Okay, so I need to compute 0.7 repeating minus 0.4 repeating plus 0.2 repeating and express the answer as a common fraction. Hmm, repeating decimals can sometimes be tricky, but I remember that there's a method to convert them into fractions. Let me think about how to approach each of these individually first.

Starting with 0.7 repeating. I think the way to convert a repeating decimal to a fraction is to set it as a variable, like x = 0.7777..., then multiply both sides by 10 to shift the decimal point. So, 10x = 7.7777..., right? Then subtract the original equation from this new one: 10x - x = 7.7777... - 0.7777..., which gives 9x = 7. So, x = 7/9. Therefore, 0.7 repeating is 7/9. That makes sense.

Next, 0.4 repeating. Using the same method. Let y = 0.4444..., multiply by 10: 10y = 4.4444..., subtract: 10y - y = 4.4444... - 0.4444... => 9y = 4 => y = 4/9. So, 0.4 repeating is 4/9. Got that.

Then, 0.2 repeating. Let z = 0.2222..., multiply by 10: 10z = 2.2222..., subtract: 10z - z = 2.2222... - 0.2222... => 9z = 2 => z = 2/9. So, 0.2 repeating is 2/9.

Now, substitute these fractions back into the original expression: 7/9 - 4/9 + 2/9. Let me compute this step by step. First, 7/9 minus 4/9 is (7 - 4)/9 = 3/9. Then, adding 2/9 gives 3/9 + 2/9 = 5/9. Wait, that seems straightforward. So the result is 5/9? Let me check my steps again to make sure I didn't make a mistake.

Converting each repeating decimal to fractions:

- 0.7 repeating: 7/9 ✔️
- 0.4 repeating: 4/9 ✔️
- 0.2 repeating: 2/9 ✔️

Substituting into the expression:

7/9 - 4/9 = 3/9 = 1/3. Then 1/3 + 2/9. Wait, hold on, 1/3 is 3/9, so 3/9 + 2/9 = 5/9. Yeah, that's correct. So the final answer is 5/9. Hmm, that seems right. Let me verify by converting the original decimals to fractions and then performing the arithmetic in decimal form to check.

Calculating 0.777... minus 0.444... plus 0.222... in decimal:

0.777... is approximately 0.777777...

0.444... is approximately 0.444444...

0.222... is approximately 0.222222...

So, 0.777777... minus 0.444444... is 0.333333..., and adding 0.222222... gives 0.555555... which is 5/9. Yep, that's 0.555555..., which is indeed 5/9. So that checks out.

Alternatively, if I add all the fractions together first: 7/9 - 4/9 + 2/9 = (7 - 4 + 2)/9 = 5/9. Same result. So, I think that's the correct answer. I don't see any errors in the steps. The key was converting each repeating decimal to a fraction using the standard method, then performing the arithmetic with those fractions. The final result simplifies to 5/9, which is a common fraction, so that's the answer.
Therefore, the result of $0.\overline{7}-0.\overline{4}+0.\overline{2}$ expressed as a common fraction is $\boxed{\frac{5}{9}}$.
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