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Worksheet on Fractions is here. Know the step-by-step procedure to solve fraction problems. Refer to addition, subtraction, multiplication, and division of fractions to know the different model problems. Follow different terminology and rules to solve fraction problems. Various methods, rules, formulae involved in fractions operations are here. Go through the below sections to know more about the fractions.

Solved Problems on Simplifying Fractions

Problem 1:

Sameer has read three-fifth of the 75 pages of his book. How many more pages he needs to read to complete the book?

Problem 2:

Ravi had Rs.675. He gave \(\frac { 13 }{ 15 } \) of the amount to Sam. Sam spent \(\frac { 9 }{ 15 } \) of the amount given to him. How much money is Sam left with?

Problem 3:

Jagadish spends \(\frac { 3 }{ 5 } \) of his income on rent and \(\frac { 1 }{ 4 } \) of remainder amount on grocery. He saves the remaining amount.

a) What fraction of income does he save?

b) If he saves Rs.1029 per month, what is his monthly income?

Problem 4:

A baby rabbit was born \(\frac { 3 }{ 4 } \) of a month early. When it was born, its weight was \(\frac { 7 }{ 8 } \) kilogram. The weight at birth was \(\frac { 9 }{ 10 } \) kilogram less than the average weight of a newborn rabbit. What is the average weight of a newborn rabbit?

Problem 5:

In the zoo, \(\frac { 3 }{ 8 } \) of the staff are male. \(\frac { 7 }{ 12 } \) of the staff works part-time at the zoo. What is the fraction of the staff that is female?

Problem 6:

The lions in the zoo are fed three times a day. During the morning feeding, \(\frac { 2 }{ 15 } \) tons of meat is fed. During the afternoon feeding, the weight of the meat will be \(\frac { 1 }{ 15 } \) ton meat fed during the morning. If the total weight of the meat in a day is \(\frac { 1 }{ 2 } \) ton, how much is fed during the feeding at night?

Problem 7:

The monkey nursery is open two times a day: \(\frac { 2 }{ 3 } \) hours at noon and \(\frac { 5 }{ 12 } \) hours in the afternoon. How much time is the monkey nursery open everyday?

Problem 8:

An octopus weighed \(\frac { 5 }{ 6 } \) kilogram. After two weeks, its weight was increased by \(\frac { 3 }{ 10 } \) kilogram. But afterwards, it lost \(\frac { 1 }{ 5 } \) kilogram in weight as it was sick. What was the current weight?

Problem 9:

Two types of fish can be found in a small tank that is 5\(\frac { 1 }{ 7 } \) feet long. A blue fish is \(\frac { 2 }{ 15 } \) feet long and an orange fish is \(\frac { 7 }{ 10 } \) feet long. How much longer is the orange fish?

Problem 10:

Ila read 25 pages of a book containing 100 pages. Lalitha read \(\frac { 2 }{ 5 } \) of the same book. Who read less?

Problem 11:

Javed was given \(\frac { 5 }{ 7 } \) of a basket of apples. What fraction of apples are left in the basket?

Problem 12:

Reena was given 1\(\frac { 1 }{ 2 } \) piece of cake and Meena was given 1\(\frac { 1 }{ 3 } \) piece of cake. Find total amount was given to both of them?

Problem 13:

Bailey’s restaurant bought 6\(\frac { 1 }{ 3 } \) pounds of onions. The restaurant bought 8 times as much potatoes as onion. How many pounds of potatoes did the restaurant buy?

Problem 14:

Betsy has \(\frac { 2 }{ 5 } \) of a cup of powdered sugar. He sprinkles \(\frac { 3 }{ 4 } \) of the powdered sugar onto a plate of brownies. How much sugar does Betsy sprinkle on the brownies?

Problem 15:

At Goshen High School, \(\frac { 1 }{ 2 } \) of the students play a sport. Of the students who play a sport, \(\frac { 1 }{ 2 } \) play football. What fraction of the students play football?

Get detailed information regarding the division of fractions here. Check all the important terms, formulae, and the usage of fractions division. Refer step by step procedure to solve the fraction division problems. Know the various methods to solve proper, improper, mixed fractions, etc. Follow the complete guide to know the day-to-day usage of fractions division problems. Go through the below sections to follow the importance and problems of division on fractions.

Division of Fractions – Introduction

Dividing the fractions also involves various methods and also a lengthy process. First of all, you have to know the difference between proper, improper, and mixed fractions. Once you come to know the difference, then you can easily analyze the method to solve them. As we have seen, for the multiplication, we have to make the fractions equivalent.

In the same way, the division of fractions requires the equivalent fractions to solve them. First, make sure you make the fractions equivalent and then follow all the steps in dividing the fractions. Generally, fraction division in the direct method requires more effort. Hence, we are providing an alternative method here.

How to Divide Fractions?

Division of fractions can be done by multiplying the fractions by writing the reciprocal of one of fraction numbers or by reversing one of two fraction integers. Reciprocal of the fraction number or reverse of the fraction number means, if the fraction is \(\frac { a }{ b } \), then \(\frac { b }{ a } \) is its reciprocal. Interchanging the position of numerator and denominator is nothing but reversing the fraction. Fractions division can be classified into 3 ways.

• Fractions division by a fraction
• Fractions division by a mixed fraction
• Fractions division by a whole number

Fractions Division by a Fraction

Fractions division can be done in three steps. For division, we convert into the multiplication of fractions and obtain the desired result. The three steps are as follows.

1. Convert the second fraction into its reciprocal and multiply it with the first fraction.
2.  Multiply the denominator and numerator of both the fractions.
3. Next, simplify the fraction numbers.

For example, suppose that \(\frac { a }{ b } \) is divided by the fraction \(\frac { c }{ d } \). We can solve the division as:

• \(\frac { a }{ b } \) ÷ \(\frac { c }{ d } \) = \(\frac { a }{ b } \) * \(\frac { d }{ c } \)
• \(\frac { a }{ b } \) ÷\(\frac { c }{ d } \) = \(\frac { a*d }{ b*c } \)
• \(\frac { a }{ b } \) ÷ \(\frac { c }{ d } \) = \(\frac { ad }{ bc } \)

As given above, we can divide the fractions in three ways. Further simplification has to be done to get the exact result.

Fractions Division by a Whole Number

Fractions Division by a whole number is the easiest process. Follow the below steps to divide a whole number.

1. The whole number is nothing but the real numbers which include zero and all positive integers. All the whole numbers can be written as the fraction values if we give the denominator value as 1.
2. Find the reciprocal of the given number
3. Now, multiply the value of the fraction by a given fraction.
4. Then, simplify the equation to get its lowest terms.

Example:

Divide the fraction \(\frac { 6 }{ 5 } \) by 10?

Solution:

As given in the question,

The equation is \(\frac { 6 }{ 5 } \) by 10.

