## All you wanted to know about Polygons and Interior Angles

Geometric figures like pentagon, hexagon, octagon, etc. are very intriguing geometric two dimensional figures with their own peculiar properties. In general, we call these figures polygons. A little later you will find the proper definition of polygons.

At times, it might look little scary or daunting to understand the properties of these polygons. If you know polygons in and out, this article may not be for you. It is for learners who would like to understand polygons in a simple common sensical manner.

So let us delve deep into the world of polygons. So, here we go with the most obvious question first –

## What is a polygon?

Polygon is any shape made up of 3 or more connecting lines on a flat sheet surface. Hence, polygons are 2-dimensional closed figures made up of straight lines. Thus the intersecting lines have to terminate at the point of intersection.

If you would like to dive deeper into the formal definition and terminology related to polygons, I recommend you check out Chegg’s page on polygons

## What is an interior angle of a polygon?

The angles formed in the interior or inside of a polygon where two pair of sides intersect are called interior angles.

## What is a regular polygon?

A polygon in which all sides are equal (equilateral) and all angles are equal (equiangular). Otherwise, it’s an irregular polygon.

## How do you find the sum of the interior angles of a polygon?

I can directly give you the formula to calculate the sum of all interior angles of a polygon. But I want you to dive a little deeper and understand how that formula is arrived.

Our goal for today is to figure out how the interior angles of a polygon change as the number of sides of the figure increases. So let us start with the polygon which has smallest number of sides, i.e. triangles.

### Triangle

To understand interior angles of a polygon, we have to keep triangles (type of polygon) as the starting point. Triangle has 3 interior angles and 3 sides. It’s a known fact that sum of these 3 interior angles of a triangle is always 180°.

Little out of context, but the moment we think of triangles Pythagoras Theorem is generally the first things which comes to our mind. In a previous post, I have tried exploring, was Pythagoras Theorem actually proved by Pythagoras?

3 sided polygon, sum of interior angles = 180°

Moving forward and looking at polygons with higher number of sides –

### Quadrilateral

Quadrilateral is polygon with 4 sides and obviously 4 interior angles.

If we connect the opposite corners of the quadrilateral (i.e. the diagonal) divides the quadrilateral into 2 triangles. If we add the sum of interior angles of theses 2 triangles, we get 180°+180°=360°. Thus the sum of interior angles in a 4 sided polygon is always 360°.

Another way of looking at it: square is a polygon with all sides equal, i.e. it’s a regular polygon. We know in a square each of the 4 angles is equal to 90°. Thus sum of all the interior angles is 90°x4 = 360°

4 sided polygon, sum of interior angles = 360°

### Pentagon

In a 5 sided polygon, also called pentagon, you can join opposite 2 vertices from any one vertex and you will get 3 triangles. Again, we know that sum of interior angles of a triangle is 180°.

Thus, sum of interior angles of 3 triangles = sum of all interior angles of a pentagon = 180°x3 = 540°

5 sided polygon, sum of interior angles = 540°

Have you noticed that, as we keep increasing the number of sides in a polygon the interior angles keep increasing.

As we move further, i.e. for each increase in number of sides of a polygon, 180° gets added to the sum of interior angles.

So the general rule or formula is,

Sum of Interior Angles = (n-2) x 180°

## How to find the interior angles of a polygon?

In a ‘n’ sided polygon, the sum of interior angles of a polygon = (n-2) x 180°

A regular polygon is equi-angular; thus each interior angle will be equal. In a ‘n’ sided or ‘n’ angled polygon,

Each Interior Angle = Sum of Interior Angles / no. of sides = [(n-2) x 180°]/n

For example, in a dodecagon (12 sided figure shown below),

Sum of interior angles

= (n-2) x 180°

= (12-2) x 180°

= 10 x 180°

= 1800°

And in a regular dodecagon, each interior angle

= 1800°/12

= 150°

## What is an exterior angle? How to find an exterior angle?

An exterior angle in any polygon is an angle formed by one side of the polygon and the extension of an adjacent side of the polygon.

Interior and exterior angles are supplementary angles.

Exterior angle = 180° – Interior angle

Thus, if interior angle is , exterior angle = 180° – X°

Another important point to keep in mind is that,

Sum of all exterior angles = 360°

So, in a regular polygon of n sides, each

Exterior angle = 360°/number of sides in a polygon

Example, in a hexagon, each exterior angle will be 360°/6 = 60°

## How to find the number of sides of a polygon, if one interior angle is given?

The easiest way to finding the number of sides, is to first find out the exterior angle.

So let’s say the interior angle given is 144°, how to find the number of sides in the polygon?

