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# Express 5.5858585858… as a rational number

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 = as x = 0.999…. and x = 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 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 ⇢ (2)

Now subtract equation (1) from equation (2) 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 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 = ⇢ (2)

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

Now subtract equation (1) from equation (2) 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 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 eq. 2

now subtract equation 1 from equation 2 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 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 ⇢ (2)

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

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

Now subtract equation (1) from equation (2) 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

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