Step 1: Conversion into the whole number

As 10 is the whole number, we convert it into fractional value i.e., we write it as \(\frac { 10 }{ 1 } \)

Step 2: Find the reciprocal of the fraction

To find the reciprocal, we get \(\frac { 1 }{ 10 } \)

Step 3: Multiply both the fractions

Multiply the fractions, \(\frac { 6 }{ 5 } \) x \(\frac { 1 }{ 10 } \)

Step 4: Now, multiply the numertors and denominators of the fractions i.e., \(\frac { 6×1 }{ 15×10 } \)

Step 5: Simplify the equation

The resutant is  \(\frac { 3 }{ 25 } \)

Fractions Division by a Mixed Fraction

Fractions Division by a mixed fraction is almost similar to fractions division by fraction. The following are the steps to divide mixed fractions.

1. Conversion of mixed fraction into an improper fraction
2. Now, find the reciprocal of the improper fraction
3. Multiply the resultant fraction by the given fraction value.
4. Simplify the fractions.

Example:

Divide the fraction \(\frac { 2 }{ 5 } \) by 3\(\frac { 1 }{ 2 } \)?

Solution:

As given in the question,

The equation is \(\frac { 2 }{ 5 } \) by 3\(\frac { 1 }{ 2 } \)

Step 1: Conversion of mixed fraction into improper fraction

Convert, 3\(\frac { 1 }{ 2 } \), we get the result as 7/2

Step 2: Now, convert the reciprocal of the improper fraction, we get \(\frac { 2 }{ 7 } \)

Step 3: Multiply \(\frac { 2 }{ 5 } \) and \(\frac { 2 }{ 7 } \)

Step 4: Multiply the numertors and denominators of the fractions i.e., \(\frac { 2×2 }{ 5×7 } \)

Step 5: Simplify the fraction

the result value is \(\frac { 4 }{ 35 } \)

Fractions Division of Decimal Values

In the above sections, we have seen how to divide the fractions into three steps. Now, we see how to divide decimals with examples.

To convert the decimal values into natural numbers, we multiply both the numerator and denominator by 10.

Example:

Divide \(\frac { 0.5 }{ 0.2 } \)

Solution:

As given in the question,

The equation is \(\frac { 0.5 }{ 0.2 } \)

To divide the decimal values, convert the values into natural numbers by multiplying both numerator and denominator with 10.

Therefore, \(\frac { 0.5×10 }{ 0.2×10 } \)

We get the solution as, \(\frac { 5 }{ 2 } \) = 2.5

Also, we use the method of dividing fractions to solve the problem.

We can write 0.5 as \(\frac { 5 }{ 10 } \) and 0.2 as \(\frac { 2 }{ 10 } \)

Therefore, to find the solution of \(\frac { 5÷10 }{ 2÷10 } \)

Take the reciprocal of the second fraction i.e., \(\frac { 10 }{ 2 } \)

= \(\frac { 5*10 }{ 10*2 } \)

= \(\frac { 50 }{ 20 } \)

= \(\frac { 5 }{ 2 } \)

= 2.5

Dividing Fractions Examples

Problem 1: Lucy has \(\frac { 1 }{ 5 } \) of a bag of dog food left. She is splitting it between her 3 dogs evenly. What fraction of the original bag does each dog get?

Solution:

As given in the question,

Amount of dog food left = \(\frac { 1 }{ 5 } \)

No of dogs = 3

The fraction of the original bag each dog gets = \(\frac { 1/5 }{ 3 } \)

= \(\frac { 1 }{ 5 } \) x \(\frac { 1 }{ 3 } \)

= \(\frac { 1 }{ 15 } \) of the bag each dog gets.

Thus the final solution is \(\frac { 1 }{ 15 } \)

Problem 2: AJ has \(\frac { 1 }{ 4 } \) of a gallon of saltwater that he is using for an experiment. He needs to evenly separate the saltwater into separate beakers. How much salt water will be in each beaker?

Solution:

As given in the question,

Amount of salt water he is using for an experiment = \(\frac { 1 }{ 4 } \)

No of beakers = 3

Amount of salt water in each beaker = \(\frac { 1/4 }{ 3 } \)

= \(\frac { 1 }{ 4 } \) x \(\frac { 1 }{ 3 } \)

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

Therefore, \(\frac { 1 }{ 12 } \) gallons of salt water will be in each beaker.

Problem 3: Devin has a board that measures 4 ft in length. The board is going to be cut into \(\frac { 1 }{ 4 } \) ft pieces. How many pieces will Devin split the board into?

Solution:

As given in the question,

Length of the board = 4 ft

No of pieces the board is going to be cut = \(\frac { 1 }{ 4 } \)

No of pieces Devin split the board = 4 ÷ \(\frac { 1 }{ 4 } \)

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

= \(\frac { 16 }{ 1 } \)

Hence, the final solution is 16 pieces.

Therefore, Devin will split the board into 16 pieces.

Have you ever wondered why certain numbers divide evenly into a number while others will not? Divisibility Rules or Tests will help you determine whether the given number actually divides the other number without actually dividing. Maths may not be easy for some of us and to help you we have listed the guidelines on whether a Number is Divisible by 6 or not. Check out Solved Examples for testing whether a number is divisible by 6 or not.

Divisibility Rule of 6 – Definition

A number is divisible by 6 if it is divisible by both 2 and 3. To know whether a number is divisible by 6 one needs to know both the Divisibility Rules of 2 and 3 in advance.

How to Check whether a Number is Divisible by 6 or not?

Go through the below Divisibility Guidelines of 6 to know whether a number is divisible by 6 or not. To know if a number is divisible by 6 it has to meet two conditions in prior and they are along the lines

• It has to be divisible by 2 i.e. the last digit in the number should be even.
• It has to be divisible by  3 i.e. the sum of digits must be a multiple of 3.

If both the conditions are met then the given number is also a multiple of 6 and divisible by 6.

Solved Examples on Divisibility Tests for 6

Find whether the following numbers are divisible by 6 or not

1. 48

Solution:

Given Number is 48

To check whether it is divisible by 6 or not it has to meet two conditions

48 is divisible by 2 since the units digit of the number i.e. 8 is divisible by 2

48 is divisible by 3 since the sum of digits is a multiple of 3 i.e. 4+8 = 12 is divisible by 3

Therefore, 48 is divisible by both 2 and 3

Thus, 48 is divisible by 6.

2. 258

Solution:

Given Number is 258

258 is divisible by 2 since the units digit of the number i.e. 6 is divisible by 2

258 is divisible by 3 since the sum of digits is divisible by 3 i.e. 2+5+8 = 15 is divisible by 3

Since 258 is divisible by both 2, 3 it is also divisible by 6

3. 154

Solution:

Given Number is 154

154 is divisible by 2 since the units digit of the number i.e. 4 is divisible by 2

154 is not divisible by 3 as the sum of the digits is not divisible by 3 i.e. 1+5+4 = 10

10 is not divisible by 3

Since the given number is not divisible by 3 the given number is not divisible by 6.

