write Number Names in Hindu-Arabic Number System.

Example: 999999
Solution:
First, we have to separate the numbers by using commas.
9,99,999
Word sentence for 9,99,999 is Nine Laks Ninety Nine Thousand Nine Hundred and Ninety Nine.

Do Check Similar Articles:

• Worksheet on Indian Numbering System
• Worksheet on Numbering Systems

Indian System of Numeration Examples

Example 1.
Write the place values for 10987654?
Solution:
In the number 10987654, the place value of each digit is
4 – units place
5 – tens place
6 – hundreds place
7 – thousands place
8 – ten thousands place
9 – lakhs place
0 – ten lakhs place
1 – crore place
1,09,87,654

Example 2.
Rewrite the number 18659 in Indian Numeral System?
Solution:
Use separators or commas to write the given number in the Indian Numeral System
18659 – 18,659

Example 3.
How do you write 6 lakhs 50 thousand in the Indian Number System?
Solution:
The method to writing 6 lakhs 50 thousand in Indian Number System is 6, 50, 000

Example 4.
How to write 100 hundred in Indian Numeral System?
Solution:
The method to write 100 hundred in Indian Numeral System 1,00, 000 (1 lakh)

Example 5.
How do you write 10 million in Indian Numeral System?
Solution:
10 Million can be written as 1,00,00,000

FAQs on Indian Numbering System

1. How do you write numbers in the Indian and International System?

In the international numbering system, millions are written after thousands while in the Indian system, lakhs are written after thousands

2. What do you mean by the Indian Number System?

The Indian numbering system corresponds to the Western system for the zeroth through the fourth powers of ten.

3. What is the difference between the Indian and International Numeral System?

The main difference between the Indian and International numeral system is the placement of commas

Roman Numerals are the collection of Roman symbols that are used by the ancient Romans in the number system. Roman numerals are used for ranking as I, II, III, IV, and so on. These Roman Numerals are used to write classroom numbers. Also, it is used for writing the numbers on the clock. Know about various applications of Roman Numerals and why they are important, etc. in the further modules.

Do Refer:

• Conversion of Numbers to Roman Numerals
• Roman Numerals
• Roman Symbols
• Worksheet on Roman Numerals

Roman Numerals – Definition

Roman Numerals is the mathematical notation that does not follow the place value system. These Roman Numerals are used instead of Natural Numbers. C, D, I, L, M, V, X are the Roman Symbols that are used to express Roman Numerals. For better understanding, we have given the Roman Numerals Chart below.

Uses of Roman Numerals in Everyday Life

• The uses of Roman Numerals include year numbers on monuments and buildings, copyright dates on the title screens of movies and TV programs.
• We also use Roman Numerals in writing the year numbers instead of Arabic Numbers.
• Appendices or introduction of numbers is numbered with Roman Numerals.
• The Roman Numerals are used to write the numbers in watches, clocks, etc.
• Roman Numerals are used for writing classroom numbers like I, II, III, IV, V, VI, VII, VIII, IX, X.
• Sequels of books are numbered with Roman Numerals.

Related Articles:

• Worksheet on Roman Symbols
• Worksheet on Operations of Roman Numerals

FAQs on Uses of Roman Numerals

1. Why are roman numerals important?

Thinking about numbers in different ways can also help them form connections or see patterns. Writing a number as a Roman numeral is another way to represent the numbers.

2. Where do we see Roman Numerals in our day to day life?

We can find Roman Numerals on Clocks, Watches, Books, Classroom numbers, etc.

3. When to Use Roman Numerals Instead of Numbers?

We use Roman Numerals instead of numbers in the introduction part or chapter numbers in books.

Worksheet on Exact Divisibility helps the students to score better grades in the exams. Any number which leaves the remainder as 0 is known as Exact Divisibility. We have conducted Divisibility Tests for various numbers to make you understand what the exact divisibility is. There are different methods in the division. Go through this page to learn the simple methods of exact divisibility. Check Divisibility Tests for 2, 3, 4, 5, 6, 7, 8, 9, and 10 on the Exact Divisibility Worksheet. 

