The Study Zone: 2020

Trigonomatry-2

Right angled triangle identity: In trigonometry, the most widely used triangle is Right Angled triangle. There are a number of formulas or identities related to right angled triangles, which we will learn about 'Pythagoras Theorem’ in this lesson.

Division (Word problems)-1G5

This worksheet is based on division word problems for grade 5, check this out:

Division-WS1G5

This worksheet is based on dividing 3-digits numbers with 1-digit number for grade 5, check this page:

Least Common Multiple (LCM)- Lesson

LCM is the smallest factor which is common in the given numbers or the smallest common number which can be divided with all the given numbers completely.

Example: LCM of 4 and 8

8 is the least common multiple of 4 and 8

Greatest Common Factor (GCF)- Lesson

GCF the greatest common factor among all the given numbers or the greatest factor which divides all the given numbers completely (zero remainder).

Rules to find GCF :

(1) Factor method: follow the steps given below:

Step1: find all the factors of given numbers. (To know how to find factors, Click here.)

Step 2: Identify the greatest common factors.

Exponents- Lesson

‘Exponent’ is a number which tells us that “how many times a number is multiplied with itself” or Exponent represents the process of repeated multiplication. The exponent is also called ‘power’ and the number being multiplies is called the ‘base’.

Trigonometry- Lesson 1

Trigonometry (from Greek trigōnon, ‘triangle’ and metron, ‘measure’) is all about triangles. It is a branch of Mathematics that studies relationships between side lengths and angles of triangles. The most commonly used triangle is ‘Right Angles Triangle’.

Pronouns- Lesson

Pronoun: A Pronoun is a word which replaces a noun or a word which can be used in place of a noun is a pronoun. The words such as I, he, she, they, what, where, whose etc. are pronouns.

Review: Probability quiz

Complete this quiz to review the concept of probability and to quick remind the facts of probability:

Probability- WS 2 (G5,G6)

Tset your knowledge about Probability, check this worksheet:

Probability-Lesson G5 G6

Before talking about probability, we will learn about some important definitions which are given below:

Experiment: an experiment or trial is any procedure that can be infinitely repeated and has a well-defined set of possible outcomes, known as the sample space. In simple words an Experiment is any activity with an observable result, such as tossing a coin, rolling a dice etc.

Outcome: outcome is the result of a single trial of an experiment, such as head or tail on a coin, numbers 1 to 6 on a dice etc.

Event: an event is one or more outcome or result of an experiment, such as getting a head on coin, getting any number on a dice etc.

Probability is the chances or likelihood of the occurrence of an event. The probability of event E can be written as P(E). Probabilities are always numbers between 0 and 1, inclusive i.e.

\(\mathbf{0}\le{P}({E})\le\mathbf{1}\)

Formula to find the Probability:

\(P\left(E\right)=\frac{Number\ of\ ways\ an\ event\ can\ happen}{Total\ number\ of\ possible\ outcomes}\)

Example: A coin is tossed ones. What are the possible outcomes? What is the probability of each outcome?

Solution: in this example, tossing a coin is the Experiment and possible outcomes are Head (H) and Tail (T)

number of ways that Head or Tail can occur = 1

Total number of possible outcomes = H, T= 2

So, Total number of possible outcomes = 2

\(P\left(H\right)=\frac{number\ of\ ways\ head\ can\ happen}{Total\ number\ of\ possible\ outcomes}\)

\(P\left(H\right)=\frac{1}{2}\)

\(P\left(T\right)=\frac{number\ of\ ways\ Tail\ can\ happen}{Total\ number\ of\ possible\ outcomes}\)

\(P\left(T\right)=\frac{1}{2}\)

Types of events: Events in a probability are of following types:

(1) Likely

(2) Unlikely

(3) Impossible

(4) Certain

1.Likely Events: events which have same probability to happen, such as getting head and tail after tossing a coin, when a die is tossed each number has same chances to happen.

2.Unlikely Events: events which may not happen or which is not likely to happen, such as tossing 5 heads in a row, getting 7 fours in a row after rolling a die.

3.Impossible Events: an event that cannot happen such as getting 7 after rolling a die, getting both head and tail after tossing a coin. Probability of an impossible event is 0.

4.Certain Events: an event which is sure to happen, such as getting head or tail on a coin, getting a number from 1 to 6 on a die. Probability of a certain event is 1.

Events can also be:

(A) Dependent events

(B) Independent events

(A) Dependent Events: the events which are affected by previous event, such as drawing two cards from a deck. In this case, after drawing first card, there will be less cards in the deck and probabilities will change.

(B) Independent Events: the events which are not affected by previous event, such as tossing a coin many times does not affect the probability of getting head or tail on next toss.

(C) Mutually exclusive events: the events which cannot happen at the same time, such as getting head or tail on a coin (both cannot occur at same time), getting any number from 1 to 6 on a die (only one number occurs after rolling a die.)

