What is 0.78888 ... . . converted into a fraction? ( 0.7 8 ¯ )

Question

What is       0.78888    ...    .    .   converted into a fraction?             (      0.7              8        ¯            )

progutannx7f · Accepted Answer

In order to accomplish this, we can make the number

(0.7 \bar{8})

equivalent to a pro numeral, and for this example, we'll use x

\bar{x}

just means that x is a reccurring /repeating number, then

0.7 \bar{8} = 0.788888888888888888888 ...

.
Thus, we have:

x = 0.7888 ... . = 0.7 \bar{8}

To get 100x, we can multiply x by 100, but we also need to do that on the other side.

x \times 100 = 78 . \bar{8} \times 100

100 x = 78 . \bar{8}

10 x = 7 . \bar{8}

The reason why we do this is because now have two numbers,

100 x = 78 . \bar{8}

, so now we can cancel out the two repeating decimals, and then subtract the second number from the first, to get a whole integer.

(100 x = 78 . 88 \bar{8}) - (10 x = 7 . 88 \bar{8}) = (90 x = 71)

Now, we can use algebra to find x.

90 x = 71

Divide each side by 90 to find x

\frac{90 x}{90} = \frac{71}{90}

x = \frac{71}{90}

0.7 \bar{8} = \frac{71}{90}

What is 0.78888..... converted into a fraction?