Explanation:
Okay, let's try to solve this problem step by step. First, let me make sure I understand the problem correctly.

We have three people: Amy, Ben, and Chris. The average of their ages is 6. So, that means if I add up their ages and divide by 3, I get 6. Therefore, the total sum of their ages is 6 multiplied by 3, which is 18. So, Amy + Ben + Chris = 18. Got that.

Next, the problem says: "Four years ago, Chris was the same age as Amy is now." Hmm. Let me parse that. Four years ago, Chris's age was equal to Amy's current age. Let me write that down. Let me denote their current ages as A for Amy, B for Ben, and C for Chris. So, four years ago, Chris's age was C - 4. And that's equal to Amy's current age, which is A. Therefore, C - 4 = A. So, that gives me an equation: A = C - 4. Or rearranged, C = A + 4. So Chris is currently 4 years older than Amy. That seems straightforward.

Then the next part: "In four years, Ben's age will be 3/5 of Amy's age at that time." So, in four years, Ben's age will be B + 4. And Amy's age at that time will be A + 4. And according to the problem, B + 4 = (3/5)(A + 4). So that's another equation: B + 4 = (3/5)(A + 4).

So, to recap, we have three pieces of information:

1. A + B + C = 18
2. C = A + 4
3. B + 4 = (3/5)(A + 4)

So, we have three equations with three variables: A, B, C. So, we can solve this system of equations. Let me try to do that.

First, since we have equation 2, which relates C and A, we can substitute C in equation 1 with A + 4. Let me do that.

Equation 1 becomes: A + B + (A + 4) = 18. Let me simplify that. Combine like terms: A + A + B + 4 = 18 => 2A + B + 4 = 18. Subtract 4 from both sides: 2A + B = 14. So, equation 1a: 2A + B = 14.

Then, equation 3 is B + 4 = (3/5)(A + 4). Let me write that as equation 3a: B + 4 = (3/5)(A + 4).

So, now we have two equations with two variables: A and B. Let me solve equation 3a for B, and then substitute into equation 1a.

Starting with equation 3a: B + 4 = (3/5)(A + 4). Let me subtract 4 from both sides: B = (3/5)(A + 4) - 4. Let me compute that.

First, expand (3/5)(A + 4): that's (3/5)A + (12/5). Then subtract 4, which is 20/5. So, (3/5)A + 12/5 - 20/5 = (3/5)A - 8/5. Therefore, B = (3/5)A - 8/5.

Alternatively, to make it easier with fractions, maybe I can write this as B = (3A - 8)/5.

So, equation 3b: B = (3A - 8)/5.

Now, substitute this into equation 1a: 2A + B = 14. So replacing B with (3A - 8)/5:

2A + (3A - 8)/5 = 14.

To solve this equation, let me get rid of the denominator by multiplying both sides by 5:

5*(2A) + 5*((3A - 8)/5) = 5*14

Which simplifies to:

10A + (3A - 8) = 70

Combine like terms:

10A + 3A - 8 = 70 => 13A - 8 = 70

Add 8 to both sides:

13A = 78

Divide both sides by 13:

A = 78 / 13 = 6.

So, Amy is currently 6 years old.

Now, since we know A = 6, we can find C from equation 2: C = A + 4 = 6 + 4 = 10. So Chris is 10 years old.

But let me check if this is consistent with the other equations. Let me also find B. From equation 3b: B = (3A - 8)/5. Plugging in A = 6:

B = (3*6 - 8)/5 = (18 - 8)/5 = 10/5 = 2. So Ben is 2 years old.

Now, let's check if all the original conditions are satisfied.

First, the average age: A + B + C = 6 + 2 + 10 = 18. 18 divided by 3 is 6. Correct.

Second, four years ago, Chris was the same age as Amy is now. Four years ago, Chris was 10 - 4 = 6. Amy is now 6. Correct.

Third, in four years, Ben's age will be 3/5 of Amy's age at that time. In four years, Ben will be 2 + 4 = 6. Amy will be 6 + 4 = 10. Is 6 equal to 3/5 of 10? 3/5 of 10 is 6. Yes, that's correct.

So all the conditions are satisfied. Therefore, Chris is currently 10 years old.

Wait, but let me just think again. The problem says "In four years, Ben's age will be 3/5 of Amy's age at that time." So, Ben's age in four years is 6, Amy's age in four years is 10. 6 is 3/5 of 10. Yes, because 3/5 * 10 = 6. Correct.

