IX grade mathematics
Friends today we all are going to understand the basic concept behind few of the most important topics of mathematics that are Intervals and percents. Interval concept need to be known by each of the student because it plays an important role in graphing, number line and other important topics. So before going further I would like to discuss about Intervals in mathematical world.
Intervals are basically a chunk of the real line. Most commonly we define for real numbers that are “a” and “b”.
The above mentioned notations may be a little confusing, but what student needs to remember that square brackets used in the notations mean the end point is included whereas the round parentheses used in the notations mean it is excluded. If both the end points are included in the interval then it is said to be closed and if they are both excluded then we can call it to be open. In any of the case if one of them is included and the other one is excluded then the interval is half open or we can also call it as half closed, depending on student preference.
Moving further the things are getting a little dark because the above mentioned notation is also used with “b” replaced with Ñ„ or “a” replaced with - Ñ„ and only the round parentheses are placed at the end. This notation states that the interval is unlimited or we can say that interval is infinite on the right or left respectively.
Let's take an example to understand it better. In the notation (1 , infinity (Ñ„)) is basically a complicated way of describing the set of all numbers greater than 1, and (- Ñ„, pie ) means that the set of all numbers less than or equal to
. The set of all real numbers can be expressed as (-Ñ„ , Ñ„)
Now we have done with Intervals so moving towards the main topic that is percents in mathematical world. Before proceeding further let's just discuss about basic definition of Percent. In simplest of mathematical manner we can say that percent means per hundred. The basic way is to understand it is that student divide something into 100 equal parts and after dividing consider so many parts of it.
The most important word and the item that creates the greatest confusion in percentage calculations is
“something”. The something of which you calculate a certain number of percent is rarely identified, and it often changes even within the same problem.
Let's take an example to understand it better. Suppose a person earns Rs. 10 an hour and acquires a raise of 20 percent, followed by another raise of 20 percent. The first raise is basically applies on the person hourly wage of Rs. 10 and the second to the new hourly wage of the person. In the above values 20 percent of Rs. 10 is Rs. 2 and so the new hourly wage of the person is Rs. 12. And now the second raise of Rs. 12 which gives Rs. 2.40 and so the hourly wage of the person after second raise is Rs. 14.40. What student need to note is the phrase that receive "a raise of 20%". It's pretty clear that it's 20 percent of the person current salary and his or her CEO's salary, or the gross national product, but that fact is not stated explicitly. One of the most noticeable thing is that even though both raises are 20 percent and they translate into different Rupees amount because they are applied to different hourly wages.
Now let's take an another example along the same lines : If any of the person is making a certain income and he or she receive a raise of about 100 percent, followed by a pay cut of 100 percent, then the new income of the employee or person is not the same what he/she started with, but rather it is zero.
In real life it is well possible to have more than 100 percent of something. The most real example is the bosses of company which probably makes more than 100 percent of the employee income.
Before proceeding further let's discuss the basic concept behind the percent calculations. The number of which we compute the percent is called the base number. In the above example Rs. 10 and then Rs. 12 are the two base numbers. The number of percent is the rate (in above example 20 is the rate) and the rate applied to the base number is the part. These all depends on the application the part can have many different names. For example, when the problem is about money, it could be a raise, a discount, a fee, a commission, etc.
Now the basic formula of calculating percentage is denoting the whole by “P”, the rates by “R” and the base by “b” and the basic formula is:
P = R / 100 x b
Generally percentage problems become distinct by what is known and what needs to be calculated, and to recheck their difficulty usually axis from describing the base number only inherently. Another difficulty which might come is that commonly the part is not of interest in itself, but what is required that it needs to be added or subtracted to the base number to get the result of interest.
Let's take an example to understand it better:
“R” = 25%
So : P = 25/100 x 2000 = 500
Finally the purchase price = 2000 – 500 = Rs. 1500.
2. Unknown Part: If any person invest money at an annual interest rate of P percent. The interest is paid monthly. Thus according to the rules every month the bank adds:
to C which is the amount of money in your account at the beginning of the month. Equivalently, every month your money is multiplied with the factor
1 + p / 12,00
Unknown base number. If any person buys a car for Rs. 4 lakh and the dealer tells the buyer that you have purchased it at a 25% discount. What is the list price of the car? As we all know that buyer received a 25 percent discount buyer purchased the car for 75 percent of its list price. Let's just denote the list price by L we get
400,000 = 75/100 x L
Now we get :
L = 400,000 x 100/75 = 533333.33
R = 678 x 100/17000 = 4 (approximately)
The growth rate in your town is about 4 percent.
Generally problems involving percentage involve more than one base, rate and part.
Now moving towards another concept that is never divide by zero. Generally problems involving percentage involve more than one base, rate and part. I suggest you all that a daily practice and hard work is what more commonly needed. The most important thing is that there is no alternative to hard work.
Division by zero is undefined. If anyone going to assign a value or number to the outcome of dividing by zero he/she might run into conflict and mathematical world would become useless. In next session we would go through this in detail.
This is all for today in the next session we will go through the above concept along with some complicated percentage problems. Division by zero is an interesting topic which should tell you all the basic fundamentals of mathematics and also tell you that how a zero can change the whole course of the mathematics. We all know without zero there is no mathematical terms and here with zero the mathematics becomes useless. So it would be one of the most interesting chapter.
Friends today we all are going to understand the basic concept behind few of the most important topics of mathematics that are Intervals and percents. Interval concept need to be known by each of the student because it plays an important role in graphing, number line and other important topics. So before going further I would like to discuss about Intervals in mathematical world.