First step is to calculate the exterior angle, which in this case = 180° – 144° = 36°

Now since it’s known that sum of all exterior angles is 360°, thus the formula is

Number of sides in a polygon = 360°/exterior angle

In above case, number of sides = 360°/36° = 10 sides. Thus it’s a decagon.

For your further reading, you can check this previous post where I’ve explained the ratio of area and volume derived from ratio of sides.

Objective of QuickerMaths is to make mathematics fun, quick and simple. I would love to hear from you in comments below.

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## Eggs-cellent Riddles & Brain Teasers

You always knew that eggs are full of nutrition but here on QuickerMaths, it will help you strengthen your brain cells too. Here are some riddles & puzzles around eggs for you to solve.

First Egg Riddle

If one and a half hens lay one and a half eggs in one and a half days, how many eggs does one hen lay in one day?

Leave your answers below.

Second Egg Riddle

2 fathers and 2 sons sat on the table to eat eggs for breakfast. They ate exactly three eggs, each person had an egg. Now you need to explain how that’s possible?

Third Egg Riddle

Let say you started selling a basket of eggs.

First customer buys one-half of your eggs plus one-half of an egg. Second customer buys one-half of your eggs plus one-half of an egg.

Third customer buys one-half of your eggs plus one half an egg.

At this point you have sold all of your eggs, and you never broke an egg. How many eggs did you start with?

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## 7 Best Ways to Master GMAT Quantitative Ability Section

Muddled up as how to prepare for the Quantitative ability section of GMAT? The Quantitative ability section of the exam is one of the most feared sections of any entrance. Although the difficulty level isn’t too much, the test is built to put students in a pinch. It is a computer adaptive test, i.e., the difficulty of every next question changes according to your rate of correct answer.

Get all information for GMAT in French here

Correct answers will result in points being awarded and difficulty being raised for the next question on-screen, while wrong answers will result in similar difficulty level and no points awarded for that question. So, those planning to appear for the exam must have a look below to get an idea of the best ways to master the section.

The test evaluates four key skills of a student, i.e. analytical writing, quantitative analysis, verbal skills, and reading skills. The language of the exam is English, with emphasis on grammar, algebra, geometry, and arithmetic. The exam also assesses analytical writing and problem-solving abilities of the students. GMAC believes that data sufficiency, logic, and critical reasoning skills are extremely vital to businesses in the real world.

GMAT 2018 Paper Pattern

To prepare for any exam, you must first be familiar with the structure of that exam. There are four sections in a GMAT exam. These are analytical writing assessment, integrated reasoning section, quantitative section, and verbal section. The exam is 3 hours and 7 minutes long, with time divided unequally amongst all the three sections.

 GMAT Test Section Questions Question Types Timing Analytical Writing Assessment 1 Topic Analysis of Argument 30 Minutes Integrated Reasoning 12 Questions Multi-Source Reasoning Graphics Interpretation Two-Part Analysis Table Analysis 30 Minutes Quantitative 31 Questions Data Sufficiency Problem Solving 62 Minutes Verbal 36 Questions Reading Comprehension Critical Reasoning Sentence Correction 65 Minutes

As you can see in the table above, there are two types of questions in the Quantitative section:

• Problem Solving (PS)
• Data Sufficiency (DS)

The Problem Solving section has the same multiple choice format that is popular for every standardized test in today day and age. There are five options, out of which only one can be the correct answer.

The other format, Data Sufficiency, is unique to the GMAT exam, with unique rules too, which require different strategies as to all other tests. This section, thus, requires the most amount of practice and work.

7 Best Tips for GMAT Quantitative Ability preparation

If you are preparing for the GMAT exam, you will need all the help you can get to clear the exam. Provided below are a few tips you can follow in order to better prepare for the Quantitative section of the exam.

Strengthen your Basics

The mathematical concepts tested in GMAT are extremely simple, consisting of basic arithmetic, algebra, and geometry. The only problem is that students tend to forget the basics as time moves on. Your GMAT preparation should first and foremost cover the basics, and only after completing those should you think about going ahead with further preparations. The best way to remember formulae is to create flashcards and stick them around in your room. That way, every time you walk by a formula, your eyes will tend to hover over the flashcard.

Practise tests and mock tests

It should be obvious that practise will make you better eventually. The more you practise, the easier you will find the test to be. Thankfully, practise tests don’t have to be very expensive. Several online resources provide free practice content to use.

After every iteration of a practice or a mock test, you would also need to analyse your performance. Review the results and note the questions that you have answered incorrectly. Improve upon these particular areas identify your area of weakness. Majority of the GMAT exam questions revolve around students’ familiarity with different types of questions and avoiding common mistakes. With ample practice, you will be able to realise which questions are trick questions, thus also saving you a lot of time. One practice every week should be a comfortable place to start.