4. 243

Solution:

Given Number is 243

243 is not divisible by 2 since the units digit of the number is not divisible by 2 i.e. 3 is not divisible by 2

243 is divisible by 3 since the sum of digits is a multiple of 3 i.e. 2+4+3 = 9 and 9 is divisible by 3

Since the given number is not divisible by 2 it is not divisible by 6

Therefore, 243 is not divisible by 6.

Complex Number is a combination of both Real and Imaginary Numbers. In other words, Complex Numbers are defined as the numbers that are in the form of x+iy where x, y are real numbers and i =√-1.

z = x+iy here x is the real part of the Complex Number and is denoted by Re Z and y is called the Imaginary Part and is denoted as Im Z. In the later sections, you will find What is a Complex Number and Properties of Complex Numbers. We tried explaining each and every Property of Complex Number in detail with Proofs.

What are Complex Numbers?

If x, y ∈ R, then ordered pair (x, y) = x + iy is called a complex number. It is denoted by z. Where x is the real part and is denoted as Re(z) and y is the imaginary part of the complex number and represented as Im(z).

(i) If Re(z) = x = 0, then the number z is a purely imaginary number

(ii) If Im(z) = y = 0 then the number z is a purely real number.

Properties of Complex Numbers

1. If x, y are two real numbers and x+iy =0 then x = 0 and y = 0

Proof:

Since, x + iy = 0 = 0 + i0, thus by the definition of equality of two complex numbers we can say that, x = 0 and y = 0.

2. If x, y, p, q are real and x + iy = p + iq then x = p and y = q

Proof:

Given x + iy = p + iq

rearranging the equation we get x − p = -i(y − q)

⇒ (x − p)² = i² (y − q)²

⇒ (x − p)² + (y − q)² = 0 (We know i² = -1)……..(1)

Since x, y, p, q are real, and (x − p)² and (y − q)² are both non-negative. Equation (1) is satisfied if each square is separately zero.

Thus we can write the equation as follows

(x − p)² = 0 or x = p and (y − q)² = 0 or y = q.

3. Similar to real numbers, the set of complex numbers also satisfy the commutative, associative, and distributive laws

Proof:

If z₁, z₂ and z₃ be three complex numbers then,

z₁ + z₂ = z₂ + z₁ (commutative law of addition) and z₁. z₂ = z₂. z₁ (commutative law of multiplication)

(z₁ + z₂) + z₃ = z₁ + (z₂ + z₃) (associative law of addition) and (z₁. z₂) z3 = z₁ (z₂. z₃) (associative law of multiplication)

z₁(z₂ + z₃) = z₁ z₂ + z₁ z₃ (distributive law)

4. Sum and Product of Two Conjugate Complex Quantities are both Real.

Proof:

Consider z = x + iy is a complex number where x, y are real.

Then, the conjugate of z is = x − iy.

Now, z + \(\overline {z}\)= x + iy + x − iy = 2x, is real.

and z. \(\overline {z}\) = (x + iy)(x − iy) = x² − i²y² = x² + y² is also real.

5. For two complex quantities z₁ and z₂,  |z₁+ z₂| ≤ |z₁ | + |z₂ |

Proof:

Let z₁ = r₁(cosθ₁ + isinθ₁ ) and z₂ = r₂(cosθ₂ + isinθ₂ ).

Hence |z₁ | = r₁ and |z₂ | = r₂

Now

z₁ + z₂ = r₁(cosθ₁isinθ₁) + r₂(cosθ₂ + isinθ₂)

= (r₁(cosθ₁+ r₂cosθ₂ )+ i(r₁sinθ₁+ r₂sinθ₂)

Hence |z₁+ z₂ | = √(r₁cosθ₁+ r₂cosθ₂)₂ + (r₁sinθ₁+ r₂sinθ₂)₂

= √r₁2(cos₂θ₁+ sin2₁) + r₂2(cos2θ₂+ sin2θ₂) + 2r₁r₂ (cosθ₁ cosθ₂+ sinθ₁ sinθ₂)

= √r1₂ + r2₂ + 2r₁r₂cos (θ₁– θ₂)

Now, |cos(θ₁– θ₂)| ≤ 1

Hence |z₁+ z₂| ≤ √r1₂ + r2₂ + 2r₁r₂ or |z₁+ z₂ | ≤ |z₁| + |z₂ |

6. If the sum of two complex numbers is real and the product of two complex numbers is also real then the complex numbers are conjugate to each other.

Proof:

Let us consider z₁ = a + ib and z₂ = c + id are two complex quantities (a, b, c, d and real and b ≠ 0, d ≠0).

As per the property,

z₁ + z₂ = a+ ib + c + id = (a + c) + i(b + d) is real.

Therefore, b + d = 0

⇒ d = -b

and,

z₁.z₂ = (a + ib)(c + id) = (a + ib)(c +id) = (ac – bd) + i(ad + bc) is real.

Therefore, ad + bc = 0

⇒ -ab + bc = 0, (Since, d = -b)

⇒ b(c – a) = 0

⇒ c = a (Since, b ≠ 0)

Hence, z₂ = c + id = a + i(-b) = a – ib

Thus, we can say that z₁ and z₂ are conjugate to each other.

Are you looking for help regarding the Units of Time Conversion i.e. from Hours to Minutes? You have come the right way and we will guide you all throughout on How to Convert from Hours to Minutes. Check out the basic definitions of hours, minutes, hrs to min conversion table, the procedure for hours to mins conversion, etc. Have a glance at the Solved Examples on Hours to Minutes Conversions and understand how to solve related problems easily. You can also check out our Math Conversion Chart to learn more about length, mass, capacity conversions along with Time Conversions.

Hours & Minutes Definitions

Hour(hr)

An Hour is a Unit of Time Conventionally defined as 1/24th of a day or 60 Minutes. The Hour is the SI unit of time accepted for the Metric System. Hours can be abbreviated as hr. As per the conversion base, 1 hr = 60 Minutes or 3600 Seconds.

Minutes(min)

A Minute is a period of time conventionally equal to 1/60 of an hour or 60 seconds. The Minute is the SI unit of time accepted for the Metric System. Minutes can be abbreviated as min. According to the Conversion Base, 1 Min = 60 Seconds.

Hours to Minutes Conversion Chart | Hrs to Mins Conversion Table

Below is the list of hours converted to minutes. Refer to them and have an idea of How many minutes are there in a Particular Hour.

How to Convert Hours to Minutes?

To convert hours measurement to minutes multiply time by the conversion ratio. Use the simple formula listed below to change between Hours to Minutes easily as a part of your conversions.

We know 1 hour = 60 minutes

Time in Minutes is equal to hours multiplied by 60.

Therefore, the Hours to Minutes Conversion Formula is given as

Now that you know the formula to Convert Between Hours to Minutes simply substitute the given hrs value and calculate the minutes in no time. To understand more about hr to min Conversions check out a few worked out examples provided by us in the below modules and know how to convert Hours to Minutes easily and quickly.