--Lesson by The Study Zone

English quiz- 1 G4

Complete this quiz based on the nouns:

Quiz-2 G4

Take this test based on pattern rules:

Quiz- 1 G4

Complete this multiple choice quick test to review your knowledge about pattern and addition:

Angles- G4 WS1

Try to count the different types of angles, check this worksheets:

If you need any help with the types of angles, go to this link.

Fractions-G5 WS 1

Redistribute the fractions, don't forget to show your work. Check this worksheet:

Number system- WS G7

Practice your knowledge of 'order of operations' to solve these word problems, check this worksheet:

Future Tense- Lesson

Future Tenses: Future tenses are used to describe an event not having happened yet, but expected to happen in the future. These are of 4 types:

(1) Future Simple

(2) Future Progressive

(3) Future Perfect

(4) Future Perfect Progressive

1. Future Simple: simple future has two different forms in English: ‘will’ and ‘be going to’. Both ‘will’ and ‘be going to’ refer to a specific time in the future.

Rule: Subject + will/shall + 1st form of verb

Example: She will read a book.

I will walk.

He will run 3 miles.

You will sleep.

2. Future Progressive: The future continuous tense describes a continuing action, something that will be happening, going on, at some point in the future.

Rule: Subject + will/shall be + Present participle

Example: She will be reading a book.

I will be walking.

They will be running.

You will be sleeping.

3. Future Perfect: Future perfect tense indicates an action will have been completed (finished) at some point in the future.

Rule: Subject + will have + Past participle

Example: She will have read a book.

I will have walked.

He will have run 3 miles.

You will have slept.

4. Future Perfect Progressive: Future perfect progressive tense is used to describe a continuous action that will be completed at some point in the future.

Rule: Subject + will/shall have been + Present participle

Example: She will have been reading.

I will have been walking.

He will have been running.

You will have been sleeping.

Note: • Present participle: The 1st form of verb ending with ‘ing’ is called Present participle.

• Past participle: The 3rd form of verb is called Past participle.

-- The Study Zone

Learn about Past Tense and Present Tense

Make sentences G3- WS2

Try to make sentences by using the given words, check this worksheet:

Find the words G3- WS1

Find the given words in a puzzle to learn new words, check this worksheet:

Match definition to the word- WS1

Learn new words and their meanings, enhance your vocabulary and English language skills. Check this worksheet:

Make sentences- WS 1 (G4-G8)

Learn new words and use them to make sentences, it will help you to improve your English language skill, check this worksheet:

Spelling and Vocabulary- WS1

Learn new words to improve your vocabulary and to avoid spelling mistakes, check this worksheet:

Past Tense- Lesson

Past Tenses: Past tenses describe an action which has happened in the past.

These are of 4 types:

(1) Past Simple

(2) Past Progressive

(3) Past Perfect

(4) Past Perfect Progressive

1. Past Simple: the simple past tense is used to describe an action which has started and finished in the past.

Rule: Subject + 2nd form of verb

Example: She read a book.

I walked

He ran 3 miles.

You slept

2. Past Progressive: The past continuous describes an action or event in a time before now, which began in the past and was still going on when another event occurred.

Rule: Subject + was/were + Present participle

Example: She was reading a book.

I was walking.

They were running.

You were slept.

3. Past Perfect: Past perfect tense indicates that an action was completed (finished) at some point in the past before something else happened means It is used to make it clear that one event happened before another in the past.

Rule: Subject + had + Past participle

Example: She had read a book.

I had walked

He had run 3 miles.

You had slept.

4. Past Perfect Progressive: Past perfect progressive tense is used to describe a continuous action that was completed at some point in the past.

Rule: Subject + had been + Present participle

Example: She had been reading. I had been walking. He has been running. You had been sleeping.

Note: • Present participle: The 1st form of verb ending with ‘ing’ is called Present participle.

• Past participle: The 3rd form of verb is called Past participle.

-- The Study Zone

Learn about Futur Tense and Present Tense

Present Tense- Lesson

Present Tenses: Present tenses describe an action which is happening in the present.

These are of 4 types:

Tenses- Lesson

In English language the formation of sentences is based on the tense rules. The main tenses are of three types:

(1) Present

(2) Past

(3) Future

(1) Present Tense: The Present tense describes a current event or state of being.

Example: I jump in the lake every Saturday. (Present event)

I am happy. (Present state of being)

Note: The present tense can also be used to describe past and future events.

Examples: The meeting ends at 6 o'clock. (future event)

A man walks into a bar. (past event)

(2) Past Tense: The past tense describes a past event or a state of being.

Examples: I jumped in the lake. (Past event)

I was happy. (Past state of being)

(3) Future Tense: The future tense expresses a future event or a future state of being.