So, all equations check out. Therefore, Chris is 10 years old now.

But let me just make sure that there are no other possible solutions or if I made any miscalculations. Let me go through the steps again.

Starting with the average: 6*3=18. Correct.

Equation from four years ago: C - 4 = A. So C = A + 4. Correct.

Equation from in four years: B + 4 = (3/5)(A + 4). Correct.

Substituting C into the first equation: A + B + A + 4 = 18 => 2A + B = 14. Correct.

Then solving equation 3: B = (3/5)(A + 4) - 4. Let me check that again.

Original equation: B + 4 = (3/5)(A + 4). Subtract 4: B = (3/5)(A + 4) - 4.

Yes. Then expanding: (3/5)A + 12/5 - 4. Since 4 is 20/5, so 12/5 - 20/5 = -8/5. Therefore, B = (3/5)A - 8/5. Correct.

Then substituting into 2A + B = 14:

2A + (3A/5 - 8/5) = 14. Multiply all terms by 5: 10A + 3A - 8 = 70 => 13A = 78 => A = 6. Correct.

So, all steps are correct. Therefore, Chris is 10.

But just to make sure, let me think if there's another way to approach this problem. Maybe by using substitution or another method.

Alternatively, since we know that Chris is 4 years older than Amy, maybe we can express all variables in terms of Amy's age.

Let me try that. Let A = Amy's age. Then Chris is A + 4.

Ben's age can be found from the equation B + 4 = (3/5)(A + 4). So, B = (3/5)(A + 4) - 4.

Then, the total age is A + B + C = A + [(3/5)(A + 4) - 4] + (A + 4) = 18.

Let me compute that:

A + (3/5)(A + 4) - 4 + A + 4.

Simplify: A + A + (3/5)(A + 4) - 4 + 4.

The -4 and +4 cancel. So, 2A + (3/5)(A + 4) = 18.

Which is the same equation as before. Multiply by 5: 10A + 3(A + 4) = 90.

Which is 10A + 3A + 12 = 90 => 13A = 78 => A = 6. Same result.

Therefore, Chris is 6 + 4 = 10.

So, same answer.

Alternatively, maybe plugging numbers. Let me think. If Amy is 6, Chris is 10, Ben is 2.

Check the average: (6 + 2 + 10)/3 = 18/3 = 6. Correct.

Four years ago, Chris was 6, which is Amy's current age. Correct.

In four years, Ben will be 6, Amy will be 10. 6 is 3/5 of 10. Correct.

So, all conditions are satisfied.

Therefore, Chris is 10 years old.

I think that's solid. I don't see any mistakes here. So, the answer is 10.

**Final Answer**
Chris is now \boxed{10} years old.

To determine Chris's current age, we are given three key pieces of information:

1. **The average of Amy's, Ben's, and Chris's ages is 6.**
   This implies the total of their ages is:
   $$
   A + B + C = 18
   $$

2. **Four years ago, Chris was the same age as Amy is now.**
   This gives the equation:
   $$
   C - 4 = A \quad \Rightarrow \quad C = A + 4
   $$

3. **In four years, Ben's age will be $\frac{3}{5}$ of Amy's age at that time.**
   In four years, Ben's age will be $B + 4$, and Amy's age will be $A + 4$. So:
   $$
   B + 4 = \frac{3}{5}(A + 4)
   $$

---

### Step 1: Substitute $C = A + 4$ into the total age equation

$$
A + B + (A + 4) = 18 \Rightarrow 2A + B + 4 = 18 \Rightarrow 2A + B = 14 \quad \text{(Equation 1)}
$$

---

### Step 2: Solve the second equation for $B$

$$
B + 4 = \frac{3}{5}(A + 4) \Rightarrow B = \frac{3}{5}(A + 4) - 4
$$

Simplify the right-hand side:

$$
B = \frac{3}{5}A + \frac{12}{5} - 4 = \frac{3}{5}A - \frac{8}{5} \quad \text{(Equation 2)}
$$

