IX grade mathematics
Hello Friends today we all are going to understand the basic concept behind one of the most important and a bit complex of IX standard that is Fraction. Before proceeding further, let's talk about the basic concept behind Fractions. In simple mathematics a fraction is an expression that consists of numerator and denominator. For example a/b where a is the numerator and b is the denominator and both are integers. The most important condition is that denominator can never be a zero. Here we will try to understand some of the basic facts about fractions and also try to know the reason behind these facts.
Few of the facts about fractions are:
Now lets go in detail with fractions. A fraction such as 5/4 is simplified in a kind of manner to explain the result of dividing 5 by 4. Commonly, A fraction X/Y explains the result of dividing X by Y. This explanation is true even if X and Y are not integers, an explanation of the principle that we make definitions in simple contexts and then generalize them so that all relevant rules remain true. However, if X and Y are not integer then the expression X/Y is called a ratio or quotient.
Division in a fraction can be defined as the solution of a multiplication problem.
Thus the fraction x = 5/4
4x = 5
In more general form, the fraction
x = a/b
is the solution of the equation
bx = a.
The most important thing is to understand the basic definition as all of the above rules can be derived from it
Now let's see how to add fractions with different denominators:
Whenever we are going to add two fractions like 1/3 and 3/7 then we firstly need to convert them to fractions with the same denominator by applying the above rule number 3. Therefore we need to extract a common denominator. In a practical manner the more smaller the denominator, the more easier it is to manipulate the numbers and that time we mostly like to use the least common denominator. Meanwhile, the the product of the two denominators always works, and it is often, like in this case, also the least common denominator. So let's take this example to illustrate it in a more conventional manner:
1/3 = 1*7/3*7 = 7/21 and 3/7 = 3*3/7*3 = 9/21
Therefore:
1/3 + 3/7 = 7/21 + 9/21 = 7+9/21 = 16/21.
Now the next thing is how to multiply fraction :
Suppose we have two fractions that are x = 3/5 and y = 14/9
This shows that x and y satisfy the equations:
5x = 3 and 9y = 14
Now moving further and multiplying on both the sides of the first equation with 14 but we need to call 9y on the left hand side, we obtain
5x X 9y = 3 X 14
This can be further simplified as
45 xy = 42
which is the another way of explaining this:
xy = 42/45
Additionally we can further simplify this fraction by dividing numerator and denominator by 3 which gives us : 14/15
Now the next topic to illustrate is division of fractions:
Let's take the same above example and try to perform division operations with same fractions then:
a = x divide y = 3/5 divide 14/9
In this, division is the inverse procedure of multiplication, so it satisfies the equation
That shows as ay = x
or we can explain as
a X 14/9 = 3/5 We can find out the exact answer by multiplying on both sides of this equation with the reciprocal of y that is with 9/14
a X 14/9 X 9/14 = 3/5 X 9/14
This becomes : a * (14*9 / 9* 14) = 3*9/5*14
x divide y = a = 3*9 / 5*14 = 27/70.
Now the next topic to study is Greatest Common Factors and Least Common Multiples:
In simple mathematical manner a multiple of a natural number is the product of that number with another natural number. The most common example is, 6, 18, and 333 are all multiples of 3. Another thing to understand is that a factor of a natural number n is another natural number that divides n without remainder. Example to understand it, the factors of 12 are 1, 2, 3, 4, 6, 12. Similar concept is that a common factor of two natural numbers m and n is basically a number that is a factor of m and also of n. Let's take an example, 8 is a common factor of 32 and 48. Other common factors are 1, 2, 4, 8, 16.
Now we are going to understand the basic behind greatest common factor. The greatest common factor is just what the name suggests, so the greatest common factor of the above example that is 32 and 48 is 16. Similarly, a common multiple of two numbers is a multiple of both, and the least common multiple is the smallest common multiple. For example, 24, 120, 180 are all common multiples of 4 and 6, and 12 is the least common multiple.
The importance of this statement is that a common denominator of the two fractions is basically a common multiple of the two denominators and the least common denominator is generally the least common multiple of the two denominators.