Read Similar Articles

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

Solved Examples on Conversion of Hours into Minutes

1. Convert 12 hrs to min?

Solution:

We know 1 hr = 60 minutes

The formula to convert from hrs to mins is given as follows

min = hours *60

min = 12*60

= 720

Therefore, 12 hrs converted to mins is 720 mins.

2. Convert 6 hrs 30 mins to minutes?

Solution:

Given 6hrs 30 mins

We know 1 hr = 60 minutes

6hrs 30 minutes = 6*60+30

= 360+30

= 390 minutes

Therefore, 6 hrs 30 mins converted to mins is 390 minutes.

3. Convert 8 hrs 42 mins to mins?

Solution:

We know 1 hr = 60 mins

8hrs 42 mins = 8*60+42

=480+42

= 512 mins

Therefore, 8 hrs 42 mins to 512 mins.

FAQs on Converting Hours to Minutes

1. What is the formula for converting hours to minutes?

The formula for converting hours to minutes is Minutes = Hours*60

2. How many minutes are there in 1 hour?

There are 60 minutes in an hour.

3. How to convert hr to mins?

Since there are 60 mins in an hour to convert from hrs to mins multiply with 60.

Worried about how to find the Arc Length? Don’t Panic as we will guide you completely in this and help you with the definition of Arc Length, Formula and Proof for Length of the Arc explained in detail. Check out the step by step explanation provided for determining the Arc Length. Also, refer to the solved examples for calculating the length of the arc and solve related problems in no time.

In general, Arc Length is nothing but the distance along the curved line making the arc. Remember that Arc Length is greater than the Straight Line Distance present between the Endpoints. Know about Denotations in the Arc Length Formula in the later sections.

Prove that the radian measure of any angle at the centre of a circle is equal to the ratio of the arc subtending that angle at the centre to the radius of the circle?

Let, us consider XOY to be a given angle. Now, with Centre O and any radius \(\overline {O L} \) draw a circle. Consider the circle drawn intersects \(\overline {O X} \) and \(\overline {O Y} \) at Points L and M respectively.

From the figure, it is clear that Arc LM subtends ∠LOM at Centre O. Now, take an Arc LN of length equal to the radius of the circle and join \(\overline {O N} \).
From definition, ∠LON = 1 radian.

We know the ratio of two arcs in a circle is equal to the ratio of the angles subtended by the arcs at the center of the circle,

Therefore, ∠LOM/∠LON = Arc LM/Arc LN
or, ∠LOM/1 radian = Arc LM/Radius \(\overline {O L} \)
or, ∠LOM = Arc LM/Radius OL × 1 radian = Arc LM/ Radius \(\overline {O L} \) radian.
Thus, circular measures of ∠LOM is Arc LM/Radius \(\overline {O L} \)
Let us consider θ be the circular measure of ∠LOM, Arc LM = s and Radius of the circle = \(\overline {O L} \) = r then,
θ = s/r, [i.e. theta equals s over r]

or, s = r θ, [i.e. s r theta formula]

Denotations in the Arc Length Formula

• s is the Arc Length
• r is the Radius of the Circle
• θ is the Central Angle of the Arc

Formulas to Measure the Arc Length

Solved Examples on Arc Length using the Formula s = rθ

1. Calculate the Length of the Arc if the radius of the circle is 9 cm and θ = 45°?

Solution:

We know the formula to Calculate the Arc Length s = rθ

Substituting the given input values r = 9 cm, θ =45° we get

Arc Length = 9*45°

= 9*0.785398(since 45° in radians =0.785398)

= 7.06 cm

Therefore, Arc Length is 7.06 cm

2. What would be the length of the arc formed by 65° of a circle having a diameter of 14 cm?

Solution:

From the given data

θ =65°

Diameter of the Circle = 14 cm

Radius of the Circle = \(\frac { d }{ 2 } \)

= \(\frac { 14 }{ 2 } \)

= 7 cm

We know the formula to calculate the Arc Length s = rθ

= 7*65°

= 7*1.13446

= 7.94 cm

Therefore, Arc Length is 7.94cm

Temperature is written in terms of Celsius, Fahrenheit, Kelvin. Temperature can be converted from one unit to another using certain formulas. Of all the Temperature Conversions Celsius to Fahrenheit Conversion is the most commonly used. To help you understand we have provided the formula for °C to °F Conversion, the procedure to convert from Celsius to Fahrenheit in the later modules.

Both Celsius and Fahrenheit have different freezing and boiling points of water and follow the varied unit differences between each scale. Check out Solved Examples on Celsius to Fahrenheit Temperature Conversion for a better understanding of the concept.

Celsius to Fahrenheit Conversion Formula

Conversion from Celsius to Fahrenheit can be mathematically expressed using the formula

F = \(\frac { 9 }{ 5 } \)*C+32

Where F is the Temperature in Fahrenheit

C is the Temperature in Celsius

How to Convert Celsius to Fahrenheit(°C to °F)?

Check out the quick and easiest way to perform Celsius to Fahrenheit Conversion. They are along the lines

• To convert from Celsius to Fahrenheit firstly multiply the value of the celsius degrees by 9.
• Later divide the product with 5.
• Add 32 to the result obtained in the earlier step.
• That’s the required Fahrenheit Value and place °F symbol next to it.

Celsius to Fahrenheit Conversion Table

Relation between °C and °F

The Relation between Celsius and Fahrenheit is direct. Celsius is directly proportional to Fahrenheit and it implies

• If the Temperature in Celsius increases then the Fahrenheit Equivalent Temperature also increases.
• If the Temperature in Celsius decreases then the Fahrenheit Equivalent Temperature also decreases.

Both the Scales of Temperature have their own measurements. In Fahrenheit Scale 32 °F is the freezing point of water and 212 °F is the boiling point of water. However, in Celsius Scale 0 °C is the Freezing Point of water and 100 °C is the boiling point of water.

Solved Examples on Celsius to Fahrenheit Conversion

1. Convert 40°C to °F?

Solution:

We know the formula to convert Celsius to Fahrenheit is

\(F=(\frac{9}{5}\times 1)+32\)

\(F= (\frac{9}{5}\times 40)+32\)

=104°F

Therefore 40°C converted to Fahrenheit is 104°F

2. What is 60°C in Fahrenheit?

Solution:

We know the formula to convert Celsius to Fahrenheit is

\(F=(\frac{9}{5}\times 1)+32\)

\(F=(\frac{9}{5}\times 60)+32\)

= 140°F

Therefore, 60°C in Fahrenheit is 140°F

FAQs on Celsius to Fahrenheit Conversion

1. What is the formula to convert from Celsius(°C) to Fahrenheit(°F)?

The Formula to Convert from Celsius to Fahrenheit is \(F=(\frac{9}{5}\times 1)+32\)

2. What is C equal to in F?

1 Degree in Centigrade is Equal to 33.8 Fahrenheit.

3. What are the different Temperature Conversions?

The three major temperature conversions are

• Conversion Between Celsius and Kelvin
• Conversion Between Fahrenheit and Kelvin
• Conversion Between Celsius and Fahrenheit

Greater than and Less than Symbols are used to compare any numbers. If a number is bigger than another symbol greater than symbol is used and when a number is less than the other number less than symbol is used. Refer to the entire article to learn about Greater than and Less than Symbols, Notations, and Tricks to remember them easily. Using Symbols will help you save time and space and users can understand them easily.