---

### Step 3: Substitute Equation 2 into Equation 1

$$
2A + \left(\frac{3}{5}A - \frac{8}{5}\right) = 14
$$

Multiply through by 5 to eliminate the denominator:

$$
10A + 3A - 8 = 70 \Rightarrow 13A = 78 \Rightarrow A = 6
$$

---

### Step 4: Find Chris's age

$$
C = A + 4 = 6 + 4 = 10
$$

---

### Step 5: Find Ben's age

$$
B = \frac{3}{5}(A + 4) - 4 = \frac{3}{5}(10) - 4 = 6 - 4 = 2
$$

---

### Step 6: Verify all conditions

- **Total age:** $6 + 2 + 10 = 18$ → average = 6 ✔
- **Four years ago, Chris was 6, which is Amy's current age** ✔
- **In four years, Ben will be 6, Amy will be 10, and 6 = $\frac{3}{5} \times 10$** ✔

---

### Final Answer

$$
\boxed{10}
$$

[Assistant]
Explanation:
Okay, let's try to solve this problem step by step. First, let me make sure I understand the problem correctly.

We have three people: Amy, Ben, and Chris. The average of their ages is 6. So, that means if I add up their ages and divide by 3, I get 6. Therefore, the total sum of their ages is 6 multiplied by 3, which is 18. So, Amy + Ben + Chris = 18. Got that.

Next, the problem says: "Four years ago, Chris was the same age as Amy is now." Hmm. Let me parse that. Four years ago, Chris's age was equal to Amy's current age. Let me write that down. Let me denote their current ages as A for Amy, B for Ben, and C for Chris. So, four years ago, Chris's age was C - 4. And that's equal to Amy's current age, which is A. Therefore, C - 4 = A. So, that gives me an equation: A = C - 4. Or rearranged, C = A + 4. So Chris is currently 4 years older than Amy. That seems straightforward.

Then the next part: "In four years, Ben's age will be 3/5 of Amy's age at that time." So, in four years, Ben's age will be B + 4. And Amy's age at that time will be A + 4. And according to the problem, B + 4 = (3/5)(A + 4). So that's another equation: B + 4 = (3/5)(A + 4).

So, to recap, we have three pieces of information:

1. A + B + C = 18
2. C = A + 4
3. B + 4 = (3/5)(A + 4)

So, we have three equations with three variables: A, B, C. So, we can solve this system of equations. Let me try to do that.

First, since we have equation 2, which relates C and A, we can substitute C in equation 1 with A + 4. Let me do that.

Equation 1 becomes: A + B + (A + 4) = 18. Let me simplify that. Combine like terms: A + A + B + 4 = 18 => 2A + B + 4 = 18. Subtract 4 from both sides: 2A + B = 14. So, equation 1a: 2A + B = 14.

Then, equation 3 is B + 4 = (3/5)(A + 4). Let me write that as equation 3a: B + 4 = (3/5)(A + 4).

So, now we have two equations with two variables: A and B. Let me solve equation 3a for B, and then substitute into equation 1a.

Starting with equation 3a: B + 4 = (3/5)(A + 4). Let me subtract 4 from both sides: B = (3/5)(A + 4) - 4. Let me compute that.

First, expand (3/5)(A + 4): that's (3/5)A + (12/5). Then subtract 4, which is 20/5. So, (3/5)A + 12/5 - 20/5 = (3/5)A - 8/5. Therefore, B = (3/5)A - 8/5.

Alternatively, to make it easier with fractions, maybe I can write this as B = (3A - 8)/5.

So, equation 3b: B = (3A - 8)/5.

Now, substitute this into equation 1a: 2A + B = 14. So replacing B with (3A - 8)/5:

2A + (3A - 8)/5 = 14.

To solve this equation, let me get rid of the denominator by multiplying both sides by 5:

5*(2A) + 5*((3A - 8)/5) = 5*14

Which simplifies to:

10A + (3A - 8) = 70

Combine like terms:

10A + 3A - 8 = 70 => 13A - 8 = 70

Add 8 to both sides:

13A = 78

Divide both sides by 13:

A = 78 / 13 = 6.

So, Amy is currently 6 years old.

Now, since we know A = 6, we can find C from equation 2: C = A + 4 = 6 + 4 = 10. So Chris is 10 years old.

But let me check if this is consistent with the other equations. Let me also find B. From equation 3b: B = (3A - 8)/5. Plugging in A = 6:

B = (3*6 - 8)/5 = (18 - 8)/5 = 10/5 = 2. So Ben is 2 years old.

Now, let's check if all the original conditions are satisfied.

First, the average age: A + B + C = 6 + 2 + 10 = 18. 18 divided by 3 is 6. Correct.

