Skip to content
Related Articles
Open in App
Not now

Related Articles

Express 5.5858585858… as a rational number

Improve Article
Save Article
  • Last Updated : 08 Mar, 2022
Improve Article
Save Article

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 the bar from the top of repeating digits and listing repeating digits at least twice. For sample, write x = 0.\bar9  as x = 0.999…. and x = 0.\overline{15}  as x = 0.151515……

Step 3: Determine the number of digits having a 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 obtained in the step 2 from the number obtained in step 4.

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

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

Express 5.5858585858… as a rational number

Solution:

Given: 5.5858585858 or 5.\overline{585}

lets assume x = 5.5858585858… ⇢ (1)

And there are two digits after decimal which are repeating,

So, multiply equation (1) both sides by 100,

So 100 x = 558.\overline{5858}          ⇢ (2)

Now subtract equation (1) from equation (2)

100x - x   =  558.\overline{5858} - 5.\overline{5858}

         99x = 553

             x = 553/99

                = 553/99

5.5858585858 can be expressed 553/99 as rational number

Similar Problems

Question 1: Rewrite the decimal as a rational number. 0.666666666…?

Solution:

Given: 0.66666..  or 0.\bar{6}

Let’s assume x = 0.66666… ⇢ (1)

And there are one digits after decimal which are repeating, so we will multiply equation 1 both sides by 10.

So 10x = 6.\bar{6}                     ⇢ (2)

Now subtract equation (1) from equation (2)

10x - x =  6.\bar{6} - 0.\bar{6}

       9x = 6

         x = 6/9     

           = 2/3                 

0.666666…  can be expressed 2/3 as rational number

Question 2: Rewrite the decimal as a rational number. 0.69696969…?

Solution:

Given: 0.696969.. or 0.\overline{69}

Let’s assume x = 0.696969… ⇢ (1)

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

So 100x = 69.\overline{69}    ⇢ (2)

Now subtract equation (1) from equation (2)

100x - x = 69.\overline{69} - 0.\overline{69}

       99x = 69

           x = 69/99

             = 23/33                    

0.69696969… can be expressed 23/33 as rational number

Question 3: Express 1.3737237… as a rational number, in form p/q where p and q have no common factors.

Solution: 

Given : 1.373737… or 1.\overline{37}

lets assume x = 1.373737….   eq. 1

And there are two digits after decimal which are repeating

so we will multiply equation 1 both sides by 100

so 100 x = 137.\overline{37}                             eq. 2

now subtract equation 1 from equation 2

100x - x = 137.\overline{37}-  1.\overline{37}

       99x = 136

           x = 136/99                    

1.373737…. can be expressed 126/99 in form of p/q as rational number 

Question 4: 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 5: Rewrite the decimal as a rational number. 0.79797979…?

Solution:

Given: 0.797979.. or 0.\overline{79}

Let’s assume x = 0.797979… ⇢ (1)

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

So 100x = 79.\overline{79}    ⇢ (2)

Now subtract equation (1) from equation (2)

100x - x = 79.\overline{79} - 0.\overline{79}

       99x = 79

           x = 79/99

              = 79/33                    

0.79797979… can be expressed 79/33 as rational number

Question 6: Rewrite the decimal as a rational number. 0.555555…?

Solution:

Given: 0.555555..  or 0.\bar5

Let’s assume x = 0.555555… ⇢ (1)

And there are one digits after decimal which are repeating, so we will multiply equation 1 both sides by 10.

So 10x = 5.\bar5                    ⇢ (2)

Now subtract equation (1) from equation (2)

10x - x =  5.\bar{5} - 0.\bar{5}

       9x = 5

         x = 5/9    

            = 5/9               

0.555555…  can be expressed 5/9 as rational number

Question 7: Express 6.684684684… as a rational number, in the form p/q where p and q have no common factors.

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


My Personal Notes arrow_drop_up
Related Articles

Start Your Coding Journey Now!