Pay special attention to Data Sufficiency questions

The Quantitative section is the most difficult section of GMAT primarily due to the Data Sufficiency portion. They would require you to think a little differently, but the more you practice them, the easier they become. There are several key points to remember when working on Data Sufficiency questions. Read the provided statements individually, and very carefully. Only after carefully evaluating the statements, make your answer choice.

Data sufficiency requires only sufficiency, not the actual answer, which means that if a problem states if the value of a variable can be determined, you only have to see whether it can be or cannot be determined, without actually solving for the value. You are just trying to find out if there’s enough information to answer the question, but you don’t actually have to find the answer.

Memorize the Five Answer Choices

There are always the same five answer choices for every Data Sufficiency Question. These answer choices are:

• Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
• Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
• Both statements together are sufficient, but neither statement alone is sufficient.
• Each statement alone is sufficient.
• Statements (1) and (2) together are not sufficient.

If you were to memorise these statements, you could save precious few seconds for every problem you attempt. You would only need to read the statement provided alongside the question and place judgement based on them, by clicking on the answer choices.

Be careful with Graphs, Charts, and Tables

A lot of questions in GMAT quantitative section will require you to read and interpret information provided on charts, graphs, and tables. It is extremely important that you read the axes, the key, units of measurement, etc. correctly so that you don’t misinterpret the data.

Use the rough paper in exam

Even if you feel like the GMAT quantitative section is too easy for you, it would only benefit you to use a paper for calculations as much as possible. Writing down your calculations will help you notice any mistake you might have made before you press the answer and move on to the next question. Writing also forces you to make sure you’re thinking through progress in steps instead of leaps, which can help reduce mistakes further. Remember, use of a calculator is forbidden in the GMAT exam.

Read the Questions carefully

Last but not the least, this is the most crucial yet the most ignored piece of advice that one can offer to any student aspiring to crack any national level exam. It is simply because students tend to make more mistakes when they are fatigued.

One of the most common mistakes on the GMAT exam is to misinterpret or read the question incorrectly. The GMAT exam purposefully throws in questions with difficult language, or questions that can mistakenly be read differently.

Instead of asking “Which of the following may be false?” GMAT will present the question as “Which of the following may not be true?” which might be misunderstood as “Which of the following may be true?” While this may sound a bit far-fetched right now, the atmosphere of pressure inside the hall and the fatigue during the exam itself could easily lead to such mistakes.

Thus, make sure that you read every question carefully so you can save yourself from these easily avoidable mistakes.

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## Top colleges offering Bachelors in Mathematics in India

There are various colleges which offer Bachelors in Mathematics course for the aspiring students. It is the responsibility of the students to find out the best colleges for the courses they wish to opt for. Below is the list of some of the top colleges offering Bachelors in Mathematics course:

## Indian Statistical Institute

It is an institute which is of national importance that has been established and recognized under the act of Indian parliament in the year 1959. It has been established in the year 1931 which is a public university and is considered one of the oldest and prestigious universities which focus on the study of stastics and mathematics.
It is one of the leading training and research institute in the fields of computer science, quantitative aptitude, statics, and related sciences and so on. The headquarters of the university is located in West Bengal, Kolkata.

Courses offered

• B Stat (Hons)
• B Math (Hons)
• M Stat
• M Math
• MS in Quantitative Economics
• MS in Library and Information Science
• M Tech in Computer Science
• M Tech in Quality, Reliability and Operations Research
• JRF in Statistics
• JRF in Computer Science
• JRF in Mathematics

## Maharishi Markandeshwar University, Ambala

It is a deemed to be university in Mulana in the state of Haryana. It was established in the year 1993 in the name of Maharishi Markandeshwar and was founded by Tarsem Garg. The institute is the leading symbol of education in terms of technical, medical and other professional streams. It has been accredited by the NAAC team with an A grade.

The university is committed to excel in research and innovation and the skill development of the students. It has developed an industry oriented education system which helps the students in order to make the students leaders in the professional world.

Courses offered

• Electrical Engineering
• Biotechnology
• Civil Engineering
• Computer Science & Engineering
• Mechanical Engineering which includes Specialization in Automobile Engineering and Mechatronics
• Electronics & Communication Engineering
• Computer Science with Specialization in Software Development Program (American Pattern)

## Chennai Mathematical Institute

It is an institute of central excellence which has been formed for the teaching and research in the mathematical sciences which is established in the year 1989 with the motto of bridging the gap between the teaching and the research in the fields of mathematics and various other allied subjects.

