将 6.684684684... 表示为有理数

有理数的形式是 p/q，其中 p 和 q 是整数，q ≠ 0。由于数字的基本结构，p/q 形式，大多数人发现很难区分分数和有理数。当一个有理数被除法时，输出是十进制形式，可以是结束也可以是重复的。

3、-3、4、-4、5 等是有理数的一些示例，因为它们可以用分数形式表示为 3/1、4/1 和 5/1。

A rational number is a sort of real number that has the form p/q where q≠0. When a rational number is split, the result is a decimal number, which can be either a terminating or a recurring decimal.

十进制数到有理数的转换

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 = 0.9bar as x = 0.999…. and x = 0.15bar as x = 0.151515……

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 number come in second step from the number obtained in step 4

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

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

将 6.684684684... 表示为有理数。

解决方案：

Given: 6.684684684 or 6.684bar

Step 1: Write the number in decimal form by removing bar from the top of repeating digits and listing repeating digits at least twice

Lets assume x = 6.684684684… ⇢ (1)

Step 2: There are three digits after decimal which are repeating, So, multiply equation (1) both sides by 1000,

So 1000 x = 6684.684684       ⇢ (2)

Now subtract equation (1) from equation (2)

1000x – x   =  6684. 684684 – 6.684684684

         999x = 6678

Divide both sides of the equation by the x coefficient.

          999x/999 = 6678/999

                       x = 6678/999

                          = 2226/ 333

                          = 742/111

6.684684684 can be expressed 742/111 as rational number

类似问题

问题 1：将 7.765765765... 表示为 p/q 形式的有理数，其中 p 和 q 没有公因数。

解决方案：

Given: 7.765765765 or $7.\overline{765}$

Let’s assume x = 7.765765765… ⇢ (1)

And, there are three digits after decimal which are repeating

So multiply equation (1) both sides by 1000

So, $1000x = 7765.\overline{765}$⇢ (2)

Now subtract equation (1) from equation (2)

$1000x - x = 7765.\overline{765}- 7.\overline{765}$

       999x = 7758

             x = 7758/999

7.765765765 can be expressed 7758/999 as rational number

问题 2：将 10.827827827... 表示为一个有理数，形式为 p/q，其中 p 和 q 没有公因数。

解决方案：

Given: 10.827827827… or $10.\overline{827}$

Let’s assume x = 10.827827827…        ⇢ 1

And there are three digits after decimal which are repeating

So multiply equation 1 both sides by 1000

So $1000 x = 10827.\overline{827}$          ⇢ (2)

Now subtract equation (1) from equation (2)

$1000x - x = 10827.\overline{827}- 10.\overline{827}$

       999x = 10817

             x = 10817/999

10.927927927 can be expressed 10817/999 in form of p/q as rational number

问题 3：将 2.272727... 表示为有理数，形式为 p/q，其中 p 和 q 没有公因数。

解决方案：

Given: 2.272727… or $2.\overline{27}$

Let’s assume x = 2.272727…. ⇢  (1)

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

So $100 x = 227.\overline{27}$          ⇢ (2)

Now subtract equation (1) from equation (2)

$100x - x = 227.\overline{27} - 2.\overline{27}$

       99x = 225

           x = 225/99    

              = 75/33

              = 25/11                

2.272727…. can be expressed 25/11 in form of p/q as rational number

问题 4：将 15.527527527... 表示为一个有理数，形式为 p/q，其中 p 和 q 没有公因数。

解决方案：

Given: 15.527527527… or $15.\overline{527}$

Let’s assume x = 15.527527527… ⇢ (1)

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

So $1000 x = 15527.\overline{527}$    ⇢ (2)

Now, subtract equation (1) from equation (2)

$1000x - x = 15527.\overline{527} - 15.\overline{527}$

       999x = 15512

             x = 15512/999

                = 15512/999

15.527527527 can be expressed 15512/999 in form of p/q as rational number .

问题 5：将 16.625625625... 表示为一个有理数，形式为 p/q，其中 p 和 q 没有公因数。

解决方案：

Given: 16.625625625… or $16.\overline{625}$

Let’s assume x = 16.625625627… ⇢ (1)

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

So $1000 x = 16625.\overline{625}$    ⇢ (2)

Now, subtract equation (1) from equation (2)

$1000x - x = 16625.\overline{625} - 16.\overline{625}$

      999x = 16609

            x = 16609/999

               = 16609/999

16.625625627 can be expressed 16609/999 in form of p/q as rational number .

问题 6：将 0.272727... 表示为有理数，形式为 p/q，其中 p 和 q 没有公因数。

解决方案：

Given: 0.272727… or $0.\overline{27}$

Let’s assume x = 0.272727…. ⇢  (1)

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

So $100 x = 27.\overline{27}$          ⇢ (2)

Now subtract equation (1) from equation (2)

$100x - x = 27.\overline{27} - 0.\overline{27}$

       99x = 27

           x = 27/99    

              = 9/33

              = 3/11                

0.272727…. can be expressed 3/11 in form of p/q as rational number

问题 7：将 8.765765765……表示为 p/q 形式的有理数，其中 p 和 q 没有公因数。

解决方案：

Given: 8.765765765 or $7.\overline{765}$

Let’s assume x = 8.765765765… ⇢ (1)

And, there are three digits after decimal which are repeating

So multiply equation (1) both sides by 1000

So, $1000x = 8765.\overline{765}$⇢ (2)

Now subtract equation (1) from equation (2)

$1000x - x = 8765.\overline{765}- 8.\overline{765}$

       999x = 8757

             x = 8757/999

8.765765765 can be expressed 8757/999 as rational number