Examples: I will jump in the lake. (Future event)

I will be happy. (Future state of bring)

There are further 4 types of each tense:

(1) Simple (2) Progressive (3) Perfect (4) Perfect Progressive

-- The Study Zone

Learn in detail: Past Tense Present Tense Future Tense

Cause and effect- WS1

Grade 5 and 6 students, use your brain to find the effect of a given cause, check this worksheet:

Reading & Writing- WS3

Basic Geometry Terms
This worksheets, is based on cross-curriculum of Math and English, students will read and learn about Basic geometry terms and then they have to answer the questions:

Trace and write the words- WS2

Trace and learn the new words, check this worksheet:

Make sentences G3- WS1

Use the words given and make sentences, check this worksheet:

Trace and write the words- WS1

Trace and learn the new words, check this worksheet:

Match the columns- WS1

Match the column A with the correct word in column B, check this worksheet:

Numbers 1 to 10- WS

Learn the numbers from 1 to 10, trace the numbers and words. check this worksheet:

Addition- WS 1

Solve these questions of number sense, check this worksheet:

Months of The Year- WS

Learn the months of the year, practice their spellings, check this worksheet:

Read & Write G1- WS2

Learn some new spellings, first read the try to write by yourself, check this worksheet:

Read and write- G1 WS

Read and then try to write by yourself, check this worksheet:

Numbers 1 to 20- WS

Practice to write the numbers from 1 to 20, check this worksheet:

Capacity- WS1

This worksheet is based on the concept of Capacity. Use your knowledge to estimate and compare the capacity, Answers are also provided with this, check this worksheet:

Big and Small-WS

Use your brain to identify big and small objects, check this worksheet:

Read & Write G2- WS1

Read the words and write them by yourself to learn the spellings, check this worksheet:

Match the column G2 - WS1

Try to match the words in two different columns, check this worksheet:

Rounding off the numbers- Lesson

Rounding means making a number simpler but keeping its value close to what it was. The result is less accurate, but easier to use.

We can round off the numbers to the:

Nearest to Ten

Nearest to Hundred

Nearest to Thousand

Nearest to Ten Thousand etc.

Subtraction 3-digit numbers G3- WS

this worksheet is based on the subtraction of 3-digit numbers, practice your knowledge check this worksheet:

Noun: A word which names a person, place, object, animal or an idea is called a noun. Example:

• Person: Max, Andy, Tom, Sarah, Julie etc.
• Place: Canada, School, Store, India, London etc.
• Object: Home, Car, Bus, Table, Chair etc.
• Animal: Dog, Cow, Rat, Rabbit, Cat etc.
• Idea: Happiness, Devotion, Faith, Kindness, Superstition etc.

Types of Nouns: there are following 8 types of nouns:

Addition (Double digit) G2- WS

try to add some double digit numbers, check this worksheet:

Subtraction Double Digit G2- WS

subtract 2-digit numbers, check this worksheet:

Comparing Numbers G3- WS 1

try to compare the numbers and then arrange in greatest to lowest order, check this worksheets

Multiplication G2- WS 1

Try to multiply by using additive strategies, check this worksheet:

Skip Counting G3- WS 1

try to count by following a pattern (skip by a number), check this worksheet:

Number sense G3- WS1

try your number skills, check this worksheet:

Skip Counting G2- WS1

Try to count by following a pattern (skip by a number), check this worksheets:

Subordinate Clause and conjunction- WS (G4 G5)-1

Try to identify Subordinate clauses and conjunctions, check this worksheet:

Creating Histogram WS (G4 & G5)-1

Try to make histograms, you can use your own scale but for your convenience, graphs are given and answers are also provided at the end of the worksheet, go n check this worksheets:

Probability WS 1-G5

Try to find the probability for some events, check this worksheets

Number sense- WS G2

let's practice some knowledge about numbers, check this worksheet:

Upper case alphabets- WS G1

Let your child play with alphabets, check this worksheet:

Lower case alphabets- WS G1

Let your child play with alphabets, check this worksheet:

Subtraction word problems- WS1

Let your child practice some subtraction skills, check this worksheets:

Addition word problems- WS1

Let your child practice some addition skills, check this worksheet:

Reading and Writing- WS 2

Branches of Government

This worksheet, is based on cross-curriculum of Social science/history and English, students will read and learn about Branches of Government and then they have to answer the questions:

Reading and Writing- WS1

Anatomy of a Circle

This worksheets, is based on cross-curriculum of Math and English, students will read and learn about Circle and then they have to answer the questions:

Verbs- WS 1

In this worksheet, student will practice their knowledge about the different forms of verb::

Verbs- WS 2

In this worksheets, students will identify Action and Linking verbs, this will help them to enhance their knowledge about the verbs: (Answers are also provided at the end of the worksheet)

Verbs- WS 3

In this worksheet, students will identify the Action verbs, it will improve their understanding about the action verbs. (Answer keys are also provided at the end of the worksheet.)

Find Mean, Median, Mode & Range- WS3

A. Find the mean, median, mode and range of the following data sets:

1. 13, 18, 13, 14, 13, 16, 14, 21, 13

Mean: _____ Median: _____

Mode: _____ Range: _____

Find Mean, Median, Mode & Range- WS2

Name:_______________________________ Date:_____________________

1. Sara Lee earned scores of 98, 100, 65, 78, 98, 35, 100, 100, 45, and 50 on her reading tests. What is the mean of her test scores?