Greater than and Less than Symbols – Definition

Greater than and Less than Symbols denotes the inequality between two values. The symbol used to represent greater than is “>” and less than is “<“.

Greater than Sign

Greater than Symbol is placed between two values if the first value is greater than the second value. In Inequality, greater than symbol will always be pointed towards the greater value, and the symbol consisting of two equal length strokes connecting in an acute angle at the right.

For Example 8>5, here 8 is greater than 5

Less than Sign

Less than symbol is placed between two values when the first value is less than the second value. In Inequality, less than symbol will always be pointed towards the lesser value, and the symbol consisting of two equal length strokes connecting in an acute angle at the left.

For Example 10<15, here 10 is less than 15.

Greater than and Less than Symbols makes it easy for the readers to understand.

All Symbols – Summary

Below is the list of Inequality Symbols and their Notations. They are along the lines

Tricks to Remember Greater than and Less than Symbols

To remember Greater than and Less than Symbols two methods are used. They are as follows

• Alligator Method
• L Method

Alligator Method: We know an Alligator or Crocodile always wants to eat a larger number of fishes. Alligator Mouth Opens always opens towards the largest number. Consider the numbers on both sides represent the number of fishes.

For Example: 10>6

Alligator mouth opens towards 10 since it wants to et 10 fishes rather than 6 fishes. It means 6 <10

L Method: Letter L appears similar to the Less than Symbol “<“.

Example: 20<40

Also, Read: Greater or Less than and Equal to

Applications of Greater than and Less than Symbols

In Mathematics we always don’t end with equality. At times, you can even have greater than and less than symbols. The Statements can be expressed using a Mathematical Expression.

Example: Consider there are x number of students in a class and they are more than 50 students if 10 more joined the class then they are 50 students in the class. Expressing this statement in mathematical expression we get

x+10>50

Solving Inequalities is similar to solving equations. While dealing with Inequalities Problems you should pay attention to the inequality sign direction. Some of the tricks that don’t affect the direction of inequalities are as listed below

• Multiply or Divide the Inequalities on both the sides with the same Positive Number.
• Add or Subtract the same number on both sides of the Inequality Expression.

Worked out Examples on Greater than and Less than Symbols

Question 1.

Diya has thirteen bananas and Manvi has twenty bananas. Find out who has more bananas?

Solution:

No. of Bananas Diya has = 13

No. of Bananas Manvi has = 20

Since 20>13

Thus, Manvi has more Bananas.

2. Dheeraj sleeps for thirty minutes and Manav sleeps for forty-five minutes every day in the afternoon. Find out who sleeps for less time?

Solution:

No. of Minutes Deeraj Sleeps = 30 min

No. of Minutes Manav Sleeps = 45 min

since 30<45

Dheeraj Sleeps for Less time compared to Manav.

FAQs on Greater than and Less than Symbols

1. What are the Different Inequality Symbols?

The different Inequality Symbols are

Greater than (>)
Less than (<)
Not equal to (≠)
Greater than or equal to (≥)
Less than or equal to (≤)

2. How to Remember Greater than and Less than Symbols?

You can use different methods like Alligator Method and L Method to remember the Greater than and Less than Symbols easily.

3. When can we use Greater than and Less than Symbols?

We can use Greater than and Less than Symbols to represent the Inequality Expressions. The Symbols used to denote greater than and less than are >, <.

Are you searching everywhere to find Word Problems on Division? If so, you have come the right way where you can find the complete list of Division Word Problems here. Practice using the Division Word Problems available here and test your level of understanding. For the sake of your convenience, we even listed Step by Step Solutions to all the Division Questions available. Have a glance at them and learn how to solve Word Problems on Division easily and quickly.

Examples of Word Problems Involving Division

1. There are 150 grade 4 students, in a school. The students are to be equally divided into 6 classes. How many students do we have in each class?

Solution:

No. of Grade 4 Students = 150

Since 150 Students are to be divided equally among 6 Classes we have the equation as such

= \(\frac { 150 }{ 6 } \)

= 25

Therefore, 6 Classes will have 25 Students each if 150 students divided equally.

2. Mary made 323 cupcakes. She packed 5 cupcakes into each box. How many boxes of cupcakes did she pack? How many cupcakes were left unpacked?

Solution:

No. of Cupcakes mary made = 323

She packed 5 Cupcakes into each box = 5

No. of boxes of cupcakes she packed =\(\frac { 323}{ 5 } \)

= 64 remainder 3

Therefore, 3 cupcakes are left unpacked.

3. If 9000 kg of wheat is packed in 85 bags, how much wheat will each bag contain?

Solution:

Since 85 bags contain wheat of 9000kg

Therefore, 1 bag contains wheat (9000 ÷ 85) kg

= 105kg

Each bag contains 105kg.

4. 85 people have been invited to a banquet. The caterer is arranging tables. Each table can seat 10 people. How many tables are needed?

Solution:

Since 85 people have been invited to a banquet and 10 people can be seated in  1 table

No. of Tables needed = 85 ÷ 10

Quotient = 8

Remainder = 5

80 People can be organized in 8 Tables and 5 People will be left. So the Caterer has to arrange one more table so that the 5 People can sit.

Thus, the total number of tables needed = 8+1

= 9

5. How many hours are there in 1800 minutes? 

Solution:

We know 1 hr = 60 minutes

To know how many hours are there in 1800 minutes we have to divide by 60 minutes

= 1800 ÷ 60

= 30 hours

Therefore, there are 30 hours in 1800 minutes.

6. A bus can hold 96 passengers. If there are 8 rows of seats on the bus, how many seats are in each row?

Solution:

The capacity of the bus that it can hold = 96 Passengers

No. of Rows in the Bus = 8

No. of Seats in each row = Capacity of Bus ÷ No. of Rows

= 96 ÷ 8

= 12

Therefore, there are 12 seats in each row.

7. $5,800 is distributed equally among 20 men. How much money will each person get?

Solution:

From the given data

Money received by 20 men = $5,800

Money received by 1 man = $5,800 ÷ 20

=

Thus $290 is received by each man when a sum of $5800 is distributed equally among 20 men.

8. Tarun had 56 apples. He divides all apples evenly among 7 friends. How many apples did Tarun give to each of his friends?

Solution:

Tarun had 56 apples

He divided all apples evenly among 7 friends

No. of Apples Tarun gave to each friend = Total Apples Tarun had ÷ No. of Friends he equally distributed the Apples among

= 56 ÷ 7

= 8

Therefore, Tarun gave 8 Apples each to his 7 friends on distributing 56 apples equally.

