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Decimals

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Express 0.\overline{2123} as a base 10 fraction in reduced form.

Jun 14, 2024

#1
+1280
0

In order to do this, we can set some variables.

Let's set $$x=0.\overline{2123}$$

If x is equal to that, then notice that $$10000 x = 2123.\overline{2123}$$

This is very important to the problem.

Subtracting x from 10000x, we get

\eqalign{10000 X &= &\hfill2123.2123...\cr X &= &\hfill0.2123...\cr \hline 9999X &= &2123\cr}

Notice that the repeating decimal cancels out.

Now, we solve for x. We get

$$X = \frac{2123}{9999}\\X = \frac{193}{909}$$

So our final answer is $$\frac{193}{909}$$

Thanks! :)

Jun 14, 2024

#1
+1280
0

In order to do this, we can set some variables.

Let's set $$x=0.\overline{2123}$$

If x is equal to that, then notice that $$10000 x = 2123.\overline{2123}$$

This is very important to the problem.

Subtracting x from 10000x, we get

\eqalign{10000 X &= &\hfill2123.2123...\cr X &= &\hfill0.2123...\cr \hline 9999X &= &2123\cr}

Notice that the repeating decimal cancels out.

Now, we solve for x. We get

$$X = \frac{2123}{9999}\\X = \frac{193}{909}$$

So our final answer is $$\frac{193}{909}$$

Thanks! :)

NotThatSmart Jun 14, 2024