+0

# Number Theory

0
3
2
+734

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

Jun 15, 2024

#2
+1204
+1

In order to solve this problem, we can use a very simple trick.

First, let's set $$x = 0.\overline{2123}$$

If we have this for the value of x, we get $$10000 x = 2123.\overline{2123}$$

Now, we subtract x from 10000x. We get

$$10000x-x= 2123.\overline{2123} -\overline{2123}\\ 9999x=2123$$

This means we get

$$x = \frac{2123}{9999}$$

Simplifying, we get

$$x = \frac{193}{909}$$

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

Thanks! :)

Jun 16, 2024

#1
+1710
0

Jun 15, 2024
#2
+1204
+1

In order to solve this problem, we can use a very simple trick.

First, let's set $$x = 0.\overline{2123}$$

If we have this for the value of x, we get $$10000 x = 2123.\overline{2123}$$

Now, we subtract x from 10000x. We get

$$10000x-x= 2123.\overline{2123} -\overline{2123}\\ 9999x=2123$$

This means we get

$$x = \frac{2123}{9999}$$

Simplifying, we get

$$x = \frac{193}{909}$$

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

Thanks! :)

NotThatSmart Jun 16, 2024