Prasanna

Know the probability of tossing three coins here. Follow the various terminology and methods involved in probability. There are various methods for tossing three coins. Check all the important formulae, fundamental counting theorem to find the probability in the later modules. Furthermore, you can get an idea of how to find the number of possible choices.  Also, find the solved examples of finding the probability of 3 coins with solutions.

Tossing of 3 Coins

When you toss 3 coins simultaneously, the possibility of outcomes are (HHT), or (TTH) or (HHH) or (THT) or (THH) or (HTH) or (HTT) or  (TTT), where H is called the heads and T is called the tails.

Therefore, the total number of toss for outcome = 2³=8

Important Formulae

• The possible outcomes of tossing 3 coins are {(HHT), or (TTH) or (HHH) or (THT) or (THH) or (HTH) or (HTT) or  (TTT)}
• The total no of possible outcomes = 8
• The probability of getting at least one head = number of possibilities of heads as outcome/total no of possibilities = 3/8
• The probability of getting at least two heads = number of possibilities of 2 heads as the outcome/total no of possibilities = 3/8
• The probability of getting no tail = number of possibilities of no tail as the outcome/total no of possibilities = 1/8

Fundamental Counting Theorem to find the Probability

To find the number of possible choices, the complete event of tossing coins is divided by the events of tossing each coin. The outcomes of the event of tossing the coin multiple times are similar to the event of tossing multiple coins.

Example: Tossing the coin for 4 times is similar to tossing 4 coins at once.

How to find the Number of Possible Choices?

When the coin has tossed, there are 2 possible outcomes, which shows either a tail or head.

The event is denoted with the letter (E)

For the Total Event

The total event is the event of tossing the first coins.

1st Sub-Event

The 1st event of tossing the first coins

This event can be defined in 2 ways

n(SE1) = 2

2nd Sub-Event

The 2nd event of tossing the second coin

This event can be defined in 2 ways

n(SE2) = 2

Total number of possible choices

n(E) = n(SE1) * n(SE2) * n(SE3)…

The sequence continues as many possibilities are present.

2 * 2 * 2 * 2…

2(number of coins)

Probabilities of Tossing Coins

• The probability of coin-flipping for 2 times and getting 3 tails in a row
• In case you flip the coin 2 times, finding the probability of getting exactly 3 tails.
• The probability of getting 3 tails while flipping 2 coins.
• The probability of getting exactly 3 tails when a coin is tossed 2 times.
• Finding the probability of getting at least 3 tails when a coin is tossed 2 times.

Probability of Tossing or Flipping Three Coins Examples

Problem 1:

When two of the unbiased coins are tossed, what is the probability of both showing head?

Solution:

As given in the question,

No of unbiased coins are tossed = 2

The probability of showing both heads is

P(E) = No of favourable outcomes/Total no of outcomes

Total no of outcomes is represented with sample space (S)

S = {HH, HT, TH, TT}

n(S) = 4

P(E) = 1/4 = 0.25

Problem 2:

Three coins are tossed, then find the probability of getting at least two heads?

Solution:

As given in the question,

No of coins = 3

Sample set (S) = {HHH, HTT, HHT, HTH, THH, THT, TTH, TTT}

The set of the favourable outcomes is denoted by E

E = the probability of getting at least two heads

E = { HHH, HHT, HTH, THH} = n(E) = 4

The probability of getting at least two heads = No of favourable outcomes/Total no of outcomes

P (at least two heads) = 4/8 = 1/2 = 0.5

Problem 3:

Three coins are tossed, then find the probability of getting at least one head?

Solution:

As given in the question,

No of coins = 3

Sample set (S) = {HHH, HTT, HHT, HTH, THH, THT, TTH, TTT}

The set of the favourable outcomes is denoted by E

E = the probability of getting at least one head

E = { HHH, HHT, HTH, THH, HTH, THH, THT, TTH} = n(E) = 7

The probability of getting at least two heads = No of favourable outcomes/Total no of outcomes

P (at least two heads) = 7/8

How to find the Probability of Tossing Coins?

To find the probability of tossing coins, we have to follow various steps. They are listed in the below fashion and you can follow them to arrive at the solution easily.

