Welcome guys! Let us start with the point where we had left in previous article and that is Number system of grade VI Algebra (also covered in 8th grade math). In our last article we have discussed about the factorization technique to sort out the problems of rational numbers and today we are going to follow the Number system content whose next topic is Fractions and variety of arithmetic operations that are applied on its queries to be sorted out.
Let us start with the demonstration of Fraction first. Fraction of any number means a part of something and presented by using two terms that are numerator and denominator. These terms are related by division operator to form the fraction, for example:
2/3; here 2 is numerator and 3 is denominator. Students are well aware with the general presentation of the fraction because it was included in grade 5 math syllabus as well, so here we are going to be specific with the operations that are entrenched in this particular grade VI math content.
So now move towards the general operations that are required to learn for solving fraction queries, firstly start with Least Common Multiple. When any fraction requires the calculation of Least Common Multiple that means student has to convert the unlike terms in the denominator of the fractions into their corresponding like terms and this can be done in following ways:
1. Like general multiplication the product of two denominator value provides a common multiple that can be explained by following example:
Common multiple of 10 and 30 is 10 x30 = 300.
But 300 may not be the least common multiple of them.
So to find LCM of 10 and 30 first find multiples of each as
Multiples of 10 are: 10, 20, 30 …
Multiples of 30 are: 30, 60, 90 …
so, the least common multiple of 10 and 30 is 30.
2. Another way to find LCM is by removing the frequent occurrence of common factors. We have 5 as a factor common to both 10 and 30. We remove the repeat occurrence of 5 from the product to get the LCM.
Product = 10 x15 = ( 2 x 5) x (2 x 3 x 5) = 300
Least common multiple = ( 2 x 5) x ( 3) = 30
Its reverse procedure includes the use of prime factors as: prime factor of LCM value contains the prime factor of the numbers.
30 = 5 x 2 x 3 = (5 x 2) x 3 = 10 x 3 and 30 is a multiple of 10
30 = 5 x 2 x 3 = (5 x 3) x 2 = 15 x 2 and 30 is a multiple of 15
3. The above phenomenon can be applied by using short division technique that is executed as:
5 | 10, 15 (factor out 5)
2, 3
Since, there are no other common factors to the quotients in the bottom so multiply the common factor to the remaining quotients to evaluate the LCM of the given numbers.
LCM = 5 x 2 x 3 = 30
Let us take one more example to understand the procedure more easily:
Determine LCM of 42 and 63.
7 | 42, 63 (factor out 7 from the numbers)
3 | 6, 9 (here 3 is common factor)
2, 3 (no other common factor)
Now as told in above part, multiply the common factors with the quotient value to get the LCM as:
LCM = 7 x 3 x 2 x 3 = 126
4. When there are more than two numbers then find all the prime factors first that are common to at-least two of those numbers to find the LCM.
Example: calculate LCM of 9, 14 and 21.
Use the short division technique
3 | 9, 14, 21 ( 3 is a prime factor for all these values)
7 | 3, 14, 7 ( here we are taking 7 because it is prime to 7 and 14)
3, 2, 1 (here no prime factor can be manipulated in parallel with condition mentioned above)
So find the LCM by following the similar procedure of multiplying quotient with common factors
LCM = 3 x 7 x 3 x 2 x 1 = 126
Now check whether LCM contains the prime factors of the number or not;.
126 = 3 x 7 x 3 x 2
= (3 x 3) x 2 x 7 = 9 x 14
= (7 x 2) x 3 x 3 = 14 x 9
= (3 x 7) x 2 x 3 = 21 x 6
This is how to evaluate least common multiple of the fraction so let us move towards the next operation. Addition and subtraction like general operations evaluation on fraction is also included in grade 5 syllabus and here also we are not going to explore that operation because you guys are already aware of it, so let's start with Multiplication and division operation of fractions.
Whenever you perform multiplication or division that time the overall operation is a mixed form of division and multiplication, still there are some ways to deal out with this kind of fraction queries as explained below with proper example:
1. Multiplication of fractions may be converted to mixed operations involving multiplication and division with natural numbers.
3/4 x 5/8 = (3 ÷ 4) x (5 ÷ 8) = 3 ÷ 4 x 5 ÷ 8
In mixed operations that involve multiplication and division, we may divide the product of dividends by the product of divisors.
3 ÷ 4 x 5 ÷ 8 = (3 x 5) ÷ (4 x 8)
Similarly, in multiplication of fractions, we may directly “divide” the product of numerators by the product of denominators.
3/4 X 5/8 = (3 x 5)/ (4 x 8) = 15/32
2. Similar to mixed operations, we may cancel out factors that are common to numerators and denominators.
Example: Multiply 8/27 by 15/16
8/27 X 15/16 = 1/27 X 15/2 (factor out 8)
Now factor out 3
= 1/9 X 5/2
= 5/18
3. If there is “no denominator,” then the number is considered a “numerator.”
(a) Three fifths of a class of 35 is girls. How many girls are in that class?