It has been recognized by the government of India under the section 3 of the UGC act, 1956. The objective of the institute is to provide the excellent quality education to the students in the best possible way.

Courses offered

• B.Sc. (Hons.) in Mathematics and Computer Science (3 year integrated course).
• B.Sc. (Hons.) in Mathematics and Physics (3 year integrated course).
• M.Sc. in Data Science
• M.Sc. in Mathematics
• M.Sc. in Applications of Mathematics
• M.Sc. in Computer Science
• Ph.D. in Mathematics
• Ph.D. in Computer Science
• Ph.D. in Physics

## Indian Institute of Technology (IIT)

It is one of the autonomous public institutes of higher education situated in India. It is governed by the Institute of technology Act, 1961 which has been declared as the institution of national importance in India. There are various branches of the institute which has been established in various other cities of India to impart excellent education to the students in the fields of technology, science and mathematics. Every year, a large number of students gear up to take admission in this university.

Courses offered

### BACHELOR OF TECHNOLOGY (B.TECH.) in

• Aerospace Engineering
• Biological Sciences and Bio-Engineering
• Chemical Engineering
• Civil Engineering
• Computer Science and Engineering
• Electrical Engineering
• Materials Science and Engineering
• Mechanical Engineering

## Lovely Professional University, Jalandhar

It is a private university situated in Jalandhar, Punjab. It was established in the year 2005 by the lovely international trust under the lovely professional university act, 2005 which started its operation in the year 2006. It has the largest single campus in terms of the private university in India. The campus spreads over 600 acres of land on the outskirts of Jalandhar and accommodates more than 24,000 students.

It has been recognized by the UGC, NCTE and AIU. It was also ranked 18th as India’s best college in the year 2017. It offers various courses in Mathematics which are spread over 3 years across six semesters. In order to get admissions in such courses, the candidates should have scored 60% and above in the secondary examination or any other degree equivalent to it. The candidates also need to clear the LPUNEST Exam 2018-2019 in order to take admission in the examination.

### Mathematics:

• Master of Science in Mathematics
• M. Sc (Mathematics) – M. Tech (Computer Science Engineering) (Dual Degree)
• Doctor of Philosophy in Mathematics (Full-time & part-time)

### Physics:

• Master of Science in Physics
• M. Sc (Physics) – M. Tech (Computer Science Engineering) (Dual Degree)
• Master of Science (hons.) in Physics
• M. Sc (Hons.) (Physics) – M. Tech (Computer Science Engineering) (Dual Degree)
• Master of Philosophy in Physics
• Doctor of Philosophy in Physics (Full-time & part-time)
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## How to prepare for mathematics in UPSC examination?

Preparing for UPSC exam is a challenging task but choosing a good optional subject increases your chances for qualifying. Mathematics is one such key player. If you have a mathematics background and have interest in that subject, then prefer maths as your optional subject. Maths has always been known as a good scoring subject. For the people who hate mugging, Mathematics is a gift. There is no hit and try involved in it. If you know the formula, you know the answer.

Choosing mathematics has its own advantages such as it requires more of application part and zero mugging. Secondly not many UPSC aspirants take up maths as the optional subjects, so there is less competition. Thirdly, the study material is easily available. And lastly, this subject has a limited syllabus unlike other subjects in which current affairs are also involved. Here are few easy tips to get high marks in the exam.

1. ### Number of attempts

During the exam, if you are unable to solve the question in two attempts, leave it. You will not be able to solve that. Focus on other questions. If a problem is taking too much time, then hold on, you may be following a wrong approach to solve it.

2. ### Tips and tricks

Practice some good tips and tricks to solve the problems. The more problems you solve, you efficiency will automatically be increased. Try to adopt Vedic Maths techniques. This will help you to solve even the biggest of equations within fractions of seconds.

A few tips and tricks using Vedic Maths :

Step 1:
Let us consider multiplication of three digit numbers 208 x 206.

Step 2:
Now, deduct the last digit from the respective numerals.
208 – 8 = 200
206 – 6 = 200

Step 3:
Pick any one number and add it with the unit digit of another number.
208 + 6 = 214

Step 4:
Now, multiply the result obtained in step 1 and step 2.
214 * 200 = 42800

Step 5:
Then, multiply the unit digits of the given numbers.
8 * 6 = 48

Step 6:
Add the values obtained in step 4 and step 5.
42800 + 48 = 42848
208 x 206 = 42848

3. ### Avoid silly mistakes

Maths is one such subject where a single silly mistake can deviate you from the actual answer. Be careful of the sign the number is carrying. Practice, practice and practice till you become perfect.

