Consecutive Odd Integers

In the previous post we have discussed about How to solve Consecutive Integers and In today's session we are going to discuss about Consecutive Odd Integers. Let us first talk about the integers. The numbers which can be expressed in the form of the whole numbers and their additive inverse are called integers. All the integers can be expressed on a number line. A number line contains the series of all positive and negative numbers marked at the equal interval, where we could observe the number zero at the center, positive numbers at the right of the number line and negative numbers at the left of the number line.  Now we will learn about the Consecutive Odd Integers. By the consecutive numbers, we mean the numbers appearing one after another. Thus 1, 2, 3 . . .  are consecutive integers.  Now if we talk about the consecutive odd integers, we mean the series of odd integers which occur one after another but are odd.  So we say that the series of odd consecutive integers are 1, 3, 5, 7, 9 , . . . . . . etc.  In case of writing the series of odd integers in the form  of general series we will assume any number x. We know that 2 * x will always be the odd number. Now on another hand we say that if we add 1 to the odd number, it becomes the odd number. For this we say that the number 2x + 1 is any odd number. Now to write the series of the odd consecutive integers, we write 2x + 1, 2x + 3, 2x + 5, 2x +7 . . . . etc. (know more about Integer, here

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How to solve Consecutive Integers

Integers are the numbers which can be expressed in the series from minus infinite to plus infinite. Now let us look at the consecutive integers. Two integers which exist one after another when they are expressed on the number line are called consecutive numbers. If we have any number n, then the successor  and the predecessor of the number n forms the series of the consecutive numbers. So if we have n = 5, then we have n-1 = 5 – 1 = 4 and  n + 1 = 5 + 1 = 6. Thus the series 4 , 5 , 6  are called consecutive numbers.   There series of consecutive numbers can be used to solve many word problem related to linear equations.  In case we talk about any 3 consecutive integers, we say that series will be x, x + 1 and x + 2. On the other hand if we need to write the series of even numbers, then we know that each even number is the multiple of 2, so the first number will be 2 * x. (know more about Integers, here

Thus the series of consecutive integers will be  2x , 2x + 2, 2x + 4. Now we will assume the series of odd consecutive integer numbers. For this we will assume the first odd integer as  2x + 1. The next odd integer will be  2x + 3, 2x + 5. Thus the series of consecutive odd numbers will be  2x + 1, 2x + 3 and 2x + 5.
 If we have the problem that the sum of 3 consecutive integers is 36, find the integers. Then we will first assume the three numbers as x , x + 1 and x + 2. Thus we represent the sum as :
X + x + 1 + x + 2  = 36
Or 3x + 3 = 36
Or 3x = 36 – 3
Or 3x = 33
Or x = 33/ 3 = 11
SO  the three numbers are 11, 12 and 13.
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Problem solving strategies

To become a better problem solver you need to analyze the problem in deep first and it require a systematic approach. Problem solving strategies requires practice, the more you practice, the better you get.
 Strategies of problem solving are:
1.      Look for the clues: In problem solving strategies, first read the problem carefully and underline the clue words. Look for the facts given and what do you need to find out.

2.      Make a plan: In strategies of problem solving, firstly set a plan. Or if you have done any problem like this before then do what you did.  You can use formulae, sketches, tables or any pattern.

3.      Now solve: In problem solving strategies, whatever plan you have thought of, just solve the problem according to it.

4.      Checking your answer: In strategies of problem solving, look over the solution. Check if it is the probable answer and also check for the units to be used in the answer.

In problem solving strategies, the first thing you do is looking for a clue, for which you need to a skills in solving problems and practice is most important here.
For example: In strategies of problem solving, if we are looking for clues in addition.
Then clues can be total, sum, in all, perimeter and so on.
Now in problem solving strategies, if you are looking for the subtraction clues, then they can be difference, exceed or how much more.
Similarly, in strategies of problem solving, if we are looking for the clues of multiplication, then you should look for the words: total, area, times, product, etc.
And in problem solving strategies, if you are looking for the clues related to division, then you should look for the words: distribute, quotient, average, share, divide,etc.


In strategies of problem solving, after all of the above points , one point which you need to remember is practice as much as you can to become a good problem solver.
In The Next Session We Are Going To Discuss Grade VI, Tools To Solve Problems.