9. Nandy needs 6 lemons to make a glass of orange juice. If Nandy has 252 oranges, how many glasses of orange juice can she make?

Solution:

No. of Lemons Nandy need to make an Orange Juice = 6

Total No. of Oranges Nandy has = 252

To find out how many glasses of orange juice Nandy can make simply divide the total number of oranges she has with the. of lemons needed to make an orange juice.

= 252÷6

= 42

Therefore, Nandy can make 42 glasses of Orange Juice.

10. In your classes you counted 130 hands. How many students were in the class?

Solution:

No. of Hands Counted = 130

Each student has 2 hands

To find how many students were in the class simply divide the total no. of hands counted by the no. of hands each student has i.e. 130÷2 = 65

Therefore, there were 65 students in the Class.

The most important part of elementary education is learning multiplication tables. Check the 20 Times Table Multiplication Chart here and master the table at an early age to build a foundation in mathematics. With the help of the multiplication table of 20, you can get perfect in various topics like algebra, fractions, divisions, etc. Know the tips to memorize 20 Tables and become perfect with all the multiplication tables.

20 Times Table Multiplication Chart

For all the complex calculations, we use calculators. But for simple calculations, using the calculator is not the correct way. Memorizing the tables will help you to improve your problem-solving skills. It is always recommended for the students to memorize the tables from at least 2 to 20. If you memorize the multiplication tables, then you can quickly solve all the problems and it can be used as a building block in maths. Check how each integer gives the solution when multiplied by 20. Follow the tips and tricks to know the multiplication table of 20.

Tips to Memorise Multiplication Tables

1. To memorize the multiplication table, you have to know how many numbers are increased to get each resultant.
2. Add the same number to which you are finding the multiple.
3. To find the first number, you have to double it, and then again to find the second number, you have to add the multiplication number.
4. Break up the numbers into manageable chunks.
5. Practice each table until you get perfect for it.
6. Practice the number of multiplication facts.

Do Check:

• 14 Times Table
• 15 Times Table Multiplication Chart

20 Times Table Multiplication Chart

20 Times Table Multiplication Chart shows that the multiplication of 20 with various whole numbers present. Here is the table of 20.

To obtain the multiplication table of 20, we need to multiply each natural number by 20. The multiplication chart of 20 is nothing but the repetition of addition. As we know that 20 * 1 = 20, adding 20 to that number will give the next multiple numbers. Similarly, if we add 20 to the next multiples, we get the Table of 20. The multiples of 20 till 10 are

20 * 1 = 20

20 * 2 = 40

20 * 3 = 60

20 * 4 = 80

20 * 5 = 100

20 * 6 = 120

20 * 7 = 140

20 * 8 = 160

20 * 9 = 180

20 * 10 = 200

Easy ways to learn Multiplication Chart of 20

We will mention the easy ways to learn and remember the multiplication table.

Tip 1: The order does not Matter

When two numbers are multiplied, the order in which the numbers are multiplied does not matter. You can multiply the first number with the second or the second number with the first.

So, you don’t have to memorize both 4*3 and 3*4, just you have to remember that 4 and 3 make 12. Memorizing the numbers in such a way will cut the job in half.

Tip 2: Divide the table into “Chunks”

It is really hard to remember the whole table at once. So, divide the table into chunks and memorize it.

1. Start learning the multiplication tables with 5
2. Now, learn the multiplication numbers up to 9 times 5.
3. Learn the multiplications which you use them more and also learn simple tables like 2,3,4 etc.
4. Also, learn the numbers which multiply with the repeated numbers like 5 * 5, 7 * 7, 8 * 8 etc.

Some Patterns   

There are a few patterns that help you to remember some patterns.

2x is nothing but doubling the number. It is nothing but adding the number to itself.

Therefore, the pattern is 2,4,6,8,10…

5x comes with the pattern of ending with 0 or 5.

Therefore, the pattern is 5,10,15,20,25,30

9x comes with the pattern of “ones” place going down and “tens” place going up.

Therefore, the pattern is 9, 18, 27, 36, 45, 54, 63…

10x is the easiest way to remember. You just need to add zero to the number before it.

Therefore, the pattern is 10,20,30,40,50,60…

The Hard Ones

The hard ones are 6 * 7 = 42, 6 * 8 = 48, 7 * 8 = 56

Better you often say it in your mind and remember them.

Importance of Multiplication Tables

Being reliant on one’s memory is much brighter than something else. If you remember the multiplication chart, you will get the feeling of self-confidence. With the help of the multiplication table, you can keep the information at the fingertips to use it whenever required. It increases the memory power and stimulates the calculation speed. Students who master the multiplication tables from 2 to 20 can know the method of holding and observing things.

Tables from 2 to 10 are basic and fundamental, it helps in the calculation of simple arithmetic operations. Quick computation and a great deal of time are the major advantages of mastering multiplication charts.

Solved Examples involving Multiplication Table of 20

Problem 1:

There are 3 monkeys. Each monkey has 20 bananas. How many bananas are there in total?

Solution:

As given in the question,

No of monkeys = 3

No of bananas each monkey has = 20

To find the total number of bananas, we have to apply the law of multiplication.

Therefore, the total number of bananas = 3 * 20 = 60

Thus, all the monkeys have 60 bananas in total.

Hence, the final solution is 60 bananas.

Problem 2:

The first squirrel has 134 nuts. The second squirrel has 20 times as many nuts as the first squirrel. How many nuts does a second squirrel have?

Solution:

As given in the question,

No of nuts, the first squirrel has = 134

No of times the second squirrel has = 20

To find the no of nuts the second squirrel has, we have to apply the law of multiplication

Therefore, the nuts that the second squirrel has = 134 * 20

= 2680

Thus, the second squirrel has 2680 nuts

Hence, the final solution is 2680 nuts

Problem 3:

To find out the cost of the stolen diamonds, multiply the cost of one diamond with the total number of diamonds stolen.

Solution:

As given in the question,

Cost of one diamond = Rs. 16000

Total number of diamonds stolen: 20

Total worth of stolen diamonds = 20 * Rs 16000 = Rs. 3,20,000

Thus, the final solution is 3,20,000 Rupees

Worried about where to find instant help regarding the Capacity and Volume Conversions Chart in both Metric and Customary units? Then, halt your search as you have reached the right place and we will provide you with the necessary Capacity and Volume Units Conversion Chart. You can use them during your calculations in day-to-day life and make your problem-solving easy.

For the sake of your convenience, we have prepared the Capacity and Volume Conversion Table. In addition to the Metric and Customary Units of Capacity and Volume, we explained the concept much better by considering few examples. Check out the Solved Problems on Capacity and Volume Conversions and learn to apply the logic when you need them during your math calculations.

