将 8.765765765... 表示为有理数

可以表示或写成a/b形式的数，其中a和b为整数，b≠0，称为有理数。由于数字的基本结构，a/b 形式，大多数人发现很难区分分数和有理数。当有理数被除时，结果值是十进制形式，可以是结束或重复，7、-7、8、-8、9 等是有理数的一些示例，因为它们可以表示为分数形式为 7/1、8/1 和 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.

十进制数到有理数的转换

以下是将十进制数转换为有理数的步骤，

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. 

将 8.765765765... 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: 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

类似问题

问题 1：将 256.58585858... 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: 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

问题 2：将 61.657657657... 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: 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

问题 3：将 101.327327327... 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: 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

问题 4：将 15.373737... 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: 15.373737… 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

问题 5：将 123.327327327... 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: 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

问题 6：将 3.373737… 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: 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

问题 7：将 0.555555... 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: 0.555555… 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