1. Analyse the question and check for the sample space.
2. Now, note down the sample space (Total no of outcomes)
3. Find the favourable part by following the question and note down the event space.
4. Use the definition of probability i.e., No of favourable outcomes/Total no of outcomes and substitute the values in it.
5. Hence, get the final result and know the probability of tossing the coins.

Various Events of Tossing 3 Coins

1. A coin is tossed 3 times or 3 coins are tossed at a time and the results or the final value is recorded. Find the probability of finding the events?

A fair coin is tossed thrice or 3 unbiased coins are tossed at a time.

The sample space for the event is

Sample set (S) = {HHH, HTT, HHT, HTH, THH, THT, TTH, TTT}

Therefore, n(S) = 8

a) Event of getting exactly one head

Let C be the event of getting exactly one head

Therefore, C = { HTT, THT, TTH}

n(C) = 3

From the definition, the probability of getting exactly one head

P(C) = n(C)/n(S) = 3/8

Thus, the probability of getting exactly one head = 3/8

b) Event of getting exactly two heads

Let A be the event of getting exactly two heads

Therefore, A = {HHT, HTH, THH}

n(A) = 3

From the definition, the probability of getting exactly two heads

P(A) = n(A)/n(S) = 3/8

Thus, the probability of getting exactly two heads = 3/8

c) Event of getting all heads

Let B be the event of getting all heads

Therefore, B = (HHH)

n(B) = 1

From the definition, the probability of getting all heads

P(B) = n(B)/n(S) = 1/8

Thus, the probability of getting all heads = 1/8

d) Event of getting two or more heads

Let X be the event of getting two or more heads

Therefore, X = { HHH, HHT, HTH, THH, HTH, THH, THT, TTH}

n(X) = 7

From the definition, the probability of two or more heads

P(X) = n(X)/n(S) = 7/8

Thus, the probability of getting two or more heads = 7/8

e) Event of getting at least 1 head

Let B be the event of getting at least 1 head

Therefore, B = (HTT, THT, TTH, HHT, HTH, THH, HHH)

n(B) = 7

From the definition, the probability of at least 1 head

P(B) = n(B)/n(S) = 7/8

Thus, the probability of getting at least = 7/8

f) Event of getting atmost 2 heads

Let Y be the event of getting atmost 2 heads

Therefore, B = (HTT, THT, TTH, HHT, HTH, THH)

n(Y) = 6

From the definition, the probability of getting atmost 2 heads

P(Y) = n(Y)/n(S) = 6/8

Thus, the probability of getting atmost 2 heads = 3/4

One of the most important concepts in a Mathematical journey is Even numbers and Odd numbers. Students who learn basic maths will start with even & odd numbers after arithmetic operations. These Even and odd numbers calculations are crucial for many students who are studying computer science engineering. Those students also practice the logic behind the even and odd number concept to code and generate an easy way of determining a given number is even or odd. Here, we have shared the List of Even and Odd Numbers Between 1 and 100 along with the definition and examples.

Even and Odd Numbers Between 1 and 100 Chart

Here is the chart for Even and Odd Numbers Between 1 and 100. Even numbers from 1 to 100 are marked in blue color and the rest of the numbers are odd numbers from 1 to 100. Check out the chart and find the given number is even or odd easily.

Now, we will see the definitions for both even and odd numbers with examples for a clear understanding of the concept:

What are Even Numbers?

The numbers that are divided by 2 evenly are called Even numbers. However, even numbers cannot have decimals. One of the common ways to check the number is even or not is when the last digit of the number is having 0, 2, 4, 6, or 8. In the mathematical form, we can find an even number by using ‘2n’, where n is an integer,

Example: Find the number 28 is an even number or not?

Solution: We know that the mathematic form is 2n for finding the even number, so

2n = 28
n = 28/2 = 14

Hence, 28 is divided by 2 so it is an even number.

What are Odd Numbers?

The definition of Odd numbers is any number that cannot be divided by two. In short, a number in form of 2k+1, where k ∈ Z (i.e. integers) are called odd numbers. These numbers are not evenly divided by 2, which implies there is some remainder left after division. For example, 1, 3, 5, 7, etc.

Example: Find 85 is an odd number or not?

Solution: We know 2k + 1 are the odd numbers

To find the number is odd or not, we have to check the unit place is having an odd number.

