Step by Step Adding Fractions

Introduction to step by step adding fractions:
                  A fraction is a number that can represent part of a whole. The earliest fractions were reciprocals of integers: ancient symbols representing one part of two, one part of three, one part of four, and so on. A much later development were the common or "vulgar" fractions which are still used today (½, ⅝, ¾, etc.) and which consist of a numerator and a denominator.

                                                                                                                                                 

Step by Step Adding Fractions:

1. Add the two fractions `20/3` and `22/4`

         Solution:

                                   The  two fractions are  `20/3` and `22/4`

                 Step1:             Add two fraction

                                              =`20/3` +`22/4`

                 Step2:          There the denominator are different  so we need to take lcm  to preform addition

                                                           =`(80+66)/12`

                  Step3:           By adding 80 and 66 the answer is 146

                                                         =`146/12`

                   Step4:                This can be simplified has

                                                           =12.1

 2. Add the two fractions `40/4` and `50/5`

            Solution:

                                  The  two fractions are `40/4` and `50/5`

                   Step1:                 Add two fraction

                                                     =`40/4` +`50/5`

                   Step2:     There the denominator is sameso we need to take lcm  to preform addition

                                                            =`(200+200)/20`

                   Step3:       By adding 200 and 200 the answer is 400

                                                                =`400/20`

                   Step4:           This can be simplified has

                                                       = 1

 3. Add the two fractions `60/5` and `70/6`

          Solution:

                                  The  two fractions are `60/5` and` 70/6`

                    Step1:          Add two fraction

                                                  =`60/5` +`70/6`

                    Step2:   There the denominator is same so we need to take lcm  to preform addition

                                                        =`(360+350)/30`

                    Step3:     By adding 360 and350 the answer is 710

                                                              =`710/30`

                    Step4:        This can be simplified has

                                                              = 23.66

Step by Step Adding Fractions:

4. Add the fractions `5/6` and `25/30`

      Solution:

                                        The two fractions are `5/6` and `25/30`

                 Step1:         The given Two fractions are equivalent fractions

                                         By simplifying `25/30` we get 5/6

                  Step2:

                                          Now we need to find the sum of `5/6` and `5/6`

                                                                   = `5/6` +`5/6`

                   Step3 :            There the denominator are same so add the numerators in the fractions together

                                                                   =`(5+5)/6`

                                              The sum of 5 and 5 is 10

                                               So `(5+5)/6` can be written as `10/6`

                  Step4:

                                             This can be reduced further as `5/3`

5. Add the two fractions `80/6` and `90/6`

        Solution:

                                     The  two fractions are `80/6` and `90/6`

                  Step1:             Add two fraction

                                                =`80/6` +`90/6`

                   Step2:       There the denominator is same so add the numerators in the fractions together

                                                            =`(80+90)/6`

                      Step3:        By adding 80 and 90 the answer is 170

                                                          =`170/6`

                      Step4:          This can be simplified has

                                                        = 28.33

 6. Add the two fractions `1000/3` and `120/4`

         Solution:

                                  The  two fractions are `100/3` and `120/4`

                Step1:           Add two fraction

                                         =`100/3` +`120/4`

                Step2:     There the denominator is same so we need to take lcm  to preform addition

                                           =`(400+3690)/12`

                Step3:      By adding 400 and 360 the answer is 760

                                                  =`760/12`

                Step4:        This can be simplified has

                                                  = 63.33

Multiply Mixed Numbers

  Here in this page we are going to discuss about multiplying mixed numbers.Mixed numbers can be written in form of improper fraction. The mixed numbers have summation of whole number and proper fraction. Proper fraction is nothing but the numerator of the fraction is less that that the denominator of the fraction. By multiplying the mixed numbers, we must follow certain procedure as follow.

How to Multiply Mixed Numbers

Let's see the procedure-

Step 1: Convert the given mixed numbers to improper fraction.

To make the mixed numbers to improper fraction, we multiply the whole number and denominator of the fraction and add the result to the numerator of the fraction.

Example: 2 ¼

Multiply 2 and 4 we get 8

Add the number 8 to the numerator of the fraction 1, we get 8 + 1 = 9.

Now get the improper fraction of the mixed number 2 ¼ = `(9)/(4)`

Step 2: Multiply the numerator and denominator of the improper fraction separately.

Example: consider the two improper fractions.

`(a)/(b)` * `(c)/(d)` , where, a > b and c > d

`(a * c)/(b * d)`  = `(p)/(q)`

Step 3: Now convert the improper fraction by mixed number as following procedure.

Example: `(8)/(5)`

5) 8 ( 1

       5

_________

        3

___________

Mixed number can be written as, quotient as the whole number, remainder is the numerator of the proper fraction and divisor is the denominator of the proper fraction.

`(8)/(5)`  = 1`(3)/(5)`

Examples:

Here are the examples on Multiplying Mixed Numbers

Example 1:

Multiplying the mixed numbers 3 ½ and 6 ¼.

