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Express .123123123… as a rational number

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  • Last Updated : 02 Jun, 2022

Rational numbers are numbers that may be expressed or written as m/n, where m and n are integers and n is not equal to zero (n ≠ 0). Because of the fundamental structure of numbers, the m/n form, most individuals have difficulties differentiating between fractions and rational numbers. When a rational number is divided, it gives a decimal value that might be either ending or repeating. Rational numbers include 11, -11, 5, -5, 9, and so on, which may be expressed in fraction form as 11/1, -5/1, and 7/1, respectively. A rational number is a kind of a real number having the formula m/n, where n is not equal to zero(n≠ 0). When you divide a rational number, you get a decimal number that may be terminated or repeated.

Steps for converting decimal values to rational numbers

  • Step 1: Find the repeating decimal and keep it equal to x.
  • Step 2: Write it in decimal form by eliminating the bar at the top of the repeated numbers and listing them at least twice.

For example, x = 0.4 bar is written as x = 0.444.. while x = 0.44 bar is written as x = 0.444444…

  • Step 3: Check the number of digits with a bar.
  • Step 4: If the number with a repeating decimal has a one-place repetition, we multiply it by 10, a two-place repetition by 100, a three-place repetition by 1000, and so on.
  • Step 5: Following that, the Subtraction of the equation is obtained in step 2 from the equation obtained in step 4.
  • Step 6: Divide the rest of the equation by the x coefficient.
  • Step 7: Lastly, write the rational number in the most basic form.

Express 0.123123123… as a rational number

Solution: 

Given: 0 .123123123. or 0.\overline{123}

Lets assume x = 0 .123123123…. ⇢ (1)

As we can see there are three digits after decimal which are repeating,

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

So, 1000 x = 123.123123 ⇢ (2)

Now subtract equation (1) from equation (2)

1000x – x = 123.123123.. – 0.123123123…

999x = 123

x = 123/999

= 41/ 333

0 .123123123. can be expressed 41/333 as rational number.

Similar Questions

Question 1: Express 26.588588… as a rational number of the form p/q, where p and q have no common factors.

Solution:  

Given: 26.588588 or  26.\overline{588}

Lets assume x =  26 .588588… ⇢ (1)

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

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

So 1000 x = 26588.\overline{588}  ⇢ (2)

Now subtract equation (1) from equation (2)

1000x - x = 26588.\overline{588}- 26.\overline{588}

999x = 26562

x = 26562/999

x = 8854/333

26.588588… can be expressed 8854/333 as rational number.

Question 2: Express 3.272727… as a rational number, in form p/q where p and q have no common factors.

Solution: 

Given: 3.272727… or 3.\overline{27}

Let’s assume x = 3.272727…. ⇢  (1)

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

So 100 x =  327.\overline{27}  ⇢ (2)

Now subtract equation (1) from equation (2)

100x – x =  327.\overline{27}- 3.\overline{27}

99x = 324

x = 324/99     

= 108/33      

= 36/11         

3.272727…. can be expressed 36/11 in form of p/q as rational number.

Question 3: Express 65.232323… as a rational number, in form p/q where p and q have no common factors?

Solution: 

Given: 65.232323… or 65.\overline{23}

Let’s assume x = 65.232323…. ⇢  (1)

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

So 100 x = 6523.\overline{23}  ⇢ (2)

Now subtract equation (1) from equation (2)

100x – x =  6523.\overline{23} - 65.\overline{23}

99x = 6458

x = 6458/99                    

65.232323…. can be expressed 6458/99 in form of p/q as rational number.

Question 4: Express 11.777… as a rational number, in form p/q where p and q have no common factors?

Solution:

Given: 11 .777777… or 11.\overline{77}

Let’s assume x = 11.777777…. ⇢  (1)

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

So 100 x = 1177.\overline{77}  ⇢ (2)

Now subtract equation (1) from equation (2)

100x - x = 1177.\overline{77} - 11.\overline{77}

99x = 1166

x = 1166 /99                    

11.777777…. can be expressed 1166/99 in form of p/q as rational number.

Question 5: Express 14.555555… as a rational number, in form p/q where p and q have no common factors.

Solution:

Given: 14.555555… or 14.\overline{555}

Let’s assume x = 14.555555…. ⇢  (1)

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

So 1000 x =14555.\overline{77}   ⇢ (2)

Now subtract equation (1) from equation (2)

1000x – x = 14555.\overline{555}- 14.\overline{555}

999x = 14541

x = 14541 /999

= 4847/333                   

14.555555…. can be expressed 4847/333 in form of p/q as rational number.

Question 6: Express 2.5050… as a rational number, in form p/q where p and q have no common factors.

Solution: 

Given: 2.5050….

Let’s assume x = 2.5050….. ⇢  (1)

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

So 100 x = 250.\overline{50}  ⇢ (2)

Now subtract equation (1) from equation (2)

100x – x = 250.\overline{50}- 2.\overline{50}

99x = 248

x = 248/99                    

2.5050…. can be expressed 248/99 in form of p/q as rational number.


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