Multiplying Mixed Numbers

In the previous session we discuss about Prime and Composite Numbers and now today we will discuss about Multiplying Mixed Numbers. In the mathematic a whole number is combined with a fraction is known as mixed number.
For example: 4 1 and 3 5
                        5          8
These are the examples of mixed fraction.
Now we will see process multiplying mixed numbers.
For multiplication of mixed fraction we have to follow some steps which are:
Step1: For the multiplication of mixed fraction firstly take a mixed number.
Step2: Then after change the mixed number into an improper fraction.
Step3: Then we write the number in multiplication order.
Step4: Then we will see the fraction values, if the values are divisible then we cancel the value otherwise we have to multiply numerator value of one fraction into another and multiply the denominator value to the other fraction.
Step5: At last again we get the result.
Suppose we have any mixed number 16 14/12 multiply to 10 8/14.
For the multiplication mixed numbers we have to follow all the above following steps:
Step1: firstly write the mixed number which is given:
   16 14/12 * 10 8/14;
Step2: Now we have to convert mixed number into improper fraction.
 = 206 * 148,
     12        14
Now multiply the numerator value of one fraction by another value and multiply denominator by other denominator value.
 = 206 * 148,
     12 * 14
On further solving we get:
⇒ 30488,
     168
Now divide the numerator value to the denominator value.
On dividing we get:
⇨ 181.47
On multiplication we get the 181.47.
Suppose we have any mixed number 8 7/10 multiply 4 8/9.
For multiplication of mixed numbers we have to follow all the above following steps:
Step1: firstly write the mixed number which is given:  
   8 7/10 * 4 8/9;
Step2: Now we have to convert mixed number into improper fraction.
 = 87 * 44,
    10     9
 = 87 * 44,
       90
Step4: Now multiply both the numerator and denominator values.
On multiplication we have:
⇒ 3696,
      90
On multiplication we get the 3696/90.
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Prime and Composite Numbers

Having learnt about factors of numbers, we will learn about another classification of numbers on the basis of factors. This is classification of numbers as prime and composite numbers. Before we try to know about this concept, let us recall factors. Factors of a number are the numbers which divide the given number completely or we can say that if a number is completely divisible by some other number, then the latter is called the factor of the former number. It is also important to note here that any number is always completely divisible by 1 & itself.
Turning to our topic of discussion here, let us define prime and composite numbers.
A number which has exactly two factors is called prime number. Such numbers are completely divisible by 1 & itself only, no other number completely divides such numbers. If we take 5 & list its factors, we get only 2 factors of 5, viz.,1 & 5; so 5 is a prime number. Similarly, 2, 3, 7, 11, 13, 17,19, 23, 29,etc are all prime numbers.
A number which has at least 3 factors, i.e., other than 1 & itself, it is divisible by at least 1 other number is called a composite number. If we take 4 & list its factors, we get 3 factors of 4, viz.,1 , 2 & 4 itself; so 4 is a composite number. Similarly, 6, 8, 9, 10, 12, 14, 16, etc are composite numbers.
In listing the examples of prime & composite numbers, we notice that our numbers begin with 2; not 1. It is because 1 is the only number which has only 1 factor, i.e.,1. It is 1 itself, hence, it does not fit either of the definitions of prime or composite numbers. Thus, we say that 1 is neither prime nor composite.
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What is a Ratio

In the previous post we have discussed about Fractions and In today's session we are going to discuss about What is a Ratio, When we need to compare the two quantities, we say that we are going to find the ratio of the two numbers. Now let us learn what is a Ratio? We say that a ratio shows the relative sizes of two or more values. We can represent the ratio of the two quantities in different ways. First method to show the ratio is by colon sign “:”. Here we write two items separated by a colon. Another way to represent the ratio is as a single number by dividing one value by the total. If we take the examples of both the cases, we say that:
If there are 25 boys and 17 girls in the class and we need to represent the ratio of boys: girls, we write  25  ratio 17 or it can be expressed as 25 :17
 In case we write there is one boy and 3 girls, then we represent the ratio as 1 : 3 and we write  that for every 1 boy, there are 3 girls.
The above mentioned statement tells that there are in all 1 + 3 students, so we can also represent the above ratio as ¼ are boys and ¾ are girls. Here we can say that out of total of 4 children, we have 1 out of 4 as boys and 3 out of 4 as girls.
 So on dividing ¼, we get  0.25 are boys and 0.75 are girls. This statement leads us to the conclusion that  25 % of all the students are boys and 75 % of all the children are girls.
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Fractions

In mathematics fraction means the numbers are in the form a/b. In fraction ‘a’ is called numerator part and ‘b’ is called denominator part and in fraction both the parts numerator and denominator are integer numbers.
Now we are going to explore fractions. Basically fractions are of three types and they are given as:
1. Proper fraction.
2. Improper fraction.
3. Mixed fraction.
Proper fraction: Fractions whose numerator is smaller than denominator is called proper fraction.
For example: - 1/3, 3/5, 6/9 etc are proper fraction.
Improper fraction: Fractions in which numerator part is larger than the denominator part is called as improper fraction.
For example: - 4/2, 8/4, 7/4 etc are improper fraction.
Mixed fraction: When the numbers consist of whole number as first part and second part as fraction are called mixed fraction.
For example: - 2(1/2), 7(3/9) etc are mixed fraction.
Now see how to add fractions and there are some rules which we have to follow while adding fraction numbers. Suppose we have two fraction numbers 2/5 and 4/5 now we want to add these two numbers before adding two fractions we have to take LCM (least common factor) of both denominators
        2/5 + 4/5 = (2+4)/5
Now it is clearly seen that both the denominators are same so denominator become 5 and we add both the numerator and get:
(2+4)/5 = 6/5,
So after adding two fractions we get the result and that is 6/5.
Now take another example where both denominators are different. Two numbers are 3/2 and 5/4, now taking the LCM of denominators and the LCM is 4.
Then, 3/2 + 5/4 = (6+5)/4,
          (6+5)/4 = 11/4,
So the final result is 11/4 and by this way we are going to solve the fractions.
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And In the next session we will discuss about What is a Ratio.