The least common multiple (LCM) and the greatest common factor (GCF) of two numbers m and n are explained by the form:
LCM = m * n/ GCF
Let's take an example to understand is better, the Greatest common factor of 4 and 6 is 2 and their least common multiple is 12. So by using above formula we can formulate it:
12 = 4*6/2
Now the another area of study is Mixed Numbers or Mixed Fractions:
Mixed Numbers are basically fractions which are written as a natural number plus a fraction in which the denominator is always greater than the numerator. This area of study is important because the integer part in the fraction gives an indication of their sizes.
Mixed numbers are popular because the integer part gives an explanation of their size, but otherwise they have little to glorify them. They always form an exception (that is the only exception) to the rule that a unidentified or missing operator means multiplication, and they make the given arithmetic operations harder to carry out.
Now the another part of fractions are complex fractions. A complex fraction is a rational expression that has a fraction in its numerator, denominator or both. Example to show the complex fraction: a/b / c/d. The sign / denotes the division of two numbers or variables. Simplifying complex fractions involves following steps: firstly student needs to rewrite the numerator and denominator so that each form a single fraction. Now divide the numerator by the denominator by multiplying the numerator by the reciprocal of the denominator. In last step we need to simplify the rational expression.
So this is enough for today. In next class we all are going to learn the divisibility rules of the fractions and the use of Euclidean Algorithm while solving fractions.
Hello Friends today we all are going to understand the basic concept behind one of the most important and a bit complex of IX standard that is Fraction. Before proceeding further, let's talk about the basic concept behind Fractions. In simple mathematics a fraction is an expression that consists of numerator and denominator. For example a/b where a is the numerator and b is the denominator and both are integers. The most important condition is that denominator can never be a zero. Here we will try to understand some of the basic facts about fractions and also try to know the reason behind these facts.
Few of the facts about fractions are:
-
The integer “x” on the top of the fraction bar is the numerator and the integer “y” at the bottom is the denominator of the fraction.
Fraction = numerator / denominator
-
Any of the integer can be considered as a fraction having a denominator equal to 1.
X = X/1
-
If we are going to multiply numerator and denominator of a fraction with the same non zero factor does not manipulate the fraction. Basically in elementary school we classify it with the term equivalent fractions, but a better view is that equivalent fractions denote the same real number and need not be distinguished other than for simplicity.
X/Y = CX/CY where C is not equal to zero.
-
A fraction of having the same denominator are added or subtracted by adding or subtracting the numerators.
X/y + C/y = X + C / y
X/y - C/y = X - C / y
-
If the fractions come with distinct denominators than firstly we need to turn them into fractions with the same denominators. This can be done by multiplying numerators and denominators with suitable factors, and then added using the rule above.
X/y + c/d = Xd / yd + cy / yd = Xd + cy/ yd
Similarly we can perform subtraction :
X/y - c/d = Xd / yd - cy / yd = Xd - cy/ yd
-
In the case of multiplication, fractions are multiplied by multiplying numerators and multiplying denominators.
X/Y * C/D = XC/YD
-
We can find the reciprocal of a fraction by switching numerators and denominators.
-
In division case : If we are going to divide a fraction by another fraction which is equivalent to multiplying with the reciprocal of the second fraction.
X/Y divide C/D = XD / YC
Now lets go in detail with fractions. A fraction such as 5/4 is simplified in a kind of manner to explain the result of dividing 5 by 4. Commonly, A fraction X/Y explains the result of dividing X by Y. This explanation is true even if X and Y are not integers, an explanation of the principle that we make definitions in simple contexts and then generalize them so that all relevant rules remain true. However, if X and Y are not integer then the expression X/Y is called a ratio or quotient.
Division in a fraction can be defined as the solution of a multiplication problem.
Thus the fraction x = 5/4
4x = 5
In more general form, the fraction
x = a/b
is the solution of the equation
bx = a.