3/5 of 35 = 3/5 X 35
Factor out 5
= 3/1 X 7/1
= 21
4. To multiply mixed numbers, convert them to improper fractions first.
Example: what is 1(1/2) of 2(1/2)?
2(1/2) X 1(1/2) = 5/2 X 3/2
= 15/4
= 3 (3/4)
5. The reciprocal of a number is obtained by switching numerator and denominator. The product of a number and its reciprocal is always 1.
(a) The reciprocal of 2/3 is 3/2, and their product is 1.
2/3 X 3/2 = (cancel out the common terms) = 1/1 = 1
(b) The reciprocal of 2 is 1/2 because the denominator of whole number is 1.
2 X 1/2 = 2/1 X ½ = 1/1 = 1
6. When situation arises where mixed numbers are involved with the division operator that time they have to be converted first into normal fraction form.
Example: divide 6(2/5) by 2(2/15)
6(2/5) ÷ 2(2/15) = 32/5 ÷ 32/15
Factor out 5 cancel common terms
1/1 X 3/1 = 3/1 = 3
Some points which students should memorize while solving fraction problems:
1. “Like’ fractions are added by doing the simple addition of numerators. Subtraction is also performed in similar way by just replacing addition operation with subtraction but denominator remains always same.
2. Additional step included when unlike fractions are added or subtracted that time first convert them to like fractions.
3. To convert unlike fractions into like fractions, we first calculate the LCM (least common multiple) of all the unlike denominators of the fraction and then we will calculate the equivalent fractions for unlike fractions with LCM (as the new denominator of the fraction).
4. To multiply fractions, just simply multiply numerators together to have product of the numerators and multiply the denominators to get the denominator’s product. To divide fraction by a fraction, reciprocal of the second fraction is taken and then multiplied with the first one.
5. In general practice, a fraction in the final answer, always expressed in its lowest terms and this is done by eliminating all the common factors out of the numerator and the denominator.
6. A “division” notation is not the only notation possible to express fractions, students also can prefer “Decimal notation”.
For more tips and tricks to sort out fraction problems students can use the online math tutoring service, where proficient math tutors are always available to sort out students’ queries.
This is all about fraction that we have explored in this session with its frequent arithmetic operations and we will be continuing with rest of the topics in Number System unit of grade VI Algebra in our successive article, so keep following and practicing with the various worksheets provided on the math website.
In upcoming posts we will discuss about Circles in Grade VI. Visit our website for information on class 9 ICSE syllabus
Let us start with the demonstration of Fraction first. Fraction of any number means a part of something and presented by using two terms that are numerator and denominator. These terms are related by division operator to form the fraction, for example:
2/3; here 2 is numerator and 3 is denominator. Students are well aware with the general presentation of the fraction because it was included in grade 5 math syllabus as well, so here we are going to be specific with the operations that are entrenched in this particular grade VI math content.
So now move towards the general operations that are required to learn for solving fraction queries, firstly start with Least Common Multiple. When any fraction requires the calculation of Least Common Multiple that means student has to convert the unlike terms in the denominator of the fractions into their corresponding like terms and this can be done in following ways:
1. Like general multiplication the product of two denominator value provides a common multiple that can be explained by following example:
Common multiple of 10 and 30 is 10 x30 = 300.
But 300 may not be the least common multiple of them.
So to find LCM of 10 and 30 first find multiples of each as
Multiples of 10 are: 10, 20, 30 …
Multiples of 30 are: 30, 60, 90 …
so, the least common multiple of 10 and 30 is 30.
2. Another way to find LCM is by removing the frequent occurrence of common factors. We have 5 as a factor common to both 10 and 30. We remove the repeat occurrence of 5 from the product to get the LCM.
Product = 10 x15 = ( 2 x 5) x (2 x 3 x 5) = 300
Least common multiple = ( 2 x 5) x ( 3) = 30
Its reverse procedure includes the use of prime factors as: prime factor of LCM value contains the prime factor of the numbers.
30 = 5 x 2 x 3 = (5 x 2) x 3 = 10 x 3 and 30 is a multiple of 10
30 = 5 x 2 x 3 = (5 x 3) x 2 = 15 x 2 and 30 is a multiple of 15
3. The above phenomenon can be applied by using short division technique that is executed as:
5 | 10, 15 (factor out 5)
2, 3
Since, there are no other common factors to the quotients in the bottom so multiply the common factor to the remaining quotients to evaluate the LCM of the given numbers.
LCM = 5 x 2 x 3 = 30
Let us take one more example to understand the procedure more easily:
Determine LCM of 42 and 63.
