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# Number Theory

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Convert 0.29\overline{6} to a fraction in simplest form.

Jun 18, 2024

#1
+1252
+1

We can apply a handy trick to solve this problem.

Let's set a variable. Let's set $$x=0.29\overline{6}$$

We want to put x in terms of a fraction.

Now, for that value of x, we have $$10 x = 2.96\overline{6}$$

This is important, as now we subtract x from 10x. We get

$$10x-x = 2.96\overline{6}-0.29\overline{6}$$

Since the repeating decimal cancels out, we have

$$9 x = 2.67$$

Now, we simply solve for x. We have

$$x = \frac{2.67}{9}$$

$$x=\frac{2.67}{9}\times \frac{100}{100}= \frac{267}{900}$$

$$x = \frac{89}{300}$$

Thus, our final answer is 89/300.

Thanks! :)

Jun 18, 2024

#1
+1252
+1

We can apply a handy trick to solve this problem.

Let's set a variable. Let's set $$x=0.29\overline{6}$$

We want to put x in terms of a fraction.

Now, for that value of x, we have $$10 x = 2.96\overline{6}$$

This is important, as now we subtract x from 10x. We get

$$10x-x = 2.96\overline{6}-0.29\overline{6}$$

Since the repeating decimal cancels out, we have

$$9 x = 2.67$$

Now, we simply solve for x. We have

$$x = \frac{2.67}{9}$$

$$x=\frac{2.67}{9}\times \frac{100}{100}= \frac{267}{900}$$

$$x = \frac{89}{300}$$

Thus, our final answer is 89/300.

Thanks! :)

NotThatSmart Jun 18, 2024