The most important thing is to understand the basic definition as all of the above rules can be derived from it
Now let's see how to add fractions with different denominators:
Whenever we are going to add two fractions like 1/3 and 3/7 then we firstly need to convert them to fractions with the same denominator by applying the above rule number 3. Therefore we need to extract a common denominator. In a practical manner the more smaller the denominator, the more easier it is to manipulate the numbers and that time we mostly like to use the least common denominator. Meanwhile, the the product of the two denominators always works, and it is often, like in this case, also the least common denominator. So let's take this example to illustrate it in a more conventional manner:
1/3 = 1*7/3*7 = 7/21 and 3/7 = 3*3/7*3 = 9/21
Therefore:
1/3 + 3/7 = 7/21 + 9/21 = 7+9/21 = 16/21.
Now the next thing is how to multiply fraction :
Suppose we have two fractions that are x = 3/5 and y = 14/9
This shows that x and y satisfy the equations:
5x = 3 and 9y = 14
Now moving further and multiplying on both the sides of the first equation with 14 but we need to call 9y on the left hand side, we obtain
5x X 9y = 3 X 14
This can be further simplified as
45 xy = 42
which is the another way of explaining this:
xy = 42/45
Additionally we can further simplify this fraction by dividing numerator and denominator by 3 which gives us : 14/15
Now the next topic to illustrate is division of fractions:
Let's take the same above example and try to perform division operations with same fractions then:
a = x divide y = 3/5 divide 14/9
In this, division is the inverse procedure of multiplication, so it satisfies the equation
That shows as ay = x
or we can explain as
a X 14/9 = 3/5 We can find out the exact answer by multiplying on both sides of this equation with the reciprocal of y that is with 9/14
a X 14/9 X 9/14 = 3/5 X 9/14
This becomes : a * (14*9 / 9* 14) = 3*9/5*14
x divide y = a = 3*9 / 5*14 = 27/70.
In simple mathematical manner a multiple of a natural number is the product of that number with another natural number. The most common example is, 6, 18, and 333 are all multiples of 3. Another thing to understand is that a factor of a natural number n is another natural number that divides n without remainder. Example to understand it, the factors of 12 are 1, 2, 3, 4, 6, 12. Similar concept is that a common factor of two natural numbers m and n is basically a number that is a factor of m and also of n. Let's take an example, 8 is a common factor of 32 and 48. Other common factors are 1, 2, 4, 8, 16.
Now we are going to understand the basic behind greatest common factor. The greatest common factor is just what the name suggests, so the greatest common factor of the above example that is 32 and 48 is 16. Similarly, a common multiple of two numbers is a multiple of both, and the least common multiple is the smallest common multiple. For example, 24, 120, 180 are all common multiples of 4 and 6, and 12 is the least common multiple.
The importance of this statement is that a common denominator of the two fractions is basically a common multiple of the two denominators and the least common denominator is generally the least common multiple of the two denominators.
The least common multiple (LCM) and the greatest common factor (GCF) of two numbers m and n are explained by the form:
LCM = m * n/ GCF
Let's take an example to understand is better, the Greatest common factor of 4 and 6 is 2 and their least common multiple is 12. So by using above formula we can formulate it:
12 = 4*6/2
Now the another area of study is Mixed Numbers or Mixed Fractions:
Mixed Numbers are basically fractions which are written as a natural number plus a fraction in which the denominator is always greater than the numerator. This area of study is important because the integer part in the fraction gives an indication of their sizes.
Mixed numbers are popular because the integer part gives an explanation of their size, but otherwise they have little to glorify them. They always form an exception (that is the only exception) to the rule that a unidentified or missing operator means multiplication, and they make the given arithmetic operations harder to carry out.
Now the another part of fractions are complex fractions. A complex fraction is a rational expression that has a fraction in its numerator, denominator or both. Example to show the complex fraction: a/b / c/d. The sign / denotes the division of two numbers or variables. Simplifying complex fractions involves following steps: firstly student needs to rewrite the numerator and denominator so that each form a single fraction. Now divide the numerator by the denominator by multiplying the numerator by the reciprocal of the denominator. In last step we need to simplify the rational expression.
So this is enough for today. In next class we all are going to learn the divisibility rules of the fractions and the use of Euclidean Algorithm while solving fractions.
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