General Formula for Converting Repeating Decimals to Fractions

Suppose that the decimal number is

`x=a.d_1d_2* * *d_m bar(d_(m+1)* * *d_(m+p)d_(m+p))`,

where the `d_k' are digits, `a' is the integer part of the number, and the vinculum (overline) indicates the repeating part of the decimal. Then

`10^mx=ad_1d_2* * *d_m . bar(d_(m+1)* * *d_(m+p)d_(m+p))` .......(1)

and `10^(m+p)x=ad_1d_2* * *d_md_(m+1)* * *d_(m+p)d_(m+p) . bar(d_(m+1)* * *d_(m+p)d_(m+p))` .......(2)

Subtract (1) from (2) :

`10^(m+p)x-10^mx=ad_1d_2* * *d_m d_(m+1)* * *d_(m+p)d_(m+p)-ad_1d_2* * *d_m`

`Rightarrow (10^(m+p)-10^m)x=ad_1d_2* * *d_m d_(m+1)* * *d_(m+p)d_(m+p)-ad_1d_2* * *d_m`

`Rightarrow 10^m(10^p-1)x=ad_1d_2* * *d_m d_(m+1)* * *d_(m+p)d_(m+p)-ad_1d_2* * *d_m`

`Rightarrow x=(ad_1d_2* * *d_m d_(m+1)* * *d_(m+p)d_(m+p)-ad_1d_2* * *d_m)/((10^p-1)10^m)`

`Rightarrow x=(ad_1d_2* * *d_m d_(m+1)* * *d_(m+p)d_(m+p)-ad_1d_2* * *d_m)/(99* * * 900* * *0)`,

where in the denominator  99...9  occurs p-times and 00...0 occurs m-times.


Example :

What rational number or fraction is equal to 1.04242424242

Step 1:

x = 1.04242424242

Step 2:

After examination, the repeating digit is 42

Step 3:

To place the repeating digit ( 42 ) to the left of the decimal point, you need to move the decimal point 3 place to the right

Again, moving a decimal point three place to the right is done by multiplying the decimal number by 1000.

When you multiply one side by a number, you have to multiply the other side by the same number to keep the equation balanced

Thus, 1000x = 1042.42424242

Step 4:

Place the repeating digit(s) to the right of the decimal point

In this example, the repeating digit is not immediately to the right of the decimal point.

Look at the equation in step 1 one more time and you will see that there is a zero between the repeating digit and the decimal point

To accomplish this, you have to move the decimal point 1 place to the right

This is done by multiplying both sides by 10

10x = 10.4242424242

Step 5:

Your two equations are:

1000x = 1042.42424242

   10x = 10.42424242

1000x - 10x = 1042.42424242 − 10.42424242

990x = 1032

Divide both sides by 990

x = 1032/990

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