Similar Problems

Question 1: Rewrite the decimal as a rational number. 0.777777777…?

Solution: 

Given: 0.7777..  or [Tex]0.\bar{7} [/Tex]

Let’s assume x = 0.77777… ⇢ (1)

And there are one digits after decimal which are repeating, so we will multiply equation 1 both sides by 10.

So 10x = [Tex]7.\bar{7} [/Tex] ⇢ (2)

Now subtract equation (1) from equation (2)

10x – x = [Tex]7.\bar{7} [/Tex] – [Tex]0.\bar{7} [/Tex]

9x = 7

x = 7/9                      

0.7777777…  can be expressed 7/9 as rational number 

Question 2:  Express 3.927927927… as a rational number of the form p/q, where p and q have no common factors.

Solution: 

Given: 3.927927927 or [Tex]3.\bar{927} [/Tex]

Let’s assume x = 3.927927927… ⇢ (1)

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

So 1000 x = [Tex]3927.\bar{927} [/Tex] ⇢ (2)

Now subtract equation (1) from equation (2)

1000x – x = [Tex]3927.\bar{927} [/Tex] – [Tex]3.\bar{927} [/Tex]

999x = 3924

x = 3924/999

= 1308/333

3.927927927 can be expressed 1308/333 as rational number 

Question 3: Rewrite the decimal as a rational number 4.3232323232 …?

Solution: 

Given: 4.3232323232 or [Tex]4.\bar{32} [/Tex]

Let’s assume x = 4.3232323232… ⇢ (1)

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

So 100 x = [Tex]432.\bar{32} [/Tex] ⇢ (2)

Now subtract equation (1) from equation (2)

100x – x = [Tex]432.\bar{32} [/Tex] – [Tex]4.\bar{32} [/Tex]

99x = 428

x = 428/99

= 428/99

4.323232323 can be expressed 428/99 as rational number 

Question 4: Rewrite the decimal as a rational number. 0.69696969…?

Solution: 

Given: 0.696969.. or [Tex]0.\bar{69} [/Tex]

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 100x = [Tex]69.\bar{69} [/Tex] ⇢ (2)

Now subtract equation (1) from equation (2)

100x – x = [Tex]69.\bar{69} [/Tex] – [Tex]0.\bar{69} [/Tex]

99x = 69

x = 69/99 

= 23/33                    

0.69696969… can be expressed 23/33 as rational number

Question 5: Express 4.8568568586… as a rational number of the form p/q, where p and q have no common factors ?

Solution: 

Given: 4.8568568586… or [Tex]4.\bar{856} [/Tex]

Let’s assume x = 4.8568568586… ⇢ (1)

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

So 1000 x = [Tex]4856.\bar{856} [/Tex] ⇢ (2)

Now subtract equation (1) from equation (2)

1000x – x = [Tex]4856.\bar{856} [/Tex] – [Tex]4.\bar{856} [/Tex]

999x = 4852

x = 4852/999                        

4.8568568586 can be expressed 4852/999 as rational number

To express the repeating decimal 0.151515… as a rational number, we can represent it as the fraction 15/99. Expressing the repeating decimal as a rational number involves recognizing its pattern and converting it into a fraction.

In this article, we will learn how to Express 0.151515….. as a rational number in detail. We will also learn about number system and various types of numbers in our number system.