Capacity and Volume Conversion Charts

When it comes to Capacity and Volume the standard unit is the liter. The other standard units in the Capacity and Volume Conversion are explained below. Make the most out of them whenever you need to change between one unit of capacity or volume to the other. Having Units of Capacity or Volume in similar units makes it easy for you to solve problems. They are as follows

Customary Units for Capacity and Volume Conversion Table are explained as follows

Also, Read:

• Units of Length Conversion Charts
• Units of Mass and Weight Conversion Chart
• Math Conversion Chart

Solved Examples on Capacity and Volume Conversions

1. Convert 45 ml into l?

Solution:

We know 1 ml = \(\frac { 1 }{ 1000 } \) l

Substituting the value of 1 ml we get the equation as

= 45*\(\frac { 1 }{ 1000 } \) l

= \(\frac { 45 }{ 1000 } \) l

= 0.045 l

Therefore, 45 ml converted to l is 0.045l

2. Convert 8 hl into l?

Solution:

We know 1 hl = 100 l

Substituting the value of 1 hl we have the equation as follows

= 8*100 l

= 800 l

Therefore 8hl converted to l is 800 l

3. Convert 0.025l into ml?

Solution:

We know 1 l = 1000 ml

0.025 l = 0.025*1000 ml

= \(\frac { 25 }{ 1000 } \)*1000 ml

= 25 ml

Therefore, 0.025l converted to ml is 25ml

FAQs on Units of Capacity and Volume Conversion Chart

1. What are all the Units of Capacity?

There are five basic units in the customary system of measurement and units are as follows

• Ounce
• Cup
• Pint
• Quart
• Gallon

2. What is th Biggest Unit of Capacity?

The Biggest Unit of Capacity is Liter(L).

3. What’s the difference between capacity and volume?

Volume and capacity both are properties of three-dimensional objects. Volume is the space that a three-dimensional object occupies, whereas capacity, on the other hand, describes how much a container can hold.

Variation can be clearly understood if you know about Variables. In Mathematics, we generally deal with two types of quantities namely variables and constants. If the value of a quantity remains unchanged during different scenarios then it is said to be constant. On the other hand, if the value of a quantity changes under varying conditions then it is said to be a variable. Refer to the complete article to know about the Definition of Variation, Types of Variation, and Solved Examples on Variation, etc.

What is Variation?

In Mathematical Equations when a relationship is established there exist two types of quantities one is constant and the other is variable. Constant will not alter with the change in other parameters of the equation and the other is a variable that changes with different situations. The Changing of variable parameters is called a Variation.

You can understand the concept of variation with a simple equation y = mx

Here m is the constant and if we consider m = 2 then y becomes 2x

If x = 1, then y becomes y = 2*1 = 2

If x = 2, then y becomes y = 2*2 = 4

By changing the value of y, x gets different values. This is the variation of y with different values of x and similarly, you can get different values of y if the value of x changes.

Types of Variation

Variations can be of different types according to the pattern of changing or relationships of variables. We have explained the most common types of variation here and they are as follows

Direct Variation: If variables change proportionately i.e. either decrease or increase then it is said to be a direct variation. If A is in direct variation with B then it can be symbolically written as A α B.

Indirect Variation: In the case of Indirect or Inverse Variation variables change disproportionately i.e. when one variable increases the other one decreases. The behavior of Variables is exactly the opposite of direct variation. Thus, it is called an Indirect or Inverse Variation. If A is in indirect variation with B then it is symbolically written as A α \(\frac { 1 }{ B } \)

Joint Variation: If Two or more variables are related directly or one variable change with change product of two or more variables then it is said to be a Joint Variation. If A is in joint variation with B and C then it is symbolically represented as such A α BC

Combined Variation: Combined Variation is a combination of direct, indirect, or joint variation. In this kind of variation, three or more variables exist. If A is in combination with B, C then it can be symbolically represented as A α \(\frac { B }{ C } \) or A α \(\frac { C }{ B } \)

Partial Variation: If two variables are related using a formula or a variable is related by the sum of two or more variables then it is said to be a Partial Variation. y = mx+c is a straight line equation and is an example of Partial Variation.

FAQs on Variation

1. How do you define Variation?

Changing of Variable Parameters is called a Variation.

2. What are the Types of Variation?

There are different types of variation and they are as follows

• Direct Variation
• Indirect Variation
• Joint Variation
• Combined Variation
• Partial Variation

3. What is a direct variation proportion?

A Relations is said to be in direct variation when one variable changes and the second variable changes proportionally.

Subsets fall under the mathematics concept Sets. A Set is a collection of objects or elements enclosed within curly braces {}. If Set A is a Collection of Odd Numbers and Set B includes { 1, 3, 5} then B is said to be a subset of A and is denoted by B⊆A whereas A is the Superset of B.

Elements of Set can be anything such as variables, constants, real numbers, whole numbers, etc. It can include a null set too at times. Learn about Subsets Definition, Symbol, Formula, Types, and Examples in the later modules.

What is a Subset?

Set A is said to be a subset of Set B if all the elements present in Set A are also present in Set B. In other words, we can say Set A is contained within Set B.

Example:  If Set A has {a, b} and set B has {a, b, c}, then A is the subset of B because elements of A are also present in set B.

Subset Symbol

In Set theory, the Subset is denoted by the symbol ⊆ and is usually read as the Subset of. Using the symbol let us denote the subsets as follows

A ⊆ B; which denotes Set A is a subset of Set B.

Remember a Subset may include all the elements that are present in the Set.

Also, Read:

• Intersection of Sets

All Subsets of a Set

Subsets of any set consist of all possible sets including its elements and even null set. Let us provide you an example and provide the possible combinations of a set so that you can better understand.

Example:

What are the Subsets of Set A = {5, 6, 7, 8}?

Solution:

Given Set A = {5, 6, 7, 8}

Subsets include

{}

{5}, {6}, {7}, {8},

{5,6}, {5, 7}, {5,8}, {6,7},{6,8}, {7,8},

{5,6,7}, {6,7,8}, {5,7,8}, {5,6,8}

{5,6,7,8}

Types of Subsets

Subsets are Classified into two types namely

• Proper subsets
• Improper subsets

Proper Subsets: Proper Set includes only a few elements of the original set.

Improper Subsets: Improper Set includes all the elements of the original set along with the null set.

Example:

If set A = { 4, 6, 8}, then,

Number of subsets: {4}, {6}, {8}, {4,6}, {6,8}, {4,8}, {4,6,8} and Φ or {}.

Proper Subsets: {}, {4}, {6}, {8}, {4,6}, {6,8}, {4,8}

Improper Subset: {4,6,8}

There is no proper formula to find how many subsets are there. You just need to write each one of them and distinguish whether it is a proper subset or improper subset.

What are Proper Subsets?

Set A is said to be a Proper Subset of Set B if Set B has at least one element that doesn’t exist in Set A.

Example: If Set A has elements as {4, 6} and set B has elements as {4, 6, 16}, then Set A is the Proper Subset of B because 16 is not present in the set A.