7 | 42, 63 (factor out 7 from the numbers)
3 | 6, 9 (here 3 is common factor)
2, 3 (no other common factor)
Now as told in above part, multiply the common factors with the quotient value to get the LCM as:
LCM = 7 x 3 x 2 x 3 = 126
4. When there are more than two numbers then find all the prime factors first that are common to at-least two of those numbers to find the LCM.
Example: calculate LCM of 9, 14 and 21.
Use the short division technique
3 | 9, 14, 21 ( 3 is a prime factor for all these values)
7 | 3, 14, 7 ( here we are taking 7 because it is prime to 7 and 14)
3, 2, 1 (here no prime factor can be manipulated in parallel with condition mentioned above)
So find the LCM by following the similar procedure of multiplying quotient with common factors
LCM = 3 x 7 x 3 x 2 x 1 = 126
Now check whether LCM contains the prime factors of the number or not;.
126 = 3 x 7 x 3 x 2
= (3 x 3) x 2 x 7 = 9 x 14
= (7 x 2) x 3 x 3 = 14 x 9
= (3 x 7) x 2 x 3 = 21 x 6
This is how to evaluate least common multiple of the fraction so let us move towards the next operation. Addition and subtraction like general operations evaluation on fraction is also included in grade 5 syllabus and here also we are not going to explore that operation because you guys are already aware of it, so let's start with Multiplication and division operation of fractions.
Whenever you perform multiplication or division that time the overall operation is a mixed form of division and multiplication, still there are some ways to deal out with this kind of fraction queries as explained below with proper example:
1. Multiplication of fractions may be converted to mixed operations involving multiplication and division with natural numbers.
3/4 x 5/8 = (3 ÷ 4) x (5 ÷ 8) = 3 ÷ 4 x 5 ÷ 8
In mixed operations that involve multiplication and division, we may divide the product of dividends by the product of divisors.
3 ÷ 4 x 5 ÷ 8 = (3 x 5) ÷ (4 x 8)
Similarly, in multiplication of fractions, we may directly “divide” the product of numerators by the product of denominators.
3/4 X 5/8 = (3 x 5)/ (4 x 8) = 15/32
2. Similar to mixed operations, we may cancel out factors that are common to numerators and denominators.
Example: Multiply 8/27 by 15/16
8/27 X 15/16 = 1/27 X 15/2 (factor out 8)
Now factor out 3
= 1/9 X 5/2
= 5/18
3. If there is “no denominator,” then the number is considered a “numerator.”
(a) Three fifths of a class of 35 is girls. How many girls are in that class?
3/5 of 35 = 3/5 X 35
Factor out 5
= 3/1 X 7/1
= 21
4. To multiply mixed numbers, convert them to improper fractions first.
Example: what is 1(1/2) of 2(1/2)?
2(1/2) X 1(1/2) = 5/2 X 3/2
= 15/4
= 3 (3/4)
5. The reciprocal of a number is obtained by switching numerator and denominator. The product of a number and its reciprocal is always 1.
(a) The reciprocal of 2/3 is 3/2, and their product is 1.
2/3 X 3/2 = (cancel out the common terms) = 1/1 = 1
(b) The reciprocal of 2 is 1/2 because the denominator of whole number is 1.
2 X 1/2 = 2/1 X ½ = 1/1 = 1
6. When situation arises where mixed numbers are involved with the division operator that time they have to be converted first into normal fraction form.
Example: divide 6(2/5) by 2(2/15)
6(2/5) ÷ 2(2/15) = 32/5 ÷ 32/15
Factor out 5 cancel common terms
1/1 X 3/1 = 3/1 = 3
Some points which students should memorize while solving fraction problems:
1. “Like’ fractions are added by doing the simple addition of numerators. Subtraction is also performed in similar way by just replacing addition operation with subtraction but denominator remains always same.
2. Additional step included when unlike fractions are added or subtracted that time first convert them to like fractions.
3. To convert unlike fractions into like fractions, we first calculate the LCM (least common multiple) of all the unlike denominators of the fraction and then we will calculate the equivalent fractions for unlike fractions with LCM (as the new denominator of the fraction).
4. To multiply fractions, just simply multiply numerators together to have product of the numerators and multiply the denominators to get the denominator’s product. To divide fraction by a fraction, reciprocal of the second fraction is taken and then multiplied with the first one.
5. In general practice, a fraction in the final answer, always expressed in its lowest terms and this is done by eliminating all the common factors out of the numerator and the denominator.
6. A “division” notation is not the only notation possible to express fractions, students also can prefer “Decimal notation”.
For more tips and tricks to sort out fraction problems students can use the online math tutoring service, where proficient math tutors are always available to sort out students’ queries.
This is all about fraction that we have explored in this session with its frequent arithmetic operations and we will be continuing with rest of the topics in Number System unit of grade VI Algebra in our successive article, so keep following and practicing with the various worksheets provided on the math website.
In upcoming posts we will discuss about Circles in Grade VI. Visit our website for information on class 9 ICSE syllabus
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