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Express 8.765765765… as a rational number

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  • Last Updated : 09 Mar, 2022
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The number which can be expressed or written in the form a/b, where a and b are integers and b ≠ 0 are known as Rational numbers. Due to the underlying structure of numbers, a/b form, most individuals find it difficult to distinguish between fractions and rational numbers. When a rational number is divided then the resulting value is in a decimal form which can be either ending or repeating, 7,-7, 8, -8, 9, and so on are some examples of rational numbers as they can be expressed in fraction form as 7/1, 8/1, and 9/1.

A rational number is a sort of real number that has the form a/b where b≠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 the Decimal number to Rational number

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

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 = 0.4 bar as x = 0.444… and x = 0.11 bar as x = 0.111111…

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 equation come in second step from the equation obtained in step 4.

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

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

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

Solution: 

Given: 8.765765765 or 8.\overline{765}

Lets assume x = 8.765765765… ⇢ (1)

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

So, multiply equation (1) both sides by 1000,

So, 1000 x = 8765.765765 ⇢ (2)

Now subtract equation (1) from equation (2)

1000x – x = 8765.765765.. – 8.765765765..

999x = 8757

x = 8757/999

= 2919/ 333

= 973/111

8.765765765.. can be expressed 973/111 as rational number

Similar Problems 

Question 1: Express 256.58585858… as a rational number of the form p/q, where p and q have no common factors.

Solution:   

Given: 256.58585858 or 256.\overline{58}

Lets assume x =  256 .58585858… ⇢ (1)

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

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

So 100 x = 25658.\overline{58} ⇢ (2)

Now subtract equation (1) from equation (2)

100x – x = 25658.\overline{58}- 256.\overline{58}

99x = 25402

x = 25402/99

256.58585858… can be expressed 25402/99 as rational number

Question 2: Express 61.657657657… as a rational number of the form p/q, where p and q have no common factors.

Solution:  

Given: 61.657657657 or 61.\overline{657}

Let’s assume x = 61.657657657… ⇢ (1)

And, there are three digits after decimal which are repeating

So multiply equation (1) both sides by 1000

So, 1000x = 61657.\overline{657}⇢ (2)

Now subtract equation (1) from equation (2)

1000x – x = 61657.\overline{657}- 61.\overline{657}

999x = 61596

x = 61596/999

= 20532/333

= 6844/111

61.657657657 can be expressed 6844/111 as rational number

Question 3: Express 101.327327327… as a rational number, in form p/q where p and q have no common factors.

Solution:

Given: 101.327327327… or 101.\overline{327}

Let’s assume x = 101.327327327… ⇢ 1

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

So multiply equation 1 both sides by 1000

So 1000 x = 101327.\overline{327}⇢ (2)

Now subtract equation (1) from equation (2)

1000x – x = 101327.\overline{327}- 101.\overline{327}

999x = 101226

x = 101226 / 999

= 33742/333

101.327327327 can be expressed 33742/333 in form of p/q as rational number

Question 4: Express 15.373737… as a rational number, in form p/q where p and q have no common factors.

Solution:

Given: 15.373737… or 15.\overline{27}

Let’s assume x = 15.373737…. ⇢  (1)

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

So 100 x = 1537.\overline{37} ⇢ (2)

Now subtract equation (1) from equation (2)

100x – x = 1537.\overline{37}- 15.\overline{37}

99x = 1522

x = 1522/99                    

15.373737…. can be expressed 1522/99 in form of p/q as rational number

Question 5: Express 123.327327327… as a rational number, in form p/q where p and q have no common factors.

Solution:

Given: 123.327327327… or 123.\overline{327}

Let’s assume x = 123.327327327… ⇢ 1

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

So multiply equation (1) both sides by 1000,

So 1000 x = 123327.\overline{327} ⇢ (2)

Now subtract equation (1) from equation (2)

1000x – x = 123327.\overline{327}- 123.\overline{327}

999x = 123204

x = 123204/999

= 41068/333

= 41068 /333

123.327327327. can be expressed 41068 /333 in form of p/q as rational number

Question 6:  Express 3.373737… as a rational number, in form p/q where p and q have no common factors.

Solution:

Given: 3.373737… or 1.\overline{27}

Let’s assume x = 3.373737…. ⇢  (1)

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

So 100 x = 337.\overline{37}⇢ (2)

Now subtract equation (1) from equation (2)

100x – x = 337.\overline{37}- 3.\overline{37}

99x = 334

x = 334/99                    

3.373737…. can be expressed 334/99 in form of p/q as rational number

Question 7: Express 0.555555… as a rational number, in form p/q where p and q have no common factors.

Solution:

Given: 0.555555… or 0.\overline{55}

Let’s assume x = 0.555555…. ⇢  (1)

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

So 100 x = 55.\overline{55} ⇢ (2)

Now subtract equation (1) from equation (2)

100x – x = 55.\overline{55} - 0.\overline{55}

99x = 55

x = 55/99    

= 5/9

0.555555…. can be expressed 5/9 in form of p/q as rational number


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