2. A family decides to hold two Christmas parties, one on Christmas Eve and one on Christmas Day. The youngest person to attend the Christmas Eve party is 15 and the oldest is 29. The youngest person to attend the Christmas Day party is 27 and the oldest is 42. Which party has a larger age range of visitors?

Find Mean, Median,Mode &Range- WS1

1. What do you understand by Mean, Median, Mode and Range of a data?

Statistics- Lesson 1

Mean: The average of numbers (data values) is called mean.
Rule to find the mean of a data set: Add all the numbers in a given data set and divide the sum with the number of data values in the set.

Example: Suppose we have to find the mean of the data set: 12, 22, 24, 27, and 35.
Step 1: add all numbers to find their sum:

12 + 22 + 24 + 27 + 35 = 120

Step 2: Then we take this sum and divide it by the number of values in the set (which is 5):

120 ÷ 5 = 24

The result 24, is the mean (average) of the data set.

Median: The middle value of a sorted data set is called the median of the set.
Rule to find the mean of a data set:
(1) If number of data values in a set is odd: suppose we need to find the median of data set 5, 7, 23, 21, 15, 30, 27
First, arrange all the values in ascending (smallest to greatest) order as,
5, 7, 15, 21, 23, 27, 30
Now, we have 7 data values (or simply we have 7 numbers in the set), find the middle number of the set, that will be the median of the set.
5, 7, 15, 21, 23, 27, 30
Here 21 is the middle number in the set, so the median of the set is 21

(2) If number of data values in a set is even: In this case, we will find the middle two terms and then the mean of middle two terms will be the median of the data set.
suppose we need to find the median of data set 5, 7, 23, 21, 15, 30, 17, 27
First, arrange all the values in ascending (smallest to greatest) order as,

5, 7, 15, 17, 21, 23, 27, 30

Now, we have 8 data values (or simply we have 8 numbers in the set),

find the middle two numbers of the set, 5, 7, 15, 17, 21, 23, 27, 30
Here, the middle two terms are 17 and 21,
now find the mean of 17 and 21

\(\frac{17+21}{2}=\frac{38}{2}=19\)

So, 19 is the median of the given data set.

Mode: The data which appears most often in a data set means the value which occurs most number of times in a set, is called the mode of the set.

Example: suppose we have data set 2,7,6,1,11,15,14,2,6,6,7

Arrange the data in ascending order, (this will make question easy), as:

1, 2, 2, 6, 6, 6, 7, 7, 11, 14, 15

In this set the number 6 occurs most number of times (three times) so, the mode of the data set is 6.

Note 1: There may be no mode if no value appears more than any other.

Example: Find the mode of 5, 11, 10, 13, 14, 2, 12, 3

In the above data each values occurs same number of times, so the data has no mode.

Note 2: There may also be two modes (bi-modal), three modes (tri-modal), or four modes or more modes (multimodal).

Example 1: Find the mode of 5, 11, 10, 5, 13, 14, 2, 2, 12, 3

Arrange the data in ascending order as, 2, 2, 3, 5, 5, 10, 11, 12, 13, 14

In the above data set the values 2 and 5 occur more than other values means both 2 and 5 occur 2-times in the data set, therefore this data has two modes 2 and 5.

Example 2: Find the mode of 5, 11, 10, 5, 13, 14, 2, 2, 12, 3, 13

Arrange the data in ascending order as, 2, 2, 3, 5, 5, 10, 11, 12, 13, 13, 14

In the above data set the values 2, 5 and 13 occur more than other values means 2, 5 and 13 occur 2-times in the data set, therefore this data has three modes 2, 5 and 13.

So, a data set can have multi modes.

Range: The ranges of the data set is the difference between the highest and lowest values in the set. To find lowest and highest values, arrange the data in ascending order.

Example: Find the Range of 11, 15, 7, 3, 4, 16, 12, 13

First arrange the data in ascending order as: 3, 4, 7, 11, 12, 13, 15, 16

Highest value = 16 and lowest value = 3

Range =16-3=3

Q: Find mean median, mode and range of 13, 18, 13, 14, 13, 16, 14, 21, 13

Ans: first arrange the data in ascending order as, 13, 13, 13, 13, 14, 14, 16, 18, 21

\({Mean}=\frac{13+13+13+13+14+14+16+18+21}{9}=\frac{263}{9}=15\)

\({Median}=middle\ number\ of\ the\ data=14\)

\({Mode}=the\ value\ which\ occurs\ most\ number\ of\ times\ in\ set=13 \)

\({Range}=highest\ value-Lowest\ value=21-13=8\)

--The Study Zone

Factorization: Prime Factorization

Before factorization, we need to know that what the factor is...?

Factor: A number which divides the given number is called the factor of that number.