Do Refer:

• Exact Divisibility
• Divisibility Rules

Worksheet on Exact Divisibility with Solutions

Question 1
Circle the number which is exactly divisible by 2?
A. 24
B. 36
C. 25
D. 43
E. 7

Solution:

A. 24
A number is said to be exactly divisible by 2 only when it has the last digits as 0, 2, 4, 6, 8.
24 has the last digit. So it is exactly divisible by 2.
24 ÷ 2 = 12
24 is exactly divisible by 2.
B. 36
A number is said to be exactly divisible by 2 only when it has the last digits as 0, 2, 4, 6, 8.
36 has the last digit. So it is exactly divisible by 2.
36 ÷ 2 = 18
36 is exactly divisible by 2.
C. 25
A number is said to be exactly divisible by 2 only when it has the last digits as 0, 2, 4, 6, 8.
5 is not divisible by 2.
Thus 25 is not exactly divisible by 25.
D. 43
A number is said to be exactly divisible by 2 only when it has the last digits as 0, 2, 4, 6, 8.
3 is not divisible by 2.
Thus 43 is not exactly divisible by 43.
E. 7
A number is said to be exactly divisible by 2 only when it has the last digits as 0, 2, 4, 6, 8.
7 is not divisible by 2.
Thus A, B are exactly divisible by 2.

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Question 2
Which of the following numbers are exactly divisible by 3?
35, 15, 21, 315, 41

Solution:

35 –  3 + 5 = 8
8 is not divisible by 3
35 is not exactly divisible by 3.
15 – 1 + 5 = 6
6 is exactly divisible by 3.
15 is exactly divisible by 3.
21 – 2 + 1 = 3
3 is exactly divisible by 3.
Thus 21 is exactly divisible by 3.
315 = 3 + 1 + 5 = 9
9 is exactly divisible by 3.
That means 315 is exactly divisible by 3.
41 – 4 + 1 = 5
5 is not exactly divisible by 3.
Thus 41 is not exactly divisible by 3.

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Question 3
Check whether 1624 is exactly divisible by 4.

Solution:

Any number that is divisible by 2 will be divisible by 4.
The last digit 4 is divisible by 4 so it will be divisible by 4.
1624 ÷ 4 = 406
Thus 1624 is exactly divisible by 4.

---

Question 4
Find the number which is exactly divisible by 5?
A. 1550
B. 625
C. 1200
D. 1628

Solution:

A. 1550
A number that has the last digit as 0 or 5 then that number will be exactly divisible by 5.
1550 has the last digit as 0. That means 1550 is exactly divisible by 5.
B. 625
A number that has the last digit as 0 or 5 then that number will be exactly divisible by 5.
625 has the last digit as 5. That means 625 is exactly divisible by 5.
C. 1200
A number that has the last digit as 0 or 5 then that number will be exactly divisible by 5.
1200 has the last digit as 0. That means 1200 is exactly divisible by 5.
D. 1628
A number that has the last digit as 0 or 5 then that number will be exactly divisible by 5.
1628 has the last digit as 8.
Thus 1628 is not exactly divisible by 5.

---

Question 5
Check whether 24 and 36 are exactly divisible by 6.

Solution:

A number that is divisible by 2 and 3 will be divisible by 6.
24 and 36 are the multiples of 2 and 3.
24 ÷ 6 = 4
36 ÷ 6 = 6
Thus 24 and 36 are exactly divisible by 6

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Question 6
Check if the numbers 49 and 343 are exactly divisible by 7.

Solution:

7 is a prime number. We have to check as per the multiples of 7.
49 = 7 × 7
49 ÷ 7 = 7
Thus 49 is exactly divisible by 7.
343 ÷ 7 = 49
Thus 343 is exactly divisible by 7.
Hence both 49 and 343 are exactly divisible by 7.