Intervals are basically a chunk of the real line. Most commonly we define for real numbers that are “a” and “b”.
-
[a , b] gives the set of all the numbers “x” satisfying a ≤ x ≤ b.
-
(a , b) gives the set of all numbers “x” satisfying a < x < b.
-
[a , b) the set of all numbers “x” satisfying a ≤ x < b.
-
(a , b] the set of all numbers “x” satisfying a < x ≤ b.
The above mentioned notations may be a little confusing, but what student needs to remember that square brackets used in the notations mean the end point is included whereas the round parentheses used in the notations mean it is excluded. If both the end points are included in the interval then it is said to be closed and if they are both excluded then we can call it to be open. In any of the case if one of them is included and the other one is excluded then the interval is half open or we can also call it as half closed, depending on student preference.
Moving further the things are getting a little dark because the above mentioned notation is also used with “b” replaced with Ñ„ or “a” replaced with - Ñ„ and only the round parentheses are placed at the end. This notation states that the interval is unlimited or we can say that interval is infinite on the right or left respectively.
Let's take an example to understand it better. In the notation (1 , infinity (Ñ„)) is basically a complicated way of describing the set of all numbers greater than 1, and (- Ñ„, pie ) means that the set of all numbers less than or equal to
. The set of all real numbers can be expressed as (-Ñ„ , Ñ„)Now we have done with Intervals so moving towards the main topic that is percents in mathematical world. Before proceeding further let's just discuss about basic definition of Percent. In simplest of mathematical manner we can say that percent means per hundred. The basic way is to understand it is that student divide something into 100 equal parts and after dividing consider so many parts of it.
The most important word and the item that creates the greatest confusion in percentage calculations is
“something”. The something of which you calculate a certain number of percent is rarely identified, and it often changes even within the same problem.
Let's take an example to understand it better. Suppose a person earns Rs. 10 an hour and acquires a raise of 20 percent, followed by another raise of 20 percent. The first raise is basically applies on the person hourly wage of Rs. 10 and the second to the new hourly wage of the person. In the above values 20 percent of Rs. 10 is Rs. 2 and so the new hourly wage of the person is Rs. 12. And now the second raise of Rs. 12 which gives Rs. 2.40 and so the hourly wage of the person after second raise is Rs. 14.40. What student need to note is the phrase that receive "a raise of 20%". It's pretty clear that it's 20 percent of the person current salary and his or her CEO's salary, or the gross national product, but that fact is not stated explicitly. One of the most noticeable thing is that even though both raises are 20 percent and they translate into different Rupees amount because they are applied to different hourly wages.
Now let's take an another example along the same lines : If any of the person is making a certain income and he or she receive a raise of about 100 percent, followed by a pay cut of 100 percent, then the new income of the employee or person is not the same what he/she started with, but rather it is zero.
In real life it is well possible to have more than 100 percent of something. The most real example is the bosses of company which probably makes more than 100 percent of the employee income.
Before proceeding further let's discuss the basic concept behind the percent calculations. The number of which we compute the percent is called the base number. In the above example Rs. 10 and then Rs. 12 are the two base numbers. The number of percent is the rate (in above example 20 is the rate) and the rate applied to the base number is the part. These all depends on the application the part can have many different names. For example, when the problem is about money, it could be a raise, a discount, a fee, a commission, etc.
Now the basic formula of calculating percentage is denoting the whole by “P”, the rates by “R” and the base by “b” and the basic formula is:
P = R / 100 x b
Generally percentage problems become distinct by what is known and what needs to be calculated, and to recheck their difficulty usually axis from describing the base number only inherently. Another difficulty which might come is that commonly the part is not of interest in itself, but what is required that it needs to be added or subtracted to the base number to get the result of interest.
Let's take an example to understand it better:
-
Unknown Part: If a person buys a gadget at a 25 percent discount. The list price is Rs. 2000. What is the total price?
“R” = 25%
So : P = 25/100 x 2000 = 500
Finally the purchase price = 2000 – 500 = Rs. 1500.
2. Unknown Part: If any person invest money at an annual interest rate of P percent. The interest is paid monthly. Thus according to the rules every month the bank adds:
1 + p / 12,00
Unknown base number. If any person buys a car for Rs. 4 lakh and the dealer tells the buyer that you have purchased it at a 25% discount. What is the list price of the car? As we all know that buyer received a 25 percent discount buyer purchased the car for 75 percent of its list price. Let's just denote the list price by L we get
400,000 = 75/100 x L
Now we get :
L = 400,000 x 100/75 = 533333.33
-
Unknown Rate: The population of the town is 17,000. A year after it becomes 17,678. Find the annual rate by which the population is growing ? The rate R:
17,678 – 17,000 = 678 = R/100 x 17,000
R = 678 x 100/17000 = 4 (approximately)
The growth rate in your town is about 4 percent.
Generally problems involving percentage involve more than one base, rate and part.
Now moving towards another concept that is never divide by zero. Generally problems involving percentage involve more than one base, rate and part. I suggest you all that a daily practice and hard work is what more commonly needed. The most important thing is that there is no alternative to hard work.
Division by zero is undefined. If anyone going to assign a value or number to the outcome of dividing by zero he/she might run into conflict and mathematical world would become useless. In next session we would go through this in detail.
This is all for today in the next session we will go through the above concept along with some complicated percentage problems. Division by zero is an interesting topic which should tell you all the basic fundamentals of mathematics and also tell you that how a zero can change the whole course of the mathematics. We all know without zero there is no mathematical terms and here with zero the mathematics becomes useless. So it would be one of the most interesting chapter.
No comments:
Post a Comment