Example: The number 2 divides the number 4, therefore 2 is a factor of 4.

The number 3, divides the number 12, therefore 3 is a factor of 12.

Factorization means when we write a number as a product of smaller numbers or the method to breaking down a number into smaller numbers.

Example: The number 20 can be written as the product of smaller numbers as,

\(20=4\times5\),

\(20=2\times10\)

\(20=2\times2\times5\)

So, the numbers 2,4,5 and 10 are the factors of 20.

Prime factorization: This is a method to find which prime numbers multiply together to make the original number.

Example: Suppose we need to find prime factors of 20.

Start dividing 20 with the smallest prime number which is 2,

we will get \(\frac{20}{2}=10\),

so 20 can be written as \(2\times10\) But 10 is not a prime number,

Now divide 10 with the smallest prime number, we will get \(\frac{10}{2}=5\),

so 10 can be written as \(2\times5\).

Now 5 is a prime number means we can not divide it further, so by combining these we can say that:

20 can be written as the product of prime numbers 2,2 and 5, these are the prime factors of 20 as:

\(\mathbf{20}=\mathbf{2}\times\mathbf{2}\times\mathbf{5}\)

Another method to find Prime Factors is Factor Tree

Factor tree: This method includes to find any factors of given number and then find factors of those factors till we didn’t get all the prime numbers.

Example: Suppose we need to find prime factors of 108 by using factor tree method.

Take factors of \(108\ =\mathbf{9}\times\mathbf{12}\) and write under 108

Now find the factors of factors 9 and 12 (which we got in step 1), as:

\(9=\mathbf{3}\times\mathbf{3}\\) and

12=\mathbf{4}\times\mathbf{3}\),

write factors under the numbers 9 and 12.

now 3 is a prime number but not 4, so find factors of 4, as:

4=2\times2

Now we got all the prime factors of 108 as:

\(\mathbf{2}\times\mathbf{2}\times\mathbf{3}\times\mathbf{3}\times\mathbf{3}\)

So, \(\ \mathbf{108}=\mathbf{2}\times\mathbf{2}\times\mathbf{3}\times\mathbf{3}\times\mathbf{3}\)

--- The Study Zone

Multiplication Lesson-1

Multiplication is one of the four operations of Mathematics. It is a process to find the total sum when a number is added multiple times, so we can say that multiplication is a ‘repeated addition’. The result of multiplication operation is called ‘product’. The two numbers being multiplied are called ‘Multiplicand’ and ‘Multiplier’ and collectively they are called ‘Factors’.

Adding Fractions

Unlike Frcations: the fractions having different denominators.

Multiplication Crossword-1

Try to solve this 'Multiplication Crossword'

Multiplication: two digits' number with one-digit number

Practice your multiplication skills by solving this worksheet:

Divisibility Rules

Divisibility rules, help the students to find if a number is divisible by another number without doing lengthy calculations.
Meaning of ‘Divisible by’: ‘Divisible by’ means when we divide a number with another number, the result is a whole number, means division is a complete division (remainder is zero).
Rules:
Note: Zero (0) is divisible by any number except by itself means 0 can be divided by any number but \(0 ÷ 0\) is meaningless.
Divisibility by 1: Every integer is divisible by 1.

Divisibility by 2

: any integer whose last digit is even means any number ending with 0, 2, 4, 6, 8, is divisible by 2.
Example: look at the number 198, last digit in this number is 8, therefore 198 is divisible by 2.

Some more examples for the divisibility by 2:

Number	Divisible by 2	Reason
32	Yes	Ends with 0
332	Yes	Ends with 2
1554	Yes	Ends with 4
16676	Yes	Ends with 6
27	No	Ends with 7
123	No	Ends with 3

Divisibility by 3

: A number is divisible by 3 if the sum of the digits of the number is a multiple of 3, means if the sum of digits is divisible by 3.
Example: Suppose we need to check if the number 72 is divisible by 3 or not
The digits of 72 are 7 and 2 sum of digits = \(7+2=9\)
9 is a multiple of 3 as \(9÷3=3\)
So, 72 is divisible by 3
Some more examples for the divisibility by 3:

Number	Divisible by 3	Reason
105	Yes	sum of digits \(1+0+5=6\) which is divisible by 3
4314	Yes	sum of digits \(4+3+1+4=12\) which is divisible by 3
577	No	sum of digits \(5+7+7=19\) which is NOT divisible by 3
71	No	sum of digits \(7+1=8\) which is NOT divisible by 3
695	No	sum of digits \(6+9+5=20\) which is NOT divisible by 3
423	Yes	sum of digits \(4+2+3=9\) which is divisible by 3

Note: This rule can be repeated when needed, look at the following example:
Consider a number 89988
sum of digits of \(89988=8+9+9+8+8=42\)
now,reapeat the rule for number 42
sum of digits of \(42=4+2=6\)
and,6 is divisible by 3 so,89988 is also divisible by 3.