Hence, 85 is an Odd number.

List of Even Numbers Between 1 and 100

Below are the even numbers that exist between 1 and 100:

2             4             6             8           10

12           14           16           18           20

22           24           26           28           30

32           34           36           38           40

42           44           46           48           50

52           54           56           58           60

62           64           66           68           70

72           74           76           78           80

82           84           86           88           90

92           94           96           98           100

List of Odd Numbers Between 1 and 100

The following are the odd numbers from 1 to 100:

1             3             5             7             9

11           13           15           17           19

21           23           25           27           29

31           33           35           37           39

41           43           45           47           49

51           53           55           57           59

61           63           65           67           69

71           73           75           77           79

81           83           85           87           89

91           93           95           97           99

FAQs on Even and Odd Numbers between 1 and 100

1. Is 55 an odd number or even?

As we can see 55 number is even or odd in the list of an odd number above. If not, check the unit place of 55, if it consists of an odd number that is not evenly divisible by 2, then 55 is an odd number.

2. How do you find if a number is even or odd?

When the given number is divisible by 2 evenly then it’s an even number else odd number. If the number is bigger like thousands or millions, check the unit place. If the unit place contains an even number, then the bigger number is even else it’s an odd one.

3. Where can I get the list of even and odd numbers between 1 and 100?

You can get the even and odd numbers between 1 and 100 list elaborately from this page of a reliable website.

In the initial stages, you might feel Maths Problems complex and difficult? Once, you understand the formula you will no longer the concept or problems difficult. Finding the Fractional Part of a Whole Number may seem a bit difficult if you aren’t aware of the formula. However, the formula for finding a fraction of a whole number involves simple division and multiplication. Learn the step-by-step procedure on How to find a Fraction of a Whole Number and also check the solved examples provided for better understanding.

How to find a Fraction of a Whole Number?

Finding the Fraction of a Whole Number is the same as multiplying the number and fraction. This method is quite simple when it comes to whole numbers. However, to solve the problems you need to know about basic multiplication and division.

• Obtain the numbers.
• Multiply the whole number with the numerator of the fraction. The denominator remains unchanged during the entire process.
• Divide the product by the denominator of the fraction.

Solved Examples on Finding a Fraction of a Number

1. Find \(\frac { 2}{ 3 } \) of 22?

Solution:

To find \(\frac { 2}{ 3 } \) of 22 multiply the numerator 2 with the whole number 22 and then divide the product with denominator 3.

i.e. \(\frac { 2}{ 3 } \) . 22

= \(\frac { 2.22}{ 3 } \)

= \(\frac { 44}{ 3 } \)

2. Find \(\frac { 3}{ 5 } \) of 15?

Solution:

To find \(\frac { 3}{ 5 } \) of 15 multiply the numerator 3 with the whole number 15 and then divide the product with denominator 5 i.e.

= \(\frac { 3}{ 5} \) . 15

= \(\frac { 3.15}{ 5 } \)

= \(\frac { 45}{ 5 } \)

= 9

3. Find \(\frac { 3 }{ 6 } \) of 54?

Solution:

To find \(\frac { 3}{ 6 } \) of 54 multiply the numerator 3 with the whole number 54 and then divide the product with denominator 6 i.e.

= \(\frac { 3}{ 6} \) . 54

= \(\frac { 3.54}{ 6} \)

= 27

Are you looking for ways on how to change Fractions? Then, take the help of this article and learn how to convert an improper fraction to a mixed number or whole number, from whole number or mixed number to improper fraction easily. Refer to Procedures on Changing Fractions to understand the concept better. Practice the Example Problems for Converting Improper Fractions to Mixed Number and Vice Versa.

How to Convert an Improper Fraction to Mixed Number?

To Change an Improper Fraction to Mixed Number follow the below-listed steps. They are as follows

• Divide the numerator with the denominator
• Note down the whole number
• Later make a note of the remainder as a numerator above the denominator

Example:

Convert \(\frac { 9 }{ 4 } \) to a Mixed Fraction?

Solution:

Step 1: Divide the Numerator with Denominator i.e. 9÷4 = 2 with a remainder of 1

Step 2: Write the Quotient obtained as the Whole Number. Note down the Remainder as Numerator in the Fraction while keeping the denominator of the fraction unchanged.