4. ### Make a formula chart

Make a chart (or two) of all the formulas and paste them in your study room. Don’t be hesitant in looking through the formula in case you forget. After some practice, the formulas will automatically be memorized by you.

5. ### Try solving by hand

Rather than just reading any solution, better take a pen and paper and try solving it. There is a huge difference between solving in mind and solving on paper.

It can be illustrated through an example :

6. ### New Topics

Some new topics have been introduced such as Mechanics and Fluid dynamics. Try to understand the concepts in the beginning of your preparation period. After reading all the solved examples, then start off with previous year question papers. This will give you an idea what and how much to study for the prelims.

7. ### Writing answers

If you are writing short answers then give one or two lines for introduction and then move straight to the point. In case of long answers, give introduction and explain wherever necessary and make a perfect conclusion at the end.

8. ### Join Test Series

Joining test series is an effective way to improve your time management and self evaluation skills. If not daily, then give a mock test at least twice a week for a sound preparation.

9. ### Keep yourself relaxed

You cannot solve any mathematics problem if you lack focus or concentration. Keep all the distractions away from you and be cool & relaxed. This way you will not only be able to understand the problem but will also hit the right answer too.

Mathematics is the first love of science. If studied and applied properly, mathematics will fetch you good marks and ultimately a good UPSC score.

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## 4 Tips for Building a Good Relationship With Your Child’s Math Tutor

Today, getting a tutor for a child having difficulties with math is a normal course of action to take. If your child needs help with understanding certain concepts in geometry, algebra, calculus, or with applying the unitary method in grade 2, there is nothing wrong with asking other parents or going online to find a tutor for your little one.

Hiring a tutor or enrolling a child in a tutorial program, in general, is a popular option among parents. It is something that many parents decide to do to help their kids excel at or meet the demands of a difficult subject (or more than one). It is one of the best ways parents can show their love and support for the children who are having a hard time keeping up with their schoolwork.

## Working with Your Child’s Tutor

As a parent, your role in your child’s academic progress won’t stop or be diminished if you decide to enroll your child in an after-school math tutorial program. Your youngster will experience more benefits from a tutorial program if you have a good working relationship with his tutor.

To make sure you and your child’s tutor are on the same page regarding his improvements and continuous progress in math, follow these helpful tips:

1. ### Practice constant and open communication

Communication is one of the vital pillars of a tutoring process. From the start, ask questions if you have any, share your feedback and raise any concerns you might have about the program. Make sure the tutor has all the necessary information regarding the key needs and preferences of your child as well.
Before scheduling a talk with the tutor or learning center, come up with a list of questions regarding their procedures and policies. Get details about their assessment process and their teaching methodologies as well.

2. ### Set clear goals for the tutorial program

Before the formal tutorial program starts, create a list of goals that you want your child to achieve. Discuss these goals with the tutor. Ensure you highlight the particular concepts or areas your kid needs help with.
When coming up with your list of goals, talk to your child. Also, keep in mind that a tutor isn’t just a person you hire to help your child do his math assignments or review for an upcoming test. If your child needs help with this subject, work with the tutor to ensure that the whole problem will be tackled and not just one area will be solved or managed.
One goal that you should never forget to list is to see a change in your child’s attitude towards math. Aside from getting higher grades, your tutor should also play an important role in changing your youngster’s negative perceptions about math, and in the process inspire the student to actively participate in studying it.

3. ### Involve your child’s teacher

For your child to benefit more from the tutorial program, inform his teacher about this after-school activity. It is important that the teacher and tutor work together so that they are on the same page in terms of the lessons and goals.
If possible, schedule parent-teacher conferences and tutor meetings to coincide with the time your child gets his progress report. It is essential that the tutor communicates with your child’s teacher to get feedback on his performance in the classroom. They don’t even have to meet personally; they can correspond by email or private messages. Just make sure your kid’s tutor and teacher coordinate regularly.

4. ### Be involved

Lastly, aside from finding a good tutor and bringing your child to the tutorial center during his sessions, you also need to take a proactive role in the whole process. At the very least, at the end of each session, ask the tutor what topics they covered today and what difficulties your child encountered. Ask the tutor how you can help reinforce what your child learned from the session.
Although this may mean relearning continuity and different ability and other mathematical concepts so that you can give some exercises for your child to work on at home, your efforts will be rewarded in the end.
Your role in helping your child overcome his fear or dislike of math and getting higher grades doesn’t stop at finding a good tutorial center and bringing him here every session. For your child to get more from each tutorial lesson, you must be totally involved in the process and have a good working relationship with the tutor.