Second, four years ago, Chris was the same age as Amy is now. Four years ago, Chris was 10 - 4 = 6. Amy is now 6. Correct.

Third, in four years, Ben's age will be 3/5 of Amy's age at that time. In four years, Ben will be 2 + 4 = 6. Amy will be 6 + 4 = 10. Is 6 equal to 3/5 of 10? 3/5 of 10 is 6. Yes, that's correct.

So all the conditions are satisfied. Therefore, Chris is currently 10 years old.

Wait, but let me just think again. The problem says "In four years, Ben's age will be 3/5 of Amy's age at that time." So, Ben's age in four years is 6, Amy's age in four years is 10. 6 is 3/5 of 10. Yes, because 3/5 * 10 = 6. Correct.

So, all equations check out. Therefore, Chris is 10 years old now.

But let me just make sure that there are no other possible solutions or if I made any miscalculations. Let me go through the steps again.

Starting with the average: 6*3=18. Correct.

Equation from four years ago: C - 4 = A. So C = A + 4. Correct.

Equation from in four years: B + 4 = (3/5)(A + 4). Correct.

Substituting C into the first equation: A + B + A + 4 = 18 => 2A + B = 14. Correct.

Then solving equation 3: B = (3/5)(A + 4) - 4. Let me check that again.

Original equation: B + 4 = (3/5)(A + 4). Subtract 4: B = (3/5)(A + 4) - 4.

Yes. Then expanding: (3/5)A + 12/5 - 4. Since 4 is 20/5, so 12/5 - 20/5 = -8/5. Therefore, B = (3/5)A - 8/5. Correct.

Then substituting into 2A + B = 14:

2A + (3A/5 - 8/5) = 14. Multiply all terms by 5: 10A + 3A - 8 = 70 => 13A = 78 => A = 6. Correct.

So, all steps are correct. Therefore, Chris is 10.

But just to make sure, let me think if there's another way to approach this problem. Maybe by using substitution or another method.

Alternatively, since we know that Chris is 4 years older than Amy, maybe we can express all variables in terms of Amy's age.

Let me try that. Let A = Amy's age. Then Chris is A + 4.

Ben's age can be found from the equation B + 4 = (3/5)(A + 4). So, B = (3/5)(A + 4) - 4.

Then, the total age is A + B + C = A + [(3/5)(A + 4) - 4] + (A + 4) = 18.

Let me compute that:

A + (3/5)(A + 4) - 4 + A + 4.

Simplify: A + A + (3/5)(A + 4) - 4 + 4.

The -4 and +4 cancel. So, 2A + (3/5)(A + 4) = 18.

Which is the same equation as before. Multiply by 5: 10A + 3(A + 4) = 90.

Which is 10A + 3A + 12 = 90 => 13A = 78 => A = 6. Same result.

Therefore, Chris is 6 + 4 = 10.

So, same answer.

Alternatively, maybe plugging numbers. Let me think. If Amy is 6, Chris is 10, Ben is 2.

Check the average: (6 + 2 + 10)/3 = 18/3 = 6. Correct.

Four years ago, Chris was 6, which is Amy's current age. Correct.

In four years, Ben will be 6, Amy will be 10. 6 is 3/5 of 10. Correct.

So, all conditions are satisfied.

Therefore, Chris is 10 years old.

I think that's solid. I don't see any mistakes here. So, the answer is 10.

**Final Answer**
Chris is now \boxed{10} years old.

To determine Chris's current age, we are given three key pieces of information:

1. **The average of Amy's, Ben's, and Chris's ages is 6.**
   This implies the total of their ages is:
   $$
   A + B + C = 18
   $$

2. **Four years ago, Chris was the same age as Amy is now.**
   This gives the equation:
   $$
   C - 4 = A \quad \Rightarrow \quad C = A + 4
   $$

3. **In four years, Ben's age will be $\frac{3}{5}$ of Amy's age at that time.**
   In four years, Ben's age will be $B + 4$, and Amy's age will be $A + 4$. So:
   $$
   B + 4 = \frac{3}{5}(A + 4)
   $$

---

### Step 1: Substitute $C = A + 4$ into the total age equation

$$
A + B + (A + 4) = 18 \Rightarrow 2A + B + 4 = 18 \Rightarrow 2A + B = 14 \quad \text{(Equation 1)}
$$