Step 3: Thus, we get 2 \(\frac { 1 }{ 4 } \)

Therefore, improper Fraction \(\frac { 9 }{ 4 } \) converted to Mixed Fraction is 2 \(\frac { 1 }{ 4 } \)

How to Convert a Mixed Number to Improper Fraction?

To convert a Mixed Number to an Improper Fraction go through the below-mentioned steps. They are as such

• Multiply the whole number part with the fraction’s denominator
• Add the result to the numerator.
• Later, place the result obtained in the earlier step as a numerator while keeping the denominator unaltered.

Example:

Convert 4 \(\frac { 3 }{ 5 } \) to Improper Fraction?

Solution:

Step 1: Multiply the whole number part with the denominator of the fraction i.e. 4*5 = 20

Step 2: Add the result to the numerator of the fraction i.e. 20+3 = 23

Step 3: Place the numerator value obtained in the earlier step and keep the denominator unaltered i.e. \(\frac { 23 }{ 5 } \)

Therefore Mixed Number \(\frac { 3 }{ 5 } \)  converted to Improper Fraction is \(\frac { 23 }{ 5 } \)

How to Convert a Whole Number to Improper Fraction?

Go through the following steps to change from a whole number to Improper Fraction and they are outlined as below

• To change a whole number to Improper Fraction simply place the whole number part as the numerator over the denominator 1.
• The resultant is called a reduced improper fraction.
• If you want to change it to an unreduced improper fraction you need to multiply the whole number with a fraction equivalent of 1.

Example

Convert 13 to Improper Fraction?

Solution:

In order to express Whole numbers as Improper Fractions you just need to place the whole number as the numerator over the denominator 1.

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

13 converted to Improper Fraction is \(\frac { 13 }{ 1 } \)

FAQs on Changing Fractions

1. What is an Improper Fraction?

An Improper Fraction is a Fraction in which the numerator is greater than the denominator.

2. What is meant by Mixed Fraction?

A Fraction represented with its quotient and remainder is called a Mixed Fraction or Mixed Number. In other words, the mixed fraction is a combination of a whole number and a proper fraction.

3. How to Convert an Improper Fraction into Mixed Number?

Divide the Numerator with the Denominator. Take the Quotient obtained as a Whole Number and the Remainder as Numerator in the Proper Fraction while keeping the Denominator Unchanged.

Do you want to expand trinomials easily without any confusion and hassle? You should refer to this page. Here, we have explained how to expand the Square of a Trinomial and perfect square trinomial definition and formulas. Students who need more subject knowledge about square trinomials and solve any kind of trinomial expansions must go with this article completely. In this article, you will also get some worked-out examples on Square of a Trinomial and Perfect square trinomial. So, let’s continue your read and learn the concept of square trinomial.

Perfect Square Trinomial Definition & Formula

An expression obtained from the square of the binomial equation is a perfect square trinomial. When the trinomial is in the form ax² + bx + c then it is said to be a perfect square, if and only if it meets the condition b² = 4ac.

The Perfect Square Trinomial Formula is as follows,

(ax)²+2abx+b² = (ax+b)²
(ax)²−2abx+b² = (ax−b)²

How to Expand the Square of a Trinomial?

Here, we are discussing the expansion of the square of a trinomial (a + b + c).

Let (b + c) = x

(i) Then (a + b + c)² = (a + x)² = a² + 2ax + x²
= a² + 2a (b + c) + (b + c)²
= a² + 2ab + 2ac + (b² + c² + 2bc)
= a² + b² + c² + 2ab + 2bc + 2ca

Therefore, (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca.

(ii) (a – b – c)² = [a + (-b) + (-c)]²
= a² + (-b²) + (-c²) + 2 (a) (-b) + 2 (-b) (-c) + 2 (-c) (a)
= a² + b² + c² – 2ab + 2bc – 2ca

Therefore, (a – b – c)² = a² + b² + c² – 2ab + 2bc – 2ca.

(iii) (a + b – c)² = [a + b + (-c)]²
= a² + b² + (-c)² + 2ab + 2 (b) (-c) + 2 (-c) (a)
= a² + b² + c² + 2ab – 2bc – 2ca

Therefore, (a + b – c)² = a² + b² + c² + 2ab – 2bc – 2ca.