Proper Subset Symbol

A Proper Subset is represented by the symbol ⊂ and is read as “Proper Subset of”. Using this symbol we can denote the Proper Subset for Set A and Set B as A ⊂ B.

Proper Subset Formula

If we have to pick the “n” number of elements from a set containing the “N” number of elements you can do so in ^NC_n number of ways.

How many Subsets and Proper Subsets does a Set have?

If a Set has “n” elements then there are 2ⁿ subsets for the given set and there are 2ⁿ -1 proper subsets for the given set.

Consider an example, If set X has the elements, X = {x, y}, then the proper subset of the given subset are { }, {x}, and {y}.

Here, the number of elements in the set is 2.

We know that the formula to calculate the number of proper subsets is 2ⁿ – 1.

= 2² – 1

= 4 – 1

= 3

Thus, the number of proper subset for the given set is 3 ({ }, {x}, {y}).

What are Improper Subsets?

A Subset that contains all the elements of the original set are called Improper Sets. It is represented using the symbol ⊆.

Set A ={1,2,3} Then, the subsets of A are;

{}, {1}, {2}, {3}, {1,2}, {2,3}, {1,3} and {1,2,3}.

Where, {}, {1}, {2}, {3}, {1,2}, {2,3}, {1,3} are the proper subsets and {1,2,3} is the improper subsets. Therefore, we can write {1,2,3} ⊆ A.

Note: Empty Set is an Improper Subset of itself and is a Proper Subset of any other set.

Power Set

Power Set is said to be a collection of all subsets. It is denoted by P(A).

If Set A has elements {a, b} then the Power Set of A is

P(A) = {∅, {a}, {b}, {a, b}}

Properties of Subsets

Below is the list of Some of the Important Properties of Subsets and they are as such

• Every Set is a subset of the given set itself. It means A ⊂ A or B ⊂ B, etc.
• Empty Set is a Subset of Every Set.
• If A is a Subset of B it means A is contained within B.
• If Set A is a subset of Set B then we can infer that B is a superset of A.

Subsets Example Problems

1. How many subsets containing three elements can be formed from the set?

S = { 1, 2, 3, 4, 5, 6, 7 }

Solution:

No. of Elements in the Set = 7

No. of Elements in the Subset = 3

No. of Possible Elements containing the 3 Elements is 7C₃

=\(\frac { 7! }{ (7-3)!*3! } \)

= \(\frac { 7*6*5*4! }{ 4!*3! } \)

= 35

2. Find the number of subsets and the number of proper subsets for the given set A = {4, 5, 6, 7,8}?

Solution:

Given: A = {4, 5, 6, 7,8}

The number of elements in the set is 4

We know that,

The formula to calculate the number of subsets of a given set is 2ⁿ

= 2⁵ = 32

The number of subsets is 32

The formula to calculate the number of proper subsets of a given set is 2ⁿ – 1

= 2⁵ – 1

= 32 – 1 = 31

The number of proper subsets is 31.

FAQs on Subsets

1. What are Types of Subsets?

Subsets are classified into two types and they are as follows

• Proper Subsets
• Improper Subsets

2. What is the formula to calculate the number of proper subsets and subsets for a given set?

If a set has “n” number of elements then the number of subsets and proper subsets is given by the formulas

The formula to calculate the number of proper subsets for a given set is 2ⁿ-1

The formula to find the number of subsets is 2ⁿ

3. What is the symbol to denote Subset?

The subset is denoted by the symbol ⊆.

In Set theory, there are various types of set operations like Subsets, Difference of sets, Union of sets, Complement of sets, and Intersections of sets. Students can easily understand and learn the set theory concepts by referring to our articles. So, today we have come up with the Intersection of Sets topic. Let’s go through this page and learn about the intersection of sets definition, symbol, properties, and examples.

Intersection of Sets Definition

If the Intersection of two sets A and B, then it is denoted by A∩B, where it contains all the elements which are common to both A and B. The symbol to denote the intersection of sets is ‘∩’. All those elements that relate to both A and B denote the intersection of A and B. Hence, we can say that,
A ∩ B = {x : x ∈ A and x ∈ B}

For n sets A₁, A₂, A₃,……A _n where all these sets are the subset of Universal set (set U). The definition of the intersection is the set of all elements which are common to all these n sets.

What is the Intersection of Two Sets?

If there are two sets, Set A and Set B, then the intersection of sets is given by:

A∩B = n(A) + n(B) – n(A∪B), where

n(A) is the cardinal number of set A,
n(B) is the cardinal number of set B,
n(A∪B) is the cardinal number of a union of two sets ie., A and B.

For Example: The intersection of two sets ie., A = {1, 2, 3} and B = {2, 3, 4} then find A∩B?

We know, A∩B = n(A) + n(B) – n(A∪B)

After calculating the process of the intersection of two sets, the result is like this A∩B = {2, 3}.

Properties of Intersection of a Set

i) Commutative Law: The union of two sets A and B follow the commutative law i.e., A ∩ B = B ∩ A

ii) Associative Law: The intersection operation follows the associative law i.e., If we have three sets A, B, and C then, (A ∩ B) ∩ C = A ∩ (B ∩ C)

iii) Identity Law: The intersection of an empty set with any set A gives the empty set itself i.e., A ∩ ∅ = ∅

iv) Idempotent Law: The intersection of any set A with itself gives the set A i.e., A ∩ A = A

v) Law of U: The intersection of a universal set U with its subset A gives the set A itself. A ∩ U = A

vi) Distributive Law: According to this law: A ∩ (B ∪ C) = ( A ∩ B ) ∪ (A ∩ C)

Solved Examples on Intersection of Sets

1. Let U be the universal set consisting of all the n – sided regular polygons where 5 ≤ n ≤ 9. If set A, B, and C are defined as:

A = {pentagon, hexagon, octagon}

B = {pentagon, hexagon, nonagon, heptagon}

C = {nonagon}

Find the intersection of the sets A and B

Solution:

Given Universal Set U = {pentagon , hexagon , heptagon , octagon , nonagon}, A = {pentagon, hexagon, octagon}, B = {pentagon, hexagon, nonagon, heptagon}, C = {nonagon}

Now, calculate the intersection of sets A and B

Here, the intersection of sets means the common elements consist in both A and B:

The elements of A = {pentagon, hexagon, octagon}, and B = {pentagon, hexagon, nonagon, heptagon}

Then, A ∩ B = {pentagon, hexagon}.

2. If A = {1,3,5,7,9} and B = {2,3,5,6,8}. Find intersection of two set A and B.

Solution:

Given Set A = {1,3,5,7,9} and Set B = {2,3,5,6,8}

Here 3, 5 are the common elements in both A and B sets.

So, the intersection of two sets A and B are, A ∩ B = {3, 5}.

3. If P = {x, y, z} and Q = {ф}. Find the intersection of two given sets P and Q.

Solution:

Given sets are P = {x, y, z} and Q = {ф}

The Intersection of given sets P ∩ Q = { }.