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Question 7
Test whether 512 exactly divisible by 8.

Solution:

A number that is divisible by 2 and 4 will be divisible by 8.
512 has 2 in its last digit. Thus it is exactly by 8
5 + 1 + 2 = 8
512 ÷ 8 = 64
Thus 512 is exactly divisible by 8.

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Question 8
Check whether 108 is exactly divisible by 9.

Solution:

108 – 1 + 0 + 8 = 9
9 is exactly divisible by 9
108 ÷ 9 = 12
Thus 108 is exactly divisible by 9.

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Question 9
Check 5040 exactly divisible by 10.

Solution:

Any number that has the last digit as 0 then that number will be exactly divisible by 10.
5040 has 0 in the last digit
Thus 5040 is exactly divisible by 10.

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Question 10
Match the following numbers that are exactly divisible by 2, 3, 5, 7.

Solution:

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Question 11
A number that ends with 5 or 0 is divisible by _________.

Solution:

A number that has the last digit as 0 or 5 then that number will be exactly divisible by 5.

---

Question 12
A number that ends up with 0 is divisible by __________.

Solution:

Any number that has the last digit as 0 then that number will be exactly divisible by 10.

---

Question 13
Find the numbers that are exactly divisible by both 2 and 3.
A. 18
B. 312
C. 36
D. 512

Solution:

A. 18
A number is said to be exactly divisible by 2 only when it has the last digits as 0, 2, 4, 6, 8.
18 has the last digit. So it is exactly divisible by 2.
18 ÷ 2 = 9
1 + 8 = 9
9 is exactly divisible by 3.
18 ÷ 3 = 6
Thus 18 is exactly divisible by 2 and 3.
B. 312
A number is said to be exactly divisible by 2 only when it has the last digits as 0, 2, 4, 6, 8.
312 has the last digit. So it is exactly divisible by 2.
312 ÷ 2 = 156
312 is exactly divisible by 2.
312 = 3 + 1 + 2 = 6
6 is exactly divisible by 3.
312 ÷ 3 = 104
Thus 312 is exactly divisible by 2 and 3.
C. 36
A number is said to be exactly divisible by 2 only when it has the last digits as 0, 2, 4, 6, 8.
36 has the last digit. So it is exactly divisible by 2.
36 ÷ 2 = 18
3 + 6 = 9
9 is exactly divisible by 3.
36 ÷ 3 = 12
Thus 36 is exactly divisible by 2 and 3.
D. 512
A number is said to be exactly divisible by 2 only when it has the last digits as 0, 2, 4, 6, 8.
512 has the last digit. So it is exactly divisible by 2.
512 ÷ 2 = 256
512 is exactly divisible by 2.
5 + 1 + 2 = 8
8 is not divisible by 3.
Thus 512 is not exactly divisible by 3.

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Question 14
Find the numbers that are exactly divisible by both 5 and 10.
A. 1250
B. 5005
C. 1600
D. 900

Solution:

A number that has the last digit as 0 or 5 then that number will be exactly divisible by 5.
Any number that has the last digit as 0 then that number will be exactly divisible by 10.
To find whether the given numbers are exactly divisible by 5 and 10 we have to check the last digits.
A. 1250
It has 0 in the last digit so the number 1250 is exactly divisible by 5 and 10.
B. 5005
It has 5 in the last digit so the number 5005 is exactly divisible by 5, not 10.
C. 1600
It has 0 in the last digit so the number 1600 is exactly divisible by 5 and 10.
D. 900
It has 0 in the last digit so the number 900 is exactly divisible by 5 and 10.

