Ivan Waters

2023-03-28

What is $0.78888.....$ converted into a fraction? $\left(0.7\overline{8}\right)$

progutannx7f

In order to accomplish this, we can make the number $\left(0.7\overline{8}\right)$ equivalent to a pro numeral, and for this example, we'll use x
$\overline{x}$ just means that x is a reccurring /repeating number, then $0.7\overline{8}=0.788888888888888888888...$.
Thus, we have:
$x=0.7888....=0.7\overline{8}$
To get 100x, we can multiply x by 100, but we also need to do that on the other side.
$x×100=78.\overline{8}×100$
$100x=78.\overline{8}$
$10x=7.\overline{8}$
The reason why we do this is because now have two numbers, $100x=78.\overline{8}$, so now we can cancel out the two repeating decimals, and then subtract the second number from the first, to get a whole integer.
$\left(100x=78.\overline{)88\overline{8}}\right)-\left(10x=7.\overline{)88\overline{8}}\right)=\left(90x=71\right)$
Now, we can use algebra to find x.
$90x=71$
Divide each side by 90 to find x
$\frac{\overline{)90}x}{\overline{)90}}=\frac{71}{90}$
$x=\frac{71}{90}$
$0.7\overline{8}=\frac{71}{90}$

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