(iv) (a – b + c)² = [a + (- b) + c]²

= a² + (-b²) + c² + 2 (a) (-b) + 2 (-b) (-c) + 2 (c) (a)
= a² + b² + c² – 2ab – 2bc + 2ca

Therefore, (a – b + c)² = a² + b² + c² – 2ab – 2bc + 2ca.

Solved Examples on Square Trinomial

1. Expand (x+4y+6z)²

Solution:

Given trinomial expression is (x+4y+6z)²

We know that (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca.

Here a=1x, b=4y, c=6z

Now, substitute the values in the expression of (a + b + c)²

Then (x+4y+6z)² = (1x)² + (4y)² + (6z)² + 2(1x)(4y) + 2(4y)(6z) + 2(6z)(1x)

= x² + 16y² + 36z² + 8xy + 48yz + 12zx

Hence, (x+4y+6z)² = x² + 16y² + 36z² + 8xy + 48yz + 12zx. 

2. Is the trinomial x² – 6x + 9 a perfect square?

Solution:

Given trinomial is x² – 6x + 9, now calculate the expression and find it is a perfect square or not.

x² – 6x + 9 = x² – 3x – 3x + 9
= x(x – 3) – 3(x – 3)
= (x – 3)(x – 3)

Otherwise,

x² – 6x + 9 = x² – 2(3)(x) + 3² = (x – 3)²

The factors of the given equation are a perfect square.

Therefore, the given trinomial is a perfect square.

3. Simplify a + b + c = 16 and ab + bc + ca = 40. Find the value of a² + b² + c².

Solution:

As per the given question, a + b + c = 16

Now, by squaring both sides, we get

(a+ b + c)² = (16)²

a² + b² + c² + 2ab + 2bc + 2ca = 256

a² + b² + c² + 2(ab + bc + ca) = 256

a² + b² + c² + 2 × 40 = 256 [Given, ab + bc + ca = 40]

a² + b² + c² + 80 = 256

At this step, we have to subtract 80 from both sides

a² + b² + c² + 80 – 80 = 256 – 80

a² + b² + c² = 176

Hence, the square of a trinomial formula will help us to expand and get the result for a² + b² + c² is 176. 

Trigonometric Identities are useful when you are dealing with Trigonometric Functions in an Algebraic Expression. Usually, the Trig Identities involve certain functions of one or more angles. There are Several Identities involving the angle of a triangle and side length.  Check out all Fundamental Trigonometric Identities derived from Trigonometric Ratios using Worksheet on Trigonometric Identities. Practice the List of Trigonometric Identities, their derivation, and problems easily taking the help of the Trig Identities Worksheet with Answers.

List of Trigonometric Identities

There are several Trigonometric Identities that are used while solving Trigonometric Problems. Have a glance at the basic or fundamental trigonometric identities listed below and make your job simple. They are as follows

Pythagorean Identities

• sin² a + cos² a = 1
• 1+tan² a  = sec²a
• cosec² a = 1 + cot² a

Reciprocal Identities

• Sin θ = \(\frac { 1 }{ Csc θ } \) or Csc θ = \(\frac { 1 }{ Sin θ } \)
• Cos θ = \(\frac { 1 }{ Sec θ } \) or Sec θ = \(\frac { 1 }{ Cos θ } \)
• Tan θ = \(\frac { 1 }{ Cot θ } \) or Cot θ = \(\frac { 1 }{ Tan θ } \)

Opposite Angle Identities

• Sin (-θ) = – Sin θ
• Cos (-θ) = Cos θ
• Tan (-θ) = – Tan θ
• Cot (-θ) = – Cot θ
• Sec (-θ) = Sec θ
• Csc (-θ) = -Csc θ

Complementary Angles Identities

• Sin (90 – θ) = Cos θ
• Cos (90 – θ) = Sin θ
• Tan (90 – θ) = Cot θ
• Cot ( 90 – θ) = Tan θ
• Sec (90 – θ) = Csc θ
• Csc (90 – θ) = Sec θ

Ratio Identities

• Tan θ = \(\frac { Sin θ }{ Cos θ } \)
• Cot θ = \(\frac { Cos θ }{ Sin θ } \)