AUTHOR BIO
Maloy Burman is the Chief Executive Officer and Managing Director of Premier Genie FZ LLC. He is responsible for driving Premier Genie into a leadership position in STEM (Science, Technology, Engineering and Mathematics) Education space in Asia, Middle East and Africa and building a solid brand value. Premier Genie is currently running 5 centers in Dubai and 5 centers in India with a goal to multiply that over the next 5 years.

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## Mentally Solve Problem of Averages in Seconds

In all competitive examinations and otherwise also in our day to day calculations we encounter issues related to averages. Especially when the composition of the group changes it becomes difficult to do computations.  Let us discuss quick and easy solutions to such problems.

### When a person leaves a group and another person replaces him, then there can be 2 scenarios:

Scenario I: When the average age increases then,

Age of the new comer = age of person who left + (no. of persons in the group * increase in average age)

Scenario II: When the average decreases then,

Age of the new comer = age of the person who left – (no. of persons in the group * decrease in average age)

Let us try to understand these with a simple example –

Question: The average age of 45 persons is decreased by 1/9 year when one of them whose age is 60 years is replaced by a new comer. What is the age of the new comer.

Solution:

Age of the new comer = age of the person who left – (no. of persons in the group * decrease in average age)

Age of the new comer = 60 – (45 * 1/9) = 55 years

Isn’t that easy to calculate using the formula I just gave you. However, to intuitively answer such questions you need to keep following things in mind

Firstly, the average age is reducing, which means an older man is replaced by relatively younger person. Hence the answer has to be lesser than 60. Hence we subtract from 60 and not add to it.

Secondly, average reduction of 1/9 kg for a group of 45 people means, in total there is a wait reduction of 1/9 per person x 45 persons = 5 kg for the entire group. This is result of the replacement of an old person by a relatively younger person in the group. That’s the reason we’re subtracting 5 from 60, to get the age of the new comer.

### B. When a person joins a group without any replacement, then there can be 2 scenarios:

Scenario I: When the average age increases then,

Age of the incoming person = previous average age of the group + no. of persons including the person who joined * increase in the average age value

Scenario II: When the average age decreases then,

Age of the incoming person = previous average age of the group – no. of present persons including the person who joined * decrease in the average age value

I will explain this further with an example,

Question: The average age of 20 teachers is 45 years which is decreased by 6/7 years when a student joins the group. Then what is the age of the student?

Solution:

Age of the outgoing person = previous average age of the group – new no. of persons including the person who joined * decrease in the average age value

Therefore, age of the student = 45 – 21 * 6/7 = 27 years

Here also you need to observe that since the average age is going down when the new person is joining, the age of new comer will be lesser than the average, hence subtraction.

Moreover, the decrease is equal to 6/7 per person for each person in the new group. That is 21 times 6/7 = 18 kgs, which needs to be subtracted from the overall average to get the age of the new comer.

### C. When a person leaves a group without any replacement, then there can be 2 scenarios:

Scenario I: When the average age increases then,

Age of the outgoing person = previous average age of the group – no. of persons excluding the person who left * increase in the average age value

Scenario II: When the average age decreases then,

Age of the outgoing person = previous average age of the group + no. of present persons excluding the person who left * decrease in the average age value

Hope we’ve covered all scenarios and with the help of the above logical formulas we will be able to do our calculations.

### Practice questions for you to solve (try to solve mentally)

Question 1: In a boat there are 8 men whose average weight is increased by 1 kg when a man of 60 kg is replaced by a new man. What is the weight of the new comer?

Question 2: In a class there are 30 boys whose average weight is decreased by 200 grams when one boy whose weight was 25 kgs leaves the class and a new comer is admitted. Find the weight of the new comer?

Question 3: A cricketer has a certain average for 9 innings. In the 10th innings, he scores 100 runs, thereby increasing his average by 8 runs. His new average is?

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## How to cover important topics of Algebra for JEE Main?

To command on mathematics in IIT JEE examination, Algebra is one of the key to get through. This portion of mathematics covers various topics, such as, Set Theory, Quadratic Equations, Logarithms, Permutations, Matrices and Determinants and Probability. In order to attain success in this field it involves the understanding of the topic versus marks ratio. There are direct questions being asked on Determinants and Matrices. Permutations and Combinations as well as Probability are most important sections. In IIT JEE Main exam mostly questions are fetched on Complex numbers, Probability and progressions & series.