Divisibility by 4: A number is divisible by 4 if the number formed by its last two digits is divisible by 4 means number formed by its last two digits is a multiple of 4.
Example: Suppose we need to check if the number 172 is divisible by 4 or not
Now, the number formed by last two digits of 172 is 72
divide 72 with 4

example: The number 72 is a divisible by 4 as \(72÷4=18\) So the number 172 is divisible by 4
Some more examples for the divisibility by 4:

Number	Divisible by 4	Reason
2324	Yes	Number formed by last two digits (24) is divisible by 4 as \(24÷4=6\)
3860	Yes	Number formed by last two digits (60) is divisible by 4 as \(60÷4=15\)
175	No	Number formed by last two digits (75) is NOT divisible by 4
2461	No	Number formed by last two digits is NOT divisible by 4
695	No	Number formed by last two digits (95) is NOT divisible by 4
15664	Yes	Number formed by last two digits (64) is divisible by 4

Divisibility by 5: any integer ending with 0 or 5 means any integer which has las digit 0 or 5, is divisible by 5.
Example: Suppose we need to check if the number 90 is divisible by 5 or not
The last digit in 90 is 0, so 90 is divisible by 5.
More examples for the divisibility by 5:

Number	Divisible by 5	Reason
135	Yes	Ends with 5
3860	Yes	Ends with 0
15875	Yes	Ends with 5
2460	Yes	Ends with 0
697	No	Ends with 7
15664	No	Ends with 4

Divisibility by 6: Any integer which is divisible by both 2 and 3.
Example: Suppose we need to check if the number 162 is divisible by 6 or not
Check divisibility by 2: As the last digit of number 162 is 2, so 162 divisible by 2 (by the divisibility rule of 2).
Check divisibility by 3: sum of digits of 162=1+6+2=9, now 9 is divisible by 3, so 162 is also divisible by 3 (by the divisibility rule of 3).
Now we can see that the number 162 is divisible by both 2 and 3, so it’s also divisible by 6.
Another example:Suppose we need to check if the number 274 is divisible by 6 or not
Check divisibility by 2: As the last digit of number 274 is 4, so 274 divisible by 2 (by the divisibility rule of 2).
Check divisibility by 3: sum of digits of 274\(=2+7+4=13\), now 13 is not divisible by 3 (by the divisibility rule of 3), so 274 is not divisible by 3.
Now we can see that the number 274 is divisible by 2 but not by 3, so it’s not divisible by 6.
Another example: Suppose we need to check if the number 165 is divisible by 6 or not
Check divisibility by 2: As the last digit of number 165 is 5, so 165 is not divisible by 2 (by the divisibility rule of 2).
Check divisibility by 3: sum of digits of 165=\(1+6+5=12\), now 12 is divisible by 3 (by the divisibility rule of 3) so, 165 is also divisible by 3
Now we can see that the number 165 is not divisible by 2 but it’s divisible by 3, so it’s not divisible by 6.
Divisibility by 7: To check if a number is divisible by 7, double the last digit and subtract from the number formed by rest of the digits, if the result is a multiple of/divisible by 7 then the number is also divisible by 7.
Example: Suppose we need to check if the number 91 is divisible by 7 or not
Now, last digit of 91 is 1, when we double it means multiply 1 by 2 and we will get 2
And, number formed by rest of the digits is 9
Subtract 2 from 9 and we will get 7 as \(9-2=7\), which is divisible by 7, so 91 is divisible by 7.
Another example: Suppose we need to check if the number 682 is divisible by 7 or not
Now, last digit of 682 is 2, when we double it means multiply 2 by 2 and we will get 4
And, number formed by rest of the digits is 68
Subtract 4 from 68 and we will get 64 as \(68-4=64\), which is not divisible by/not a multiple of 7, so 682 is not divisible by 7.
Note: This rule can be repeated when needed, look at following example:
Suppose we need to check if the number 1771 is divisible by 7 or not
Now, last digit of 1771 is 1, when we double it means multiply 1 by 2 and we will get 2
And, number formed by rest of the digits is 177
Subtract 2 from 177 and we will get 175 as \(177-2=175,\)
Repeat the rule for 175:
Now, last digit of 175 is 5, when we double it means multiply 5 by 2 and we will get 10
And, number formed by rest of the digits is 17
Subtract 10 from 17 and we will get 7 as \(17-10=7\), which is divisible by 7, so 1771 is divisible by 7.
Divisibility by 8: A number is divisible by 8 if the number formed by its last three digits is divisible by 8 means number formed by its last three digits is a multiple of 8.
Example: Suppose we need to check if the number 3120 is divisible by 8 or not
Now, the number formed by last three digits of 3120 is 120
now divide 120 with 8
the number 120 is divisible by 8 as \(120÷8=15\)
so the number 3120 is divisible by 8.
More examples for the divisibility by 8:

Number	Divisible by 8	Reason
2824	Yes	Number formed by last three digits is divisible by 8; \((824÷8=103)\)
53104	Yes	Number formed by last three digits is divisible by 8; \((104÷8=6638)\)
1166	No	Number formed by last three digits is divisible by 8; (166 is not divisible by 8)
12461	No	Number formed by last three digits is divisible by 8; (461 is not divisible by 8)
15664	Yes	Number formed by last three digits is divisible by 8;\((664÷8=83)\)

Divisibility by 9: A number is divisible by 9 if the sum of the digits of the number is a multiple of 9, means if the sum of digits is divisible by 9.
Example: Suppose we need to check if the number 3672 is divisible by 9 or not
The digits of 3672 are 3, 6, 7 and 2 sum of digits = \(3+6+7+2=18\)
18 is a multiple of 9 as \(18÷9=2\)
So, 3672 is divisible by 9
More examples for the divisibility by 9:

Number	Divisible by 9	Reason
135	Yes	sum of digits\(=1+3+5=9\) and 9 is divisible by 9
9846	Yes	sum of digits\(=9+8+4+6=27\) and 27 is divisible by 9
577	No	sum of digits\(=5+7+7=19\) and 19 is not divisible of 9
74	No	sum of digits\(=7+4=11\) and 8 is not divisible by 9

Note: This rule can be repeated when needed, look at the following example:
Consider a number 16299
sum of digits of 16299\(=1+6+2+9+9=27\)
reapeat the rule for number 27
sum of digits of \(27=2+7=9\)
and,9 is divisible by 9 so,16299 is also divisible by 9.
Divisibility by 10: any integer ending with 0, is divisible by 10.
Example: Suppose we need to check if the number 170 is divisible by 10 or not
Last digit of number 170 is 0, so 170 is divisible by 10
Divisibility by 11: To check if a number is divisible by 11, subtract the last digit from the number formed by rest of the digits, if the result is a multiple of/divisible by 11, then the number is also divisible by 11.
Example: Suppose we need to check if the number 297 is divisible by 11 or not
Last digit of 297 is 7
Number formed by rest of the digits = 29
Now, subtract 7 from 29 and the result will be 29-7=22, which is a multiple of 11 (22 is divisible by 11), so 297 is divisible by 11.
Note: This rule can be repeated when needed, look at the following example:
Suppose we need to check if the number 14652 is divisible by 11 or not
Last digit of 14652 is 2
Number formed by rest of the digits = 1465
Now, subtract 2 from 1465 and the result will be \(1465-2=1463\),
Now, repeat the rule for 1463:
Last digit of 1463 is 3
Number formed by rest of the digits = 146
Now, subtract 3 from 146 and the result will be \(146-3=143\),
Again, repeat the rule for 143:
Last digit of 143 is 3
Number formed by rest of the digits = 14
Now, subtract 3 from 14 and the result will be \(14-3=11\), which is a multiple of 11 (11 is divisible by 11), so 14652 is divisible by 11.
Another example: Suppose we need to check if the number 976 is divisible by 11 or not
Last digit of 976 is 6
Number formed by rest of the digits = 97
Now, subtract 6 from 97 and the result will be \(97-6=91\), which is not a multiple of 11 (91 is not divisible by 11) so, 976 is not divisible by 11.
Another Rule to Check Divisibility By 11:
If in any integer, the sum of the digits at even position = the sum of the digits at odd position, then the number is divisible by 11.
OR if, the difference of the sum of the digits at even position and the sum of the digits at odd position =11, then the number is divisible by 11.
Example: Suppose we need to check if the number 1364 is divisible by 11 or not
Now, digits at Odd position are 4 and 3
sum of 4 and 3\(=4+3=7\)
Digits at Even position are 6 and 1
sum of 6 and 1\(=6+1=7\)
Sum of digits at Odd position = Sum of digits at Even position = 7
So, 1364 is divisible by 11.
Another example: Suppose we need to check if the number 1903 is divisible by 11 or not
Now, digits at Odd position are 3 and 9
sum of 3 and 9\(=3+9=12\)
Digits at Even position is 1 and 0
sum of 1 and 0\(=1+0=1\)
Difference of Sum of digits at Odd position and Sum of digits at Even position \(= 12-1=11\)
So, 1903 is divisible by 11.

