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Express 6.684684684… as a rational number

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  • Last Updated : 08 Mar, 2022
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Rational numbers are of the form p/q, where p and q are integers and q ≠ 0. Because of the underlying structure of numbers, p/q form, most individuals find it difficult to distinguish between fractions and rational numbers. When a rational number is divided, the output is in decimal form, which can be either ending or repeating.

3,-3, 4, -4, 5, and so on are some examples of rational numbers as they can be expressed in fraction form as 3/1, 4/1, and 5/1.

A rational number is a sort of real number that has the form p/q where q≠0. When a rational number is split, the result is a decimal number, which can be either a terminating or a recurring decimal.

Conversion of Decimal number to Rational number 

Step 1: Obtain the repeating decimal and put it equal to x 

Step 2: Write the number in decimal form by removing bar from the top of repeating digits and listing repeating digits at least twice. For sample, write x = 0.9bar as x = 0.999…. and x = 0.15bar as x = 0.151515……

Step 3: Determine the number of digits having bar ..

Step 4: If the repeating decimal has 1 place repetition, multiply by 10 , if it has a two place repetition, multiply by 100 and a three place repetition multiply by 1000 and so on.

Step 5: Subtract the number come in second step from the number obtained in step 4

Step 6: Divide both sides of the equation by the x coefficient.

Step 7: In last Write the rational number in its simplest form. 

Express 6.684684684… as a rational number.

Solution: 

Given: 6.684684684 or 6.684bar

Step 1: Write the number in decimal form by removing bar from the top of repeating digits and listing repeating digits at least twice

Lets assume x = 6.684684684… ⇢ (1)

Step 2: There are three digits after decimal which are repeating, So, multiply equation (1) both sides by 1000,

So 1000 x = 6684.684684       ⇢ (2)

Now subtract equation (1) from equation (2)

1000x – x   =  6684. 684684 – 6.684684684

         999x = 6678

Divide both sides of the equation by the x coefficient.

          999x/999 = 6678/999

                       x = 6678/999

                          = 2226/ 333

                          = 742/111

6.684684684 can be expressed 742/111 as rational number

Similar Problems

Question 1: Express 7.765765765… as a rational number of the form p/q, where p and q have no common factors.

Solution:

Given: 7.765765765 or 7.\overline{765}

Let’s assume x = 7.765765765… ⇢ (1)

And, there are three digits after decimal which are repeating

So multiply equation (1) both sides by 1000

So, 1000x = 7765.\overline{765} ⇢ (2)

Now subtract equation (1) from equation (2)

1000x - x = 7765.\overline{765}- 7.\overline{765}

       999x = 7758

             x = 7758/999

7.765765765 can be expressed 7758/999 as rational number

Question 2: Express 10.827827827… as a rational number, in form p/q where p and q have no common factors.

Solution: 

Given: 10.827827827… or 10.\overline{827}

Let’s assume x = 10.827827827…        ⇢ 1

And there are three digits after decimal which are repeating

So multiply equation 1 both sides by 1000

So 1000 x = 10827.\overline{827}            ⇢ (2)

Now subtract equation (1) from equation (2)

1000x - x = 10827.\overline{827}- 10.\overline{827}

       999x = 10817

             x = 10817/999

10.927927927 can be expressed 10817/999 in form of p/q as rational number

Question 3: Express 2.272727… as a rational number, in form p/q where p and q have no common factors.

Solution:

Given: 2.272727… or 2.\overline{27}

Let’s assume x = 2.272727…. ⇢  (1)

And there are two digits after decimal which are repeating, so multiply equation (1) both sides by 100,

So 100 x = 227.\overline{27}           ⇢ (2)

Now subtract equation (1) from equation (2)

100x - x = 227.\overline{27} - 2.\overline{27}

       99x = 225

           x = 225/99    

              = 75/33

              = 25/11                

2.272727…. can be expressed 25/11 in form of p/q as rational number

Question 4: Express 15.527527527… as a rational number, in form p/q where p and q have no common factors.

Solution:

Given: 15.527527527… or 15.\overline{527}

Let’s assume x = 15.527527527… ⇢ (1)

And there are three digits after decimal which are repeating, so multiply equation (1) both sides by 1000,

So 1000 x = 15527.\overline{527}     ⇢ (2)

Now, subtract equation (1) from equation (2)

1000x - x = 15527.\overline{527} - 15.\overline{527}

       999x = 15512

             x = 15512/999

                = 15512/999

15.527527527 can be expressed 15512/999 in form of p/q as rational number .

Question 5: Express 16.625625625… as a rational number, in form p/q where p and q have no common factors.

Solution:  

Given: 16.625625625… or 16.\overline{625}

Let’s assume x = 16.625625627… ⇢ (1)

And there are three digits after decimal which are repeating, so multiply equation (1) both sides by 1000,

So 1000 x = 16625.\overline{625}     ⇢ (2)

Now, subtract equation (1) from equation (2)

1000x - x = 16625.\overline{625} - 16.\overline{625}

      999x = 16609

            x = 16609/999

               = 16609/999

16.625625627 can be expressed 16609/999 in form of p/q as rational number .

Question 6: Express 0.272727… as a rational number, in form p/q where p and q have no common factors.

Solution: 

Given: 0.272727… or 0.\overline{27}

Let’s assume x = 0.272727…. ⇢  (1)

And there are two digits after decimal which are repeating, so multiply equation (1) both sides by 100,

So 100 x = 27.\overline{27}           ⇢ (2)

Now subtract equation (1) from equation (2)

100x - x = 27.\overline{27} - 0.\overline{27}

       99x = 27

           x = 27/99    

              = 9/33

              = 3/11                

0.272727…. can be expressed 3/11 in form of p/q as rational number

Question 7: Express 8.765765765… as a rational number of the form p/q, where p and q have no common factors.

Solution:

Given: 8.765765765 or 7.\overline{765}

Let’s assume x = 8.765765765… ⇢ (1)

And, there are three digits after decimal which are repeating

So multiply equation (1) both sides by 1000

So, 1000x = 8765.\overline{765} ⇢ (2)

Now subtract equation (1) from equation (2)

1000x - x = 8765.\overline{765}- 8.\overline{765}

       999x = 8757

             x = 8757/999

8.765765765 can be expressed 8757/999 as rational number


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