Angle Sum and Difference Identities

Consider two angles , α and β, the trigonometric sum and difference identities are as follows:

• sin(α+β)=sin(α).cos(β)+cos(α).sin(β)
• sin(α–β)=sinα.cosβ–cosα.sinβ
• cos(α+β)=cosα.cosβ–sinα.sinβ
• cos(α–β)=cosα.cosβ+sinα.sinβ
• tan(α+β) = \(\frac { tanα+tanβ }{ 1-tanα.tanβ } \)
• tan(α-β) = \(\frac { tanα-tanβ }{ 1+tanα.tanβ } \)

Prove the following Trigonometric Identities

1. (1 – cos²θ) csc²θ  =  1?

Solution:

Let us consider L.H.S =  (1 – cos²θ) csc²θ  and  R.H.S  =  1.

L.H.S =  (1 – cos²θ) csc²θ

We know sin²θ + cos²θ  =  1,

sin²θ  =  1 – cos²θ

L.H.S =  sin²θ ⋅ csc²θ

We also know csc²θ =  1/ sin²θ

L.H.S =  sin²θ ⋅  1 / sin²θ

L.H.S = 1

Hence Proved, L.H.S = R.H.S

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2. Prove tan θ sin θ + cos θ  =  sec θ

Solution:

Let L.H.S  =  tan θ sin θ + cos θ  and R.H.S =  sec θ.

L.H.S =  tan θ sin θ + cos θ

We know tanθ = \(\frac { Sin θ }{ Cos θ } \)

L.H.S = \(\frac { Sin θ }{ Cos θ } \) ⋅ sin θ + cos θ

L.H.S =  Sin² θ / Cos θ+ cos θ

L.H.S = (sin²θ/cos θ) + (cos²θ/cosθ)

L.H.S = (sin²θ + cos²θ) / cos θ

L.H.S= \(\frac { 1}{ cos θ } \)

L.H.S = 1 / cos θ

= Sec θ

Therefore, L.H.S = R.H.S

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3. Prove cot θ + tan θ  =  sec θ csc θ?

Solution:

Let L.H.S  =  cot θ + tan θ and R.H.S  =  sec θ csc θ.

L.H.S =  cot θ + tan θ

L.H.S =  \(\frac { Cos θ }{ Sin θ } \) + \(\frac { Sin θ }{ Cos θ } \)

L.H.S = (cos²θ/sin θ cos θ) + (sin²θ/sin θ cos θ)

L.H.S = (cos²θ + sin²θ) / sin θ cos θ

L.H.S = \(\frac { 1 }{ sin θ cos θ } \)

L.H.S = \(\frac { 1 }{ cos θ } \)⋅ \(\frac { 1 }{ sin θ } \)

L.H.S =  sec θ csc θ

L.H.S = R.H.S

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4. Prove sec θ √(1 – sin²θ)  =  1?

Solution:

Let L.H.S  =  sec θ √(1 – sin²θ)  and R.H.S  =  1.

L.H.S =  sec θ √(1 – sin²θ)

We know sin²θ + cos²θ  =  1, we have

cos²θ  =  1 – sin²θ

Then,

L.H.S  =  sec θ √cos²θ

L.H.S =  sec θ ⋅ cos θ

L.H.S =  sec θ ⋅ \(\frac { 1 }{ sec θ } \)

L.H.S = \(\frac { sec θ }{ sec θ } \)

L.H.S =  1

L.H.S =  R.H.S

---

5. Prove (1 – cos θ)(1 + cos θ)(1 + cot²θ)  =  1

Solution:

Let L.H.S  =  (1 – cos θ)(1 + cos θ)(1 + cot²θ)  =  1 and R.H.S  =  1.

L.H.S =  (1 – cos θ)(1 + cos θ)(1 + cot²θ)

L.H.S =  (1 – cos²θ)(1 + cot²θ)

We know sin²θ + cos²θ  =  1, we have

sin²θ  =  1 – cos²θ

Then,

L.H.S =  sin²θ ⋅ (1 + cot²θ)

L.H.S =  sin²θ  + sin²θ ⋅ cot²θ

L.H.S =  sin²θ  + sin²θ ⋅ (cos²θ/sin²θ)

L.H.S =  sin²θ + cos²θ

L.H.S =  1

Therefore, L.H.S = R.H.S

---

6. Prove tan⁴θ + tan²θ  =  sec⁴θ – sec²θ?

Solution:

Let L.H.S  =  tan⁴θ + tan²θ  and R.H.S  =  sec⁴θ + sec²θ.