Divisibility by 12: Any integer that is divisible by both 3 and 4 is also divisible by 12.
Example: Suppose we need to check if the number 660 is divisible by 12 or not
Check divisibility by 3: sum of digits of 660\(=6+6+0=12\), now 12 is divisible by 3 (because 12 is a multiple of 3 as \(12÷3=4)\) so, 660 is also divisible by 3.
Check divisibility by 4: number formed by last two digits of 660 is 60, now 60 is divisible by 4 (because 60 is a multiple of 4 as \(60÷4=15)\) so, 660 is also divisible by 4
Now we can see that the number 660 is divisible by both 3 and 4, so it’s also divisible by 12.
Another example: Suppose we need to check if the number 648 is divisible by 12 or not
Check divisibility by 3: sum of digits of \( 548=6+4+8=18\), now 18 is divisible by 3 (because 18 is a multiple of 3 as \(18÷3=6)\) so, 648 is also divisible by 3.
Check divisibility by 4: number formed by last two digits of 648 is 48, now 48 is divisible by 4 (because 48 is a multiple of 4 as \(48÷4=12)\) so, 648 is also divisible by 4
Now we can see that the number 648 is divisible by both 3 and 4, so it’s also divisible by 12.
Another example: Suppose we need to check if the number 224 is divisible by 12 or not
Check divisibility by 3: sum of digits of 224\(=2+2+4=8\), now 8 is not divisible by 3 (because 8 is not a multiple of 3) so, 224 is not divisible by 3.
Check divisibility by 4: number formed by last two digits of 224 is 24, now 24 is divisible by 4 (because 24 is a multiple of 4 as \(24÷4=6)\) so, 224 is also divisible by 4
Now we can see that the number 224 is not divisible by 3 but it is divisible by 4 only, so it is not divisible by 12.
Another example: Suppose we need to check if the number 549 is divisible by 12 or not
Check divisibility by 3: sum of digits of 549\(=5+4+9=18\), now 18 is divisible by 3 (because 18 is a multiple of 3 as \(18÷3=6)\) so, 549 is divisible by 3.
Check divisibility by 4: number formed by last two digits of 549 is 49, now 49 is not divisible by 4 (because 49 is not a multiple of 4) so, 548 is not divisible by 4
Now we can see that the number 548 is divisible by 3 but not divisible by 4, so it is not divisible by 12.
-- (Satinder Gill)

Angles and their types- Lesson

Angle: The space between two intersecting lines or surfaces at or close to the point where they meet.

Angles and their relation

The angles have around 7 types of relations, which are explained below:

Adverbs

An adverb is a word that modifies a verb, an adjective or another adverb, expressing manner, place, time or degree or certainty . Learn more at Adverbs and its kinds:

Adverbs of Manner

Adverbs of manner tell us how something happens. Click to learn more

Adverbs of Place

Adverbs of place tell us where something happens. Click to learn more:

Adverbs of Time

Adverbs of time tell us when an action happened, they also tell about how often and for how long the action happened Click to learn more:

Adverbs of Degree

Adverbs of degree tell us about the intensity of something means the degree or extent to which something happens Click to learn more:

Adverbs of Certainty

Adverbs of certainty express how certain we feel about an action or event Click to learn more:

Verbs

To learn about the Verbs and their types, check this well explained lesson:

Decimal numbers- Lesson1

A Decimal number is a number that has a decimal point.

A decimal point is a period (.) that separates a decimal number into two parts, The number before the decimal point is called the whole part and the number after the decimal point is called the decimal part.

For example, in the number 123.45, 123 is the whole part and 45 is the decimal part.

Formula Sheet- Surface Area and Volume-1

To learn all the formulas that you need to find Surface area and Volume of solid shapes, check this sheet:

Long Division

Long Division:The long division can be better learned by keeping in mind the four steps of the process:

1.) Divide

2.) Multiply

3.) Subtract

4.) Bring down

And also by understanding the four parts of division:

1.) Dividend: the number being divided

2.) Divisor: the number which divides the dividend

3.) Quotient: the number of parts the dividend has divided into

4.) Remainder: any part of the dividend which left over

Example: \(4325\div5\)

1. If you are dividing a number with a One Digit number,

look at the left-most digit of the dividend,

In our case left-most digit of dividend is 4

2. If the left-most digit is smaller than the divisor,

combine it with the number next right to it and put 0 in quotient.

Now 4 is smaller than 5,

so we will put 0 in quotient and combine 4 with the next digit to it which is 3

and we will get a new number as 43.

3. Divide: Now, check if 5 goes into 43? divide 43 lines in the group of 5

4. Multiply: find the closest and smallest number to 43 in the table of 5, which is:
\(5\times\mathbf{8}=\mathbf{40}\),

(remember step 3 where we divided 40 lines in 8 group

having 5 lines in each group and 3 were left over)

5. Subtract: write 40 under the number 43 and subtract, after subtraction we got 3

6. Bring down: the number we got in step 5 is smaller than the dividend,

so bring down the next digit of dividend and we got new number as 32

7. Repeat: now repeat the steps from 3 to 6 as: the closest and smallest number to
32 is \(5\times\mathbf{6}=\mathbf{30}\), write 30 under 32 and subtract and after subtraction we got 2, which is smaller than 5, so bring down the next digit of dividend and we will get new number as 25

8. Repeat: now repeat the steps from 3 to 6 as: the closest and smallest number to 25 is \(5\times\mathbf{5}=\mathbf{25}\), write 25 under 25 and subtract and after subtraction we will get 0.

Note: Repeat the above steps until we got remainder 0 (Zero) or there is no digit left in dividend to bring down.

--The Study Zone