Changing Infinite Repeating Decimals to Fractions

Remember: Infinite repeating decimals are usually represented by putting a line over (sometimes under) the shortest block of repeating decimals. Every infinite repeating decimal can be expressed as a fraction.

Find the fraction represented by the repeating decimal  .

Let n stand for   or 0.77777 …

So 10 n stands for   or 7.77777 …

10 n and n have the same fractional part, so their difference is an integer.

You can solve this problem as follows.

So 

Find the fraction represented by the repeating decimal  .

Let n stand for   or 0.363636 …

So 10 n stands for   or 3.63636 …

and 100 n stands for   or 36.3636 …

100 n and n have the same fractional part, so their difference is an integer. (The repeating parts are the same, so they subtract out.)

You can solve this equation as follows:

Now simplify   to  .

So 

Find the fraction represented by the repeating decimal  .

Let n stand for   or 0.544444 …

So 10 n stands for   or 5.444444 …

and 100 n stands for   or 54.4444 …

Since 100 n and 10 n have the same fractional part, their difference is an integer. (Again, notice how the repeated parts must align to subtract out.)

You can solve this equation as follows.

So 

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