L.H.S =  tan⁴θ + tan²θ

L.H.S =  tan²θ (tan²θ + 1)

We know that,

tan²θ  =  sec²θ – 1

tan²θ + 1  =  sec²θ

Then,

L.H.S =  (sec²θ – 1)(sec²θ)

L.H.S  =  sec⁴θ – sec²θ

---

7. Prove √{(sec θ – 1)/(sec θ + 1)}  =  cosec θ – cot θ?

Solution:

Let L.H.S  = √{(sec θ – 1)/(sec θ + 1)} and R.H.S  =  cosec θ – cot θ

= √{(sec θ – 1)/(sec θ + 1)}

= √[{(sec θ – 1) (sec θ – 1)}/{(sec θ + 1) (sec θ – 1)}]

= √{(sec θ – 1)² / (sec²θ – 1)}

= √{(sec θ – 1)² / tan²θ}

= (sec θ – 1)/tan θ

=  (sec θ/tan θ) – (1/tan θ)

=  {(1/cos θ)/(sin θ/cos θ)} – cot θ

= {(1/cos θ) ⋅ (cos θ/sin θ)} – cot θ

=  (1/sin θ) – cot θ

= cosec θ – cot θ

Therefore, L.H.S = R.H.S

---

Wondering what are the Terms used in Division mathematical operation? Then, this page will explain to you all about Division Terms along with some solved examples. Students and Teachers who are searching for a clear explanation about the Terms of Division can make their look at this guide and understand the Division operation so easily. Let’s start with the definition of Division and then we will discuss the Terms used in Division with a brief explanation and worked-out problems.

Also Check: Properties of Division of Integers

Definition of Division

In maths, Division is a process of distributing a group of things into equal parts. Also, the division is one of the basic arithmetic operations that provides a fair output of sharing. In other words, the division is also a repeated subtraction.

However, the division is an operation inverse of multiplication. Mathematical Notation for Division is ÷, /. For example 13 ÷ 5 or 5/7.

Each and every part concerned in the Division Equation has special names.

Dividend ÷ Divisor = Quotient & Remainder

Let’s see the explanation of each part involved in the division equation from the below module.

Terms Used in Division – Explanation

There are four terms that represent the four numbers in a division sum. They are as such,

Dividend: Number that is being divided in the division method.
Divisor: Number that divides the dividend exactly.
Quotient: Number that results after the completion of the division process.
Remainder: “Left Over Number” after the division of the given number.

Facts About Division

• Division by zero (0) is undefined.
• The division of the same numerator and denominator is always One (1). For instance: 5/5 = 1.
• If you divide any number by one then the answer will be the original number. It means, when the divisor is 1 then the quotient will be equal to the dividend. For example: 20 ÷ 1= 20.

Solved Examples of Division Terms

1. Divide 65 by 5 & identify the Terms used in Division

Solution:

Here, the division process is shown in an image along with the division terms. So check out the image and get to know the terms of division:

Here, the final values of division terms are:

Dividend = Divisor × Quotient + Remainder ⇒ 65 = 5 × 13 + 0.

2. 20 Divided by 5

Solution:

First, divide 20 by 5 and then know which value is referred to which division term,

20 ÷ 5 = 4 ,

Here, 20 is Dividend

5 is Divisor

4 is Quotient

0 is Remainder ( Because there is no leftover number after the division process).

3. Divide 49474 by 7

Solution:

Given division expression is \(\frac { 49474 }{ 7 } \)

Now, we have to divide the 49474 by 7 and get the result.

After the division process, we get the outcome as \(\frac { 49474 }{ 7 } \) = 7067 R 5.

Here, the terms used in the division of \(\frac { 49474 }{ 7 } \) are 49474 is the dividend, 7 is the divisor, 7067 is the quotient and 5 is the remainder of the division.

For more division problems click on the Examples on Division of Integers and solve all the problems easily.

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