There are many portions in Algebra section and it has a weightage which is most often equal to the weightage of Calculus and together they cover around 65 percent of entire Maths. So let’s find out the important topics of Algebra for JEE Main:

1: Complex Numbers and Quadratic Equations

Algebra of complex numbers, modulus and argument (or amplitude) of a complex number, square root of a complex number, triangle inequality, Quadratic equations in real and complex number system and their solutions, Relation between roots and coefficients etc.

2: Matrices and Determinants

Types and algebra of matrices, properties of determinants order two and three, area of triangles using determinants. Adjoint and inverse of a square matrix, Test of consistency and solution of simultaneous linear equations in two or three variables.

3: Permutations and Combinations

Fundamental principle of counting, permutation as an arrangement and combination as selection

4: Mathematical Induction and Reasoning

Statements, logical operations. ‘and’, ‘or’, ‘implies’, ‘implied by’, ‘if and only if’. Understanding of tautology, contradiction, converse and contrapositive.

5: Binomial Theorem

Binomial theorem for a positive integral index, general term and middle term, properties of Binomial coefficients and simple applications.

6: Sequences and Series

Arithmetic and Geometric progressions, Relation between A.M. and G.M. Sum up to n terms of special series: Sn, Sn2, Sn3. Arithmetico – Geometric progression.

7: Statistics and Probability

Measures of Dispersion

Calculation of mean, median, mode of grouped and ungrouped data. Calculation of standard deviation, variance and mean deviation for grouped and ungrouped data.

Probability

Probability of an event, addition and multiplication theorems of probability, Baye’s theorem, probability distribution of a random variable, Bernoulli trials and Binomial distribution.
The above topics are bifurcated into different subtopics which must be emphasised on. A few points to study and cover Algebra are:

1. Priority wise: Progressions & Series, Permutations & Combinations, Binomial, probability, Complex numbers, Matrices & determinants (these are arranged in the order of ease and weightage combined)
2. Algebra requires huge practice with the blend of understanding the concepts. Start any topic with NCERT and solve ALL the questions even if you find them extremely easy.
3. Once you are done with that move on to R D Sharma objective and then finally any one of Cengage Algebra G Tejwani or Sk Goyal Arihant publication can be considered the reference book.
4. Always go through solved examples first and then start solving the unsolved questions
5. At the end of any chapter, go for previous year questions on that chapter and this time focus on speed and accuracy.

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## 5 Tips to Easily Solve Math Problems

Mathematics is a fundamental part of life, be it in buying groceries at the market to finding the mass of a gluon in particle physics, mathematics is involved. Hence, it is quite evident that everyone should have a requisite understanding of mathematics. But we know that mathematics is not everyone’s cup of tea. From calculus to geometry, everyone has difficulty with some topic. Here I am to give you a few tips and tricks which will help you to improve your efficiency in maths helping you to perform better in your exams and certain life scenarios.

• Easy percentage calculation: We know how percentage implies to a portion of a 100. To calculate percentages first multiply the percentage to the number and then shift decimal point two places to the left. It is as simple as that but for our better understanding let us use an example.
Consider the case in a restaurant where you have to tip the waiter 20% of a bill of 1500. The tip can be calculated as => 1500 x 20 = 30000
Shifting the decimal point two places to the left => 300
• Divisibility: Checking if a number, usually a large number, is divisible by 2, 3, 4, 5, 6, 9 is something that happens now and then in daily life, like in the case of splitting a bill and so on. Let us see a few tips which tell us if a number is properly divisible.
i. Divisibility by 2: 1’s place digit will be divisible by 2.
ii. Divisibility by 3: If sum of the digits is divisible by 3. (eg: 501= 5 + 0 + 1 = 6 = 3 x 2)
iii. Divisibility by 5: If the last digit is 5 or 0.
iv. Divisibility by 6: If the condition for divisibility of 2 and 3 are satisfied.
v. Divisibility by 9: If sum of the digits is divisible by 9. (6930 = 6 + 9 + 3 + 0 = 18 = 9 x 2)
vi. Divisibility by 12: If the criteria of divisibility of 3 and 4 are satisfied.
• Temperature Conversion: To convert temperatures between Celsius and Fahrenheit, we have certain shortcuts which help us save both time and effort.
i. C->F
F = (C x 1.8) + 32
ii. F->C
C=(F-32)÷1.8
• An easy way to remember the value of Pi: Consider the sentence “May I have a large container of coffee”. Counting the number of letters in each word is 31415926 and after adding a decimal place after 3 we get 3.1415926 which is the actual value of Pi correct to 7 decimal places.
• Multiplication by 11: Let me explain this with an example, consider the problem 45 x 11; Write down the number in the 100’s place as 4 and the 1’s place as 5. Then the number in the 10’s place is the sum of the digits; 4+5 = 9. Thus, 45 x 11 = 495. If the sum of the digits exceeds 9, then just carry over the digit to the 100’s place.
For example, 89 x 11;
100’s place = 8
10’s place = 9
Sum of the digits = 17
Product =        17
8  _  9
=      979

Thus, here we have stated a few genius tricks which will help you through scenarios which require you to have quick thought processing when it comes to mathematics. For a better understanding of topics of mathematics like algebra, Quadratic Equation, geometry and more tricks check out our YouTube channel.

