Theory of Proportions

Introduction to theory of proportions:

       The theory of proportion is one of the basic topic in mathematic. In our usual life, there are a lot of occasions as we compare two quantities by means of their measurements. As soon as we compare two quantities of the same type by division, we contain a ratio of those two quantities.

Definition of Ratio: Ratio means similarity of two like quantities by division.

Definition of Theory Proportions:

      Proportions is a correspondence of two ratios.

      Consider the proportions

                a: b = c: d

      The first and fourth terms (a and d) are knoen as the extreme terms or extremes.

      The second and third terms (b and c) are known as the middle terms or means.

Important property:

      Product of extremes = Product of means.

Examples of Theory Proportions:

Let us see some examples of theory of proportions.

Example 1:

      Verify 5: 6 = 10: 12 is a proportion or not.

Solution:

      Product of extremes = 5*12 = 60

      Product of means = 6*10 = 60.

             `:.`   60 = 60

       These two products are equal.

            `:.` 5:6 = 8: 6 is a proportion.

Example 2:

       Verify 6: 7 = 12: 15 is a proportion or not.

Solution:

       Product of extremes = 6*15 = 90

       Product of means = 7*12 = 84

                          `:.`    90 = 84

       These two products are not equal.

         So, 6: 7 =12: 15 is not a proportion.

Example 3:

       If 2: 3= 6:_ is a proportion, find the missing term.

Solution:

       Let us assume the missing value is x

       Product of extremes = 2*x

       Product of means = 3*6 =18

       Since it is a proportion, 2*x =18

                                          2x = 18

       Divide both sides by 2 on both sides we get,

                                        2x/2 = 18/2

                                             x = 18/2 = 9

       `:. ` The missing term is 9

       So the proportion is 2:3 =6: 9

Example 4:

       The income and Savings of a family are into the ratio 8: 3 If the income of the family is Rs. 3,300.Find how much is being saved.

Solution:

       Let us savings be Rs. x.

          `:.` The proportion is 8: 3 = 3300: x

                     (Income: saving) = (Income: saving)

                                       11x = 9900

                                   11x/11= 9900/11

                                           x= 900

      Therefore, the savings = Rs. 900.

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)`