OPERATIONS WITH FRACTIONS
About "Operations with fractions"
Operations with fractions :
We can not imagine the subject Math without the term "Fraction". Because, we solve many problems with it. So, we must be aware of the operations which we often do in fractions.
Let us explore the following operations with fractions.
(i) Addition
(ii) Subtraction
(iii) Multiplication
(iv) Division
(v) Reciprocal
Addition and subtraction of fractions with same denominator
Example 1 :
Simplify : 2/5 + 3/5
Solution :
Here, for both the fractions, we have the same denominator, we have to take only one denominator and add the numerators.
Then, we get
2/5 + 3/5 = (2+3) / 5 = 5/5 = 1
Example 2 :
Simplify : 7/5 - 3/5
Solution :
Here, for both the fractions, we have the same denominator, we have to take only one denominator and subtract the numerators.
Then, we get
7/5 - 3/5 = (7-3) / 5 = 4/5
Addition and subtraction of fractions with different denominators
Here, we explain two methods to add two fractions with different denominators.
1) Cross multiplication method
2) L.C.M method
Cross multiplication method :
If the denominators of the fractions are co-prime or relatively prime, we have to apply this method.
Fro example, let us consider the two fractions 1/8, 1/3.
In the above two fractions, denominators are 8 and 3.
For 8 and 3, there is no common divisor other than 1. So 8 and 3 are co-prime.
Here we have to apply cross-multiplication method to add the two fractions 1/8 and 1/3 as given below.
L.C.M method :
If the denominators of the fractions are not co-prime (there is a common divisor other than 1), we have to apply this method.
Fro example, let us consider the two fractions 5/12, 1/20.
In the above two fractions, denominators are 12 and 20.
For 12 and 20, if there is at least one common divisor other than 1, then 12 and 20 are not co-prime.
For 12 & 20, we have the following common divisors other than 1.
2 & 4
So 12 and 20 are not co-prime.
In the next step, we have to find the L.C.M (Least common multiple) of 12 and 20.
12 = 2² x 3
20 = 2² x 5
When we decompose 12 and 20 in to prime numbers, we find 2, 3 and 5 as prime factors for 12 and 20.
To get L.C.M of 12 and 20, we have to take 2, 3 and 5 with maximum powers found above.
So, L.C.M of 12 and 20 = 2² x 3 x 5
= 4 x 3 x 5
= 60
Now we have to make the denominators of both the fractions to be 60 and add the two fractions 5/12 and 1/20 as given below.
Note :
We have to do the same process for subtraction of two fractions with different denominators.
Let us look at next stuff on "Fractions in mathematics"
Multiplication of a fraction by a whole number
To multiply a proper or improper fraction with the whole number,first, we have to multiply the whole number with the numerator of the fraction, keeping the denominator same.
For example,
2 x 3/5 = 6/5
3 x 7/11 = 21/11
To multiply a mixed fraction by a whole number, first convert the mixed fraction to an improper fraction and then multiply.
For example,
4 x 3 4/7 = 4 x 25/7 = 100/7 = 14 2/7
Let us look at next stuff on "Fractions in mathematics"
Converting improper fractions to mixed numbers
The picture given below clearly illustrates, how to convert improper fractions in to mixed numbers
Multiplication of a fraction by a fraction
To multiply a proper or improper fraction by another proper or improper fraction, we have to multiply the numerators and denominators.
For example,
2/3 x 4/5 = 8/15
1/3 x 7/11 = 7/33
The reciprocal of a fraction
If the product of two non-zero numbers is equal to one then each number is called the reciprocal of the other.
So the reciprocal of 3/5 is 5/3.
Note:
Reciprocal of 1 is 1 itself. 0 does not have a reciprocal.
Division of a whole number by a fraction
To divide a whole number by any fraction, multiply that whole number by the reciprocal of that fraction.
For example,
6 ÷ 2/5 = 6 x 5/2 = 30/2 = 15
While dividing a whole number by a mixed fraction, first convert the mixed fraction into improper fraction and then solve it.
6 ÷ 3 4/5 = 6 ÷ 19/5 = 6 x 5/19 = 30/19 = 1 11/19
Division of a fraction by another fraction
To divide a fraction by another fraction, multiply the first fraction by the reciprocal of the second fraction.
For example,
1/5 ÷ 3/7 = 1/5 x 7/3 = 7/15
After having gone through the stuff given above, we hope that the students would have understood "Operations with fractions".
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WORD PROBLEMS
HCF and LCM word problems
Word problems on simple equations
Word problems on linear equations
Word problems on quadratic equations
Area and perimeter word problems
Word problems on direct variation and inverse variation
Word problems on comparing rates
Converting customary units word problems
Converting metric units word problems
Word problems on simple interest
Word problems on compound interest
Word problems on types of angles
Complementary and supplementary angles word problems
Markup and markdown word problems
Word problems on mixed fractrions
One step equation word problems
Linear inequalities word problems
Ratio and proportion word problems
Word problems on sets and venn diagrams
Pythagorean theorem word problems
Percent of a number word problems
Word problems on constant speed
Word problems on average speed
Word problems on sum of the angles of a triangle is 180 degree
OTHER TOPICS
Time, speed and distance shortcuts
Ratio and proportion shortcuts
Domain and range of rational functions
Domain and range of rational functions with holes
Graphing rational functions with holes
Converting repeating decimals in to fractions
Decimal representation of rational numbers
Finding square root using long division
L.C.M method to solve time and work problems
Translating the word problems in to algebraic expressions
Remainder when 2 power 256 is divided by 17
Remainder when 17 power 23 is divided by 16
Sum of all three digit numbers divisible by 6
Sum of all three digit numbers divisible by 7
Sum of all three digit numbers divisible by 8
Sum of all three digit numbers formed using 1, 3, 4
Sum of all three four digit numbers formed with non zero digits