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Question 15
Find the numbers that are exactly divisible by both 4 and 8.
A.1640
B. 32
C. 80
D. 804

Solution:

A. 1640
16 and 40 are multiples of 4 and 8.
16 and 40 are divisible by both 4 and 8
1640 ÷ 4 = 410
1640 ÷ 8 = 205
1640 is exactly divisible by both 4 and 8.
Thus the quotient is 410 and 205 and leaves the remainder as 0.
B. 32
Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32…
Multiples of 8 are 8, 16, 24, 32…
32 ÷ 4 = 8
32 ÷ 8 = 4
Thus 32 is exactly divisible by both 4 and 8.
C. 80
80 = 8 + 0 = 8
8 is exactly divisible by 4 and 8
80 ÷ 4 = 20
80 ÷ 8 = 10
Thus 80 is exactly divisible by both 4 and 8.
D. 804
8 + 0 + 4 = 12
12 is divisible by 4 but not 8.
804 ÷ 4 = 201
804 ÷ 8 = 100.5
Thus 804 is exactly divisible by 4 but not 8.

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The division is one the basic arithmetic operation which we have to learn in the primary level. Dividing by 10, 100, and 1000 Divisors means a number is divided into 10 or 100 or 1000 equal parts. For this, you need to learn the multiples of 10 which helps you to solve the problems quickly. Go to the below section to understand the concept of Divide by 10, 100, and 1000 Divisors.

Also, Refer:

• Dividend Divisor Quotient And Remainder
• Divide by 2-digit Divisors
• Dividing 2-Digit Number by 1-Digit Number

How to Divide by 10, 100 and 1000 Divisors?

The method to Divide by 10, 100, and 1000 Divisors is given below. Hence go through the below steps and practice the questions related to Dividing by 10, 100, and 1000 Divisors. Examples for Divide by 10, 100, and 1000 Divisors with answers are given in the coming section.

• When we divide a number by 10 then the very first 2-digit from the extreme right of the dividend will be divided by 10.
• Check whether the divisor 10 is greater than the 2 digit number.
• Then multiply 10 with the suitable digit.
• Then write the quotient and remainder.
• Write the next digit beside the remainder
• Again repeat the same process till you get the remainder as 0 or less than 10.
• The same process will be applied for 100 and 1000 divisors

Division of Decimal by 10, 100, and 1000 Divisors

The division of decimals numbers is similar to the concept of fractions. When we divide the decimal number by 10, 100, or 1000 we need to shift the point to the left. If the divisor is 10 you have to shift the point one time to the left. If the divisor is 100 you have to shift the point two times to the left same with 1000.
Example:
25 ÷ 10 = 2.5
25 ÷ 100 = 0.25
25 ÷ 1000 = 0.025

Dividing by 10 100 1000 Examples

Example: 1
A dealer bought 10 tables for $38000. What will be the cost of 1 table?
Solution: Given,
A dealer bought 10 tables for $38000.
Divide 38000 by 10 and find the cost of 1 table.
38000 ÷ 10 = 3800

The quotient is 3800 and the remainder is 0.

Example 2.
Divide the given numbers by 100 and write the quotient and remainder.
i. 28800 ÷ 100
ii. 10000 ÷ 100
iii. 1128 ÷ 100
Solution:
i.
It is a 3 digit divisor. Consider 3 digits in the dividend
288 > 100
Now multiply with 100
100 × 2 = 200
Write the result below the dividend and quotient to the right.
288 – 200 = 88
Write the remainder below and add 0 to the right it becomes 880
Again multiply with 100.
100 × 8 = 800
Subtract 880 and 800 we get 80.
Write the remainder below and add 0 to the right it becomes 800
Again multiply with 100.
100 × 8 = 800
Subtract 800 and 800 we get 0.
Thus the remainder is 0.
ii.
It is a 3 digit divisor. Consider 3 digits in the dividend
100 = 100
Now multiply with 100
100 × 1 = 100
Write the result below the dividend and quotient to the right.
100 – 100 = 0
Write the remainder below and add 0 to the right it becomes 00 and write 0 near the quotient.
Thus the remainder is 0.
iii.

