Express 2.684684684… as a rational number

Rational numbers are numbers that may be represented or written in the form m/n, where m and n are integers and n ≠ 0. Most people have difficulty distinguishing between fractions and rational numbers because of the underlying structure of numbers, the m/n form. When a rational number is split, the result is a decimal value that might be either ending or recurring. Examples of rational numbers are 11, -11, 5, -5, 9, and so on, which may be written in fraction form as 11/1, -5/1, and 7/1.

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.

Conversion of the Decimal number to Rational number

Here Given 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 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 2.684684684… as a rational number

Solution:

Given: 2.684684684.. or 

Lets assume x = 2.684684684… ⇢ (1)

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

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

 So, 1000 x = 2684.684684 ⇢ (2)

Now subtract equation (1) from equation (2)

1000x – x = 2684.684684.. –  2.684684..

999x =  2682

x = 2682/999

= 894/ 333

= 298/111

2.684684684.. can be expressed 298/111 as rational number

Similar Questions

Question 1: Express 356.68686868… as a rational number of the form p/q, where p and q have no common factors.

Solution:  

Given: 356.68686868… or 

Lets assume x =  356.68686868… ⇢ (1)

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

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

So 100 x = ⇢ (2)

Now subtract equation (1) from equation (2)

100x – x =

99x = 35312

x = 35312/99

356.68686868… can be expressed 35312/99 as rational number.

Question 2:  Express 31.247247247… as a rational number of the form p/q, where p and q have no common factors.

Solution:  

Given: 31.247247247 or 

Let’s assume x = 31.247247247… ⇢  (1)

And, there are three digits after decimal which are repeating

So multiply equation (1) both sides by 1000

So, 1000x =  ⇢ (2)

Now subtract equation (1) from equation (2)

1000x – x =

999x = 31216

9x = 31216/999

31.247247247… can be expressed 31216/999 as rational number

Question 3: Express 105.357357357… as a rational number, in form p/q where p and q have no common factors.

Solution:

Given: 105.357357357… or 

Let’s assume x = 105.357357357… ⇢ 1

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

So multiply equation 1 both sides by 1000

So 1000 x =  ⇢ (2)

Now subtract equation (1) from equation (2)

1000x – x =

999x = 105252

x = 105252/ 999

= 35084/333

105.357357357 can be expressed 35084/333 in form of p/q as rational number.

Question 4: Express 14.777777… as a rational number, in form p/q where p and q have no common factors.

Solution: 

Given: 14.777777… or 

Let’s assume x = 14.777777…. ⇢  (1)

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

So 100 x =  ⇢ (2)

Now subtract equation (1) from equation (2)

100x – x = 

99x = 1463

x = 1463 /99                    

14.777777…. can be expressed 1463/99 in form of p/q as rational number .

Question 5:  Express 157.927927927… as a rational number, in form p/q where p and q have no common factors.

Solution:

Given: 157.927927927… or 

Let’s assume x = 157.927927927… ⇢ 1

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

So multiply equation (1) both sides by 1000,

So 1000 x =  ⇢ (2)

Now subtract equation (1) from equation (2)

1000x – x = 

999x = 157770

x = 157770/999

157.927927927… can be expressed 157770/999 in form of p/q as rational number.

Question 6: Express 2.252525… as a rational number, in form p/q where p and q have no common factors.

Solution:

Given: 2.252525…. 

Let’s assume x = 2.252525…… ⇢  (1)

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

So 100 x =   ⇢ (2)

Now subtract equation (1) from equation (2)

100x – x = 

99x = 223

x = 223/99                    

2.252525…. can be expressed 223/99 in form of p/q as rational number.

Question 7: Express 0.111111… as a rational number, in form p/q where p and q have no common factors.

Solution:

Given: 0.111111… 

Let’s assume x = 0.111111…. ⇢  (1)

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

So 100 x =  ⇢ (2)

Now subtract equation (1) from equation (2)

100x – x = 

99x = 11

x = 11/99    

= 1/9

0.111111…. can be expressed 1/9 in form of p/q as rational number .