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Express 1.3212121… as a rational number

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  • Last Updated : 04 Apr, 2022

A rational number is a kind of real number with the formula m/n, where n≠ 0. When a rational number is divided, the outcome is a decimal number, which can be either ended or repeated. These numbers are written as p/q, where p and q are integers and q ≠ zero. Because of the underlying structure of numbers, which is p/q form, most individuals find it difficult to distinguish between fractions and rational numbers. Whole numbers constitute fractions, while integers constitute the numerator and denominator of rational numbers.

Conversion of the Decimal number to Rational number

Below are the steps for the conversion of decimal numbers to rational numbers,

  • Step 1: Identify the repeating decimal and put it equal to x
  • Step 2: Write it in decimal form by removing the bar from the top of repeating digits and listing repeating digits at least twice.

For Example, write x = 0.3 bar as x = 0.333… and x = 0.33 bar as x = 0.333333…

  • Step 3: Examine the number of digits having a bar.
  • Step 4: If the number having a repeating decimal has 1 place repetition then we will multiply it by 10, if it has a two-place repetition then it will be multiplied by 100, and a three-place repetition after the decimal will be multiplied by 1000, and so on.
  • Step 5: After that Subtract the equation obtained in the second step from the equation obtained in step 4.
  • Step 6: Whatever is left, Divide both sides of the equation by the x coefficient.
  • Step 7: At last Write the rational number in its simplest form. 

Express 1.3\overline{21}  = 1.3212121… as a rational number

Solution:  

Given: 1.3212121… 

Let’s assume x = 1.3212121…. ⇢  (1)

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

So 100 x = 132. 12121\overline{21}   ⇢ (2)

Now subtract equation (1) from equation (2)

100x – x = 132. 12121\overline{21} - 1.3\overline{21}

99x = 130.8

x = 130.8/99    

= 1308/990

=  436/330

1.3212121…. can be expressed 436/330  in form of p/q as rational number.

Sample Questions

Question 1: Express 5.959595… as a rational number, in form p/q where p and q have no common factors.

Solution: 

Given: 5.959595…. or 5.\overline{95}

Let’s assume x = 5.959595… ⇢  (1)

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

So 100 x =  595.\overline{95}   ⇢ (2)

Now subtract equation (1) from equation (2)

100x – x = 595.\overline{95}- 5.\overline{95}

99x = 590

x = 590/99                    

5.959595… can be expressed 590/99 in form of p/q as rational number.

Question 2:  Express 26.333333… as a rational number, in form p/q where p and q have no common factors.

Solution: 

Given: 26.333333… or 26.\overline{33}

Let’s assume x = 26.333333…. ⇢  (1)

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

So 100 x = 2633.\overline{33}   ⇢ (2)

Now subtract equation (1) from equation (2)

100x – x = 2633.\overline{33}- 26.\overline{33}

99x = 2607

x = 2607 /99                    

26.333333… can be expressed 2607/99 in form of p/q as rational number.

Question 3: Express 9.969696… as a rational number, in form p/q where p and q have no common factors.

Solution: 

Given : 9.969696… or 9.\overline{9696}

Let’s assume x = 9.969696… ⇢ (1)

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

So 100 x =  996.\overline{96}   ⇢ (2)

Now subtract equation (1) from equation (2)

100x – x = 996.\overline{96}- 9.\overline{96}

99x = 987

x = 987/99       

=  329/33             

9.969696…. can be expressed 329/33 in form of p/q as rational number.

Question 4: Express 10.65656565… as a rational number, in form p/q where p and q have no common factors.

Solution: 

Given: 10.65656565… or 10.\overline{65}

Let’s assume x = 10.65656565… ⇢  (1)

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

So 100 x = 1065.\overline{65}   ⇢ (2)

Now subtract equation (1) from equation (2)

100x – x = 1065.\overline{65} - 10.\overline{65}

99x = 1055

x = 1055/99    

= 1055/99

10.656565…. can be expressed 10555/99 in form of p/q as rational number.

Question 5: Express 159.986986986… as a rational number, in form p/q where p and q have no common factors.

Solution: 

Given: 159.986986986… or 159.\overline{986}

Let’s assume x = 159.986986986… ⇢ 1

And, there are three digits after decimal which are repeating,

So multiply equation 1 both sides by 1000

So 1000 x = 159986.\overline{986}   ⇢ (2)

Now subtract equation (1) from equation (2)

1000x – x = 159986.\overline{986}- 159.\overline{986}

999x = 159827

x = 159827 / 99

159.986986986 can be expressed 159827 / 99 in form of p/q as rational number.

Question 6: Express 56.55555.. as a rational number of the form p/q, where p and q have no common factors.

Solution: 

Given: 56.55555 or 56.\overline{5}

Lets assume x = 56.55555… ⇢ (1)

And, there are one digit after decimal which are repeating,

So, multiply equation (1) both sides by 10,

So 10 x = 565.\overline{5}   ⇢ (2)

Now subtract equation (1) from equation (2)

 10x – x = 565.\overline{5}- 56.\overline{5}

9x = 509

x = 509 /9

56.55555… can be expressed 509/9 as rational number.

Question 7:  Express 0.99999… as a rational number, in form p/q where p and q have no common factors.

Solution: 

Given: 0.99999… or 0.\overline{9}

Let’s assume x = 0.99999…. ⇢  (1)

And, there are one digit after decimal which are repeating, so multiply equation (1) both sides by 10,

So 10 x = 9.\overline{9}   ⇢ (2)

Now subtract equation (1) from equation (2)

10x – x = 9.\overline{9} - 0.\overline{9}

99x = 9

x = 9/99    

= 1/11

0.99999…. can be expressed 1/11 in form of p/q as rational number.


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