It is a 3 digit divisor. Consider 3 digits in the dividend
112 > 100
Now multiply with 100
100 × 1 = 100
Write the result below the dividend and quotient to the right.
112 – 100 = 12
Write the remainder below and add 8 to the right it becomes 128.
Again multiply with 100.
100 × 1 = 100
Subtract 128 and 100 we get 28.
Thus the remainder is 28.

Example 3.
Find the quotient and remainder for the following numbers.
i. 34000 ÷ 1000
ii. 3720 ÷ 1000
iii. 1500 ÷ 1000
Solution:
i.
It is a 4 digit divisor. Consider 3 digits in the dividend
3400 > 1000
Now multiply with 1000
1000 × 3 = 3000
3400 – 3000 = 400
400 is the remainder and 3 is the quotient
Add 0 to the right side of the remainder it becomes 4000
1000 × 4 = 4000
Thus the remainder is 0 and the quotient is 34.
ii.
It is a 4 digit divisor. Consider 3 digits in the dividend
3720 > 1000
Now multiply with 1000
1000 × 3 = 3000
3720 – 3000 = 720
720 < 1000
Thus the remainder is 720 and the quotient is 3.
iii.
It is a 4 digit divisor. Consider 3 digits in the dividend
1500 > 1000
Now multiply with 1000
1000 × 1 = 1000
1500- 1000 = 500
500 < 1000
Thus the remainder is 500 and the quotient is 1.

FAQs on Divide by 10, 100, and 1000 Divisors

1. What happens when you divide a 2 digit number by 10 100 or 1000?

When we divide by 100, the 2-digits in the ones and the tens place form the remainder while the remaining digits form quotient.

2. How do you divide a number by 1000?

To divide a number by 1000, move all of its digits to thousands place value columns to the right.

3. What is the rule for dividing by 10?

Move the decimal point one place to the left. Place value is the value of a digit based on its location in the number.

Roman Symbols are the numerals used in the system of numerical notation. Roman Symbols came from Rome, where people wanted to use their own numerals to write various numbers. Roman Numerals are the collection of symbols C, D, I, L, M, V, X which are written in a specific order. A subtractive notation in Roman Symbols is found by taking the last character value and subtracting the value of the character that comes just before it.

Also Refer:

• Conversion of Roman Numeration
• Conversion of Numbers to Roman Numerals
• Roman Numeration
• Worksheet on Roman Numerals

What are Roman Numbers?

Roman Numbers are essentially a decimal or base 10 number system instead of place value notation. Roman Numbers are used in a numerical notation based on the ancient Roman system. We cannot use the same symbol more than three times in Roman Numbers.

How to Read Roman Numerals?

C, D, I, L, M, V, X are any symbols of Roman Numerals. ‘I’ can be read as 1, ‘V’ can be read as 5, ‘X’ can be read as 10, ‘C’ can be read as 100, ‘M’ can be read as 1000. If ‘I’ is placed before V then it is 4. If ‘I’ is placed is after ‘V’ then is 6. Similarly, If ‘I’ is placed before ‘X’ then it is read as 9. If ‘I’ is placed after ‘X’ then it is 11. If ‘C’ is placed before ‘X’ then it is read as 90.

 

Rules to Express Roman Symbols

1. We can use I and X 3 times in a numeral.
2. We never use V, L, D more than one time. We have to use it only once.
3. When we write a smaller symbol to the left of the greater symbol it means that the smaller numeral is subtracted from the greater one.
4. When we write a smaller symbol to the right of the greater symbol it means that the smaller numeral is added from the greater one.
5. We never write V to the left of X. Similarly, L and D are never written to the left to any other symbol.

Roman Numerals Chart

The below table shows the Roman Numerals Chart of Roman Symbols. Follow this Roman Number Chart to convert the Arabic numbers to roman symbols.

 

Roman Numbers Symbols to Numbers Conversion

To convert from roman symbols to whole numbers we need to split the numbers according to the symbols. For better understanding, we have provided a table that shows the Conversion of Roman Numbers to Arabic Numbers.