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## 7 Ways To Support Your Child’s Maths Learning

As parents, you can help your child want to learn maths in various ways apart from hiring maths tutors. That desire to learn will be the key to your child’s success. Enjoyment is also a crucial motivator for learning. In addition, you can point out how fortunate they are to have the great opportunity to learn maths today.

Having a good grasp of maths can open the doors to plenty of exciting possibilities.

# 7 Ways To Help Your Child Learn Maths

## 1. Offer insights into the different ways to approach maths

There are important things that your child must understand so they can develop more confidence in their maths ability.

Firstly, let them know that problems can be solved in many different ways. Learning maths goes beyond finding the correct answer. It is also a method of solving different problems and applying what they’ve learnt to new problems.

Secondly, point out that wrong answers are useful. Incorrect answers can be used to help them figure out where and why they made a mistake. Their explanation can help you discover if they need help with the concepts related to answering the problem or with number skills like division, subtraction, addition, and multiplication.

Lastly, help your child become a risk taker. Let them see the value of attempting to solve difficult problems. Allow them to explore various approaches and encourage them to speak up about their insights. This will strengthen their maths skills.

## 2. Promote a positive attitude towards maths

Maths can be difficult. I was not good in maths when I was a student. I did not like maths either. These statements can undermine your child’s attitude towards the subject. These comments will also give false impressions that maths is something that they can either be good at or not.

As parents, you must become a positive force in helping them learn maths. You need to let them know that solving maths problems can be satisfying, that knowledge of maths concepts is generally crucial in life, that anyone can be good at it, and that it will open up the doors to excellent career options.

## 3. Illustrate how maths works in day-to-day life

Your home is the best place to start exploring maths with your child. Integrate maths language and activities into daily routines to show them how this subject works in their daily life.

Sorting and matching activities will introduce your child to different mathematical operations like measurement and classification. For instance, let them sort the laundry to be washed. Ask them to put all the whites together, all the colored garments, and all the towels. As they sort things, let them count aloud how many shirts or towels are there. It is also helpful to give them the wrong number so they can count the items one by one and show to you that you have made a mistake.

Let them recognize that numbers are all around them. This will help them understand that numbers are important and that they can be used for various purposes.

## 4. Practice maths every day

Even when you are not at home, you can teach your child some maths concepts. In the grocery store, for instance, let them compare the prices of multipacks of vegetables and decide which of these packs offer the best value. Also, give them the opportunity to manage money by giving them some pocket money and encouraging them to budget how much will be spent. When you are on a trip, take note of the distance and the speed, and let them estimate how much time is left for your trip.

## 5. Let your child teach you maths

Instead of telling or showing your child how to add or multiply, it is better for you to let them teach you how they’ve learned to add or to multiply in school. Whenever you do not understand a part of the approach, let them know and ask for more clarifications. Each time they try to teach you something, they will definitely learn from that.

## 6. Communicate with the maths teacher

When you are concerned about your child’s learning in maths or unsure about a certain approach used in school, discuss this with their teacher.

Most educators appreciate receiving feedback, and when your child is having a hard time understanding these concepts, it is possible that other students are stuck as well.

## 7. Play games that buoy up mathematical thinking

Playing with blocks can teach basic maths skills like counting, recognizing symmetry, sorting, identifying patterns, and number recognition. Moreover, games with number cards will help your child come up with tactics for using numbers in several combinations by subtracting, dividing, multiplying and adding.

Helping your child to learn maths does not necessarily mean that both of you cannot have a good time and laugh. In fact, you can make games out of any maths concepts and skills. Use these activities to strengthen their maths skills as well as to build strong positive attitudes toward maths.

AUTHOR BIO
Bushra Manna is one of the founders and Principal of Leaps and Bounds Education Centre – Motorcity. She has 20 years experience teaching the British and American curricula internationally at primary level. Bushra has a passion for teaching and started her teaching career as an assistant teacher for 2 years, during which an autistic boy was appointed to her care within a mainstream classroom setting. Working with Ismail opened her eyes to the significance of knowing a child’s best learning style and having an individualized approach to teaching and building a child’s self confidence.

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