Solution:

Step 1: Convert the mixed numbers 3 ½ to improper fraction, we get

3 ½ = `(7)/(2)`

Step 2: Convert the mixed numbers 6 ½ to improper fraction, we get

6 ¼ =  `(25)/(4)`

Step 3: Multiplying the improper fractions  `(7)/(2)` and `(25)/(4)`, we get

`(7)/(2)` * `(25)/(4)` =  `(175)/(8)`

Step 4: Convert the improper fraction  `(175)/(8)` to mixed number.

8) 175 (21

    16

___________

       1 5

          8

___________

           7

Mixed number  `(175)/(8)` =  `21(7)/(8)`

Answer:  `21(7)/(8)`

Example 2:

Multiplying the mixed numbers 4 ½ and 2 ¾.

Solution:

Step 1: Convert the mixed numbers 4 ½ to improper fraction, we get

4 ½ = `(9)/(2)`

Step 2: Convert the mixed numbers 2 ¾ to improper fraction, we get

2 ¾ = `(11)/(4)`

Step 3: Multiplying the improper fractions `(9)/(2)` and `(11)/(4)`, we get

`(9)/(2)` * `(11)/(4)` = `(99)/(8)`

Step 4: Convert the improper fraction `(99)/(8)` to mixed number.

8) 99 (12

    8

___________

     1 9

     1 6

___________

         3

Mixed number `(99)/(8)` =   `12(3)/(8)`

Answer:   `12(3)/(8)`

Exponents Negative Numbers

Introduction:

         The exponent number is shows how many times to use the base number in a multiplication. The exponent number is placed at upper right of the base number. Exponent -x in the expression a^-x. For example, -4 is the exponent in 2^-4= 1/2⁴ = 0.0625. The base and exponent numbers may be positive or negative. Negative exponents are a way of indicating reciprocals.

Rules of Exponents Negative Numbers

Definitions

1. aⁿ = a·a·a···a  (n times)

2. a⁰ = 1  (a ≠ 0)

3. a^-1 = 1/aⁿ (a  ≠ 0)

4. a^m/n = ⁿ√a^m or (ⁿ√a)_m (a ≥ 0, m ≥ 0, n > 0)

Combining

1. Multiplication: a^x a^y = a^{x + y}

2. Division: a^x / a^y = a^x-y (a ≠ 0)

3. Powers: (a^x)^y = a^xy

Distributing   (a ≥ 0, b ≥ 0)

1. (ab)^x = a^x b^x

2. (a/b)^x = a^x/b^x  (b ≠ 0)

Careful!!

1. (a + b)ⁿ ≠ aⁿ + bⁿ

2. (a – b)ⁿ ≠ aⁿ – bⁿ

Rule for Exponents Negative Numbers:

a^-n = 1/aⁿ

Examples: 5^-2 = 1/5² = 1/25 

(2/3)^-3 = (3/2)³ = 27/8

Exponents Negative Numbers – Examples

Negative exponents numbers solved problems

Example 1: Solve this expression 8^-2

Solution:

     Here, the exponent is -2 negative exponents. Usually in positive exponent the exponent number is shows how many times to use the base number in a multiplication. In negative exponent number also shows like this. But in the negative exponent numbers we have to find the reciprocal of the numbers.

8^-2 = 1/8² = 1 / (8 × 8) = 1/64 = 0.015625

8² = 8 × 8 = 64.

Reciprocal of the number is 1/64 = 0.015625

Example 2: Solve 4^-3

Solution:

4^-3 = 1/4³ = 1 / (4 × 4 × 4) = 1/64 = 0.015625

Example 3: Solve this expression 2²/2^-3

Solution:

2²/2^-3 = 2² × 2³ = 2⁵ = 2 × 2 × 2 ×2 × 2 = 32

Example 4: Solve this expression 2(3^-1)

Solution:

2(3^-1) = 2(1/3) = 2/3 = 0.67

Example 5: Simplify this equation and solve this equation (4x)^-3, x = 2.

Solution:

(4x)^-3 = 1/64x³

Put x = 2

1/64(2³) = 1/64(8) = 1/512 = 0.001953125

Example 6: Simplify this equation and solve this equation (x^-3/y^-4)^-3, x = 1, y = 1.

Solution:

(x^-3 / y^-4)^-3 = (x^-3)^-3/(y^-4)^-3 = (y^-4)³/(x^-3)³ = y^-12/x^-9 = x⁹/y¹²

Put x =1 and y = 1 in the equation to get

1⁹/1¹² = 1/1 =1

Exponents Negative Numbers – Practice

Solve these problems for practice on negative exponents.

Problem 1: Solve 2^-2 - Answer: 0.25

Problem 2: Solve 3^-2 - Answer: 0.11

Problem 3: Solve 3²/3^-3 - Answer: 243

Problem 4: Solve this expression 4(4^-1) - Answer: 1

Problem 5: Simplify this equation and solve this equation (2x)^-2, x = 3. - Answer: 36