Roman Numeral	Calculation to get the number	Roman Symbol	Calculation to get the number
I	1	XVII	10 + 5 + 1 + 1 = 17
II	1 + 1 = 2	XVIII	10 + 5 + 1 + 1 + 1 = 18
III	1 + 1 + 1 = 3	XIX	10 – 1 + 10 = 19
IV	5 – 1 = 4	XX	10 + 10 = 20
V	5	XXX	10 + 10 + 10 = 30
VI	5 + 1 = 6	XL	50 – 10 = 40
VII	5 + 1 + 1 = 7	L	50
VIII	5 + 1 + 1 + 1 = 8	LX	50 + 10 = 60
IX	10 – 1 = 9	LXX	50 + 10 + 10 = 70
X	10	LXXX	50 + 10 + 10 + 10 = 80
XI	10 + 1 = 11	XC	100 – 10 = 90
XII	10 + 1 + 1 = 12	C	100
XIII	10 + 1 + 1 + 1 = 13	CC	100 + 100 = 200
XIV	10 – 1 + 5 = 14	CCC	100 + 100+ 100 = 300
XV	10 + 5 = 15	CD	500 – 100 = 400
XVI	10 + 5 + 1 = 16	D	500

Also Refer:

• Worksheet on Roman Symbols
• Worksheet on Roman Numeration

How to Write Date in Roman Symbols?

We can write the date or years in the form of Roman Symbols.
Example: 12-11-2020 now write this date from numbers to Roman Numbers i.e, XII – XI – MMXX.
Let us convert from numbers to Roman Numbers.
2001 – MMI
2002 – MMII
2003 – MMIII
2004 – MMIV
2005 – MMV
2006 – MMVI
2007 – MMVII
2008 – MMVIII
2009 – MMIX
2010 – MMX
2011 – MMXI
2012 – MMXII
2013 – MMXIII
2014 – MMXIV
2015 – MMXV
2016 – MMXVI
2017 – MMXVII
2018 – MMXVIII
2019 – MMXIX
2020 – MMXX

Examples of Roman Symbols

Example: 1
Convert the number 1562 to Roman Number.
Solution:
1560 = 1000 + 500 + 60 + 2
The number 1000 can be written in the roman numeral as ‘M’
The number 500 can be written in the roman numeral as ‘D’
The number 60 can be written in the roman numeral as ‘LX’
The number 2 can be written in the roman numeral as ‘II’
1560 = MDLXII

Example 2:
Convert the following Arabic Numbers to Roman Symbols.
i. 5
ii 17
iii. 28
iv. 1515
v. 2021
Solution:
i. We can write 5 in Roman Number as V.
ii. We can write 17 in Roman Symbol as XVII.
iii. We can write 28 in Roman Numeral as XXVIII.
iv. We can write 1515 in Roman Number as MDXV.
v. We can write 2021 in Roman Number as MMXXI.

Example: 3
Convert the number 2525 to Roman Numeral.
Solution:
2525 = 2000 + 500 + 20 + 5
The number 1000 can be written in the roman numeral as ‘M’
The number 500 can be written in the roman numeral as ‘D’
The number 20 can be written in the roman numeral as ‘XX’
The number 5 can be written in the roman numeral as ‘V’
Thus 2525 = MMDXXV

Example: 4
Convert from MCLV to Arabic Number.
Solution:
M stands for 1000
C stands for 100
L stands for 50
V stands for 5
So, MCLV = 1000 + 100 + 50 + 5 = 1155

FAQs on Roman Symbols

1. How many Roman Symbols are there?

There are 7 Roman Symbols they are C, D, I, L, M, V, X.

2. Is C 100 in Roman numerals?

Yes, C stands for 100 in Roman numbers.

3. Why is there no zero in Roman numerals?

Roman numerals start to count from one and had no symbol to represent 0.