# What is 0.78888.....  converted into a fraction? (0.7bar8)

Jun 11, 2017

#### Answer:

$0.7 \overline{8} = \frac{71}{90}$

#### Explanation:

How we can do this is by making the number $\left(0.7 \overline{8}\right)$ equivalent to a pro numeral, and for this example, we'll use $x$

Note: $\overline{x}$ just means that $x$ is a reccurring /repeating number, so $0.7 \overline{8} = 0.788888888888888888888 \ldots$.

So now we have:

$x = 0.7888 \ldots . = 0.7 \overline{8}$

What we can do is multiply $x$ by $100$ to get $100 x$, and obviously we have to do that to the other side.

$x \times 100 = 78. \overline{8} \times 100$

$100 x = 78. \overline{8}$

$10 x = 7. \overline{8}$

The reason why we do this is because now have two numbers, color(brown)(100x = 78.bar8, and color(brown)(10x = 7.bar8, so now we can cancel out the two repeating decimals, and then subtract the second number from the first, to get a whole integer.

$\left(100 x = 78. \cancel{88 \overline{8}}\right) - \left(10 x = 7. \cancel{88 \overline{8}}\right) = \left(90 x = 71\right)$

Now we can find $x$ by using algebra.

$90 x = 71$

Divide each side by $90$ to find $x$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{90}}} x}{\cancel{\textcolor{red}{90}}} = \frac{71}{90}$

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

color(blue)(0.7bar8 = 71/90

Because we can not simplify any further, this is our final answer.

I got all of this information from Khan Academy's videos on this , you can check it out here:
Converting Repeating Decimals to Fractions 1
Converting Repeating Decimals to Fractions 2
Hope this Helps :)