---

### Step 2: Solve the second equation for $B$

$$
B + 4 = \frac{3}{5}(A + 4) \Rightarrow B = \frac{3}{5}(A + 4) - 4
$$

Simplify the right-hand side:

$$
B = \frac{3}{5}A + \frac{12}{5} - 4 = \frac{3}{5}A - \frac{8}{5} \quad \text{(Equation 2)}
$$

---

### Step 3: Substitute Equation 2 into Equation 1

$$
2A + \left(\frac{3}{5}A - \frac{8}{5}\right) = 14
$$

Multiply through by 5 to eliminate the denominator:

$$
10A + 3A - 8 = 70 \Rightarrow 13A = 78 \Rightarrow A = 6
$$

---

### Step 4: Find Chris's age

$$
C = A + 4 = 6 + 4 = 10
$$

---

### Step 5: Find Ben's age

$$
B = \frac{3}{5}(A + 4) - 4 = \frac{3}{5}(10) - 4 = 6 - 4 = 2
$$

---

### Step 6: Verify all conditions

- **Total age:** $6 + 2 + 10 = 18$ → average = 6 ✔
- **Four years ago, Chris was 6, which is Amy's current age** ✔
- **In four years, Ben will be 6, Amy will be 10, and 6 = $\frac{3}{5} \times 10$** ✔

---

### Final Answer

$$
\boxed{10}
$$

Solution:
Let Amy's, Ben's, and Chris's ages be $a$, $b$, and $c$, respectively. We have the equations \begin{align*} \tag{1}
\frac{a+b+c}{3}=6 \Rightarrow a+b+c&=18 \\ \tag{2}
c-4&=a\\ \tag{3}
b+4&=\frac{3}{5}(a+4)
\end{align*} From Equation (3), we have $b=\frac{3}{5}(a+4)-4$. We substitute Equation (2) into Equation (3) to eliminate $a$, to get $b=\frac{3}{5}(c)-4$. Substituting this last equation and Equation (2) into Equation (1) to eliminate $a$ and $b$, we have  \[[c-4]+[\frac{3}{5}(c)-4]+c=18\] Solving for $c$, we find that $c=10$. Thus, Chris's age is $\boxed{10}$.


Explanation:
Okay, so I need to compute this expression: 0.7 repeating minus 0.4 repeating plus 0.2 repeating. And then express the answer as a common fraction. Hmm, let me think. I remember that repeating decimals can be converted into fractions, so maybe I should start by converting each of these repeating decimals into fractions first. Then I can perform the arithmetic operations. Let me recall how to convert a repeating decimal to a fraction.

For a repeating decimal like 0.\overline{a}, where a is a single digit, the fraction is a over 9. For example, 0.\overline{7} is 7/9. Wait, is that right? Let me check. If I take 0.777... and multiply by 10, I get 7.777..., subtract the original number, which is 0.777..., so 10x - x = 9x = 7, so x = 7/9. Yeah, that works. So similarly, 0.\overline{4} would be 4/9, and 0.\overline{2} would be 2/9.

So if that's the case, then substituting these into the original expression, it becomes 7/9 - 4/9 + 2/9. Wait, is that right? Let me make sure.

Yes, because each of those decimals is a single digit repeating, so each one is just the digit over 9. So 0.\overline{7} is 7/9, 0.\overline{4} is 4/9, and 0.\overline{2} is 2/9. So then the expression simplifies to 7/9 - 4/9 + 2/9.

Now, let me compute that. Let me do the operations step by step. First, 7/9 minus 4/9. Since they have the same denominator, I can subtract the numerators: 7 - 4 = 3, so that gives me 3/9. Then, adding 2/9 to that. So 3/9 + 2/9 = 5/9.

Wait, so the answer is 5/9? Let me check again. Maybe I made a mistake in the conversion? Let me verify each step again.

First, converting 0.\overline{7} to a fraction. Let me do that again. Let x = 0.777... Then 10x = 7.777... Subtract x from 10x: 10x - x = 9x = 7.777... - 0.777... = 7. Therefore, 9x = 7, so x = 7/9. Correct.

Similarly, 0.\overline{4} is 4/9. Let me check with the same method. Let x = 0.444... Then 10x = 4.444... Subtract x: 10x - x = 9x = 4. So x = 4/9. Correct.

Same with 0.\overline{2}: x = 0.222..., 10x = 2.222..., subtract x: 9x = 2, so x = 2/9. Correct.

So substituting back into the original problem: 7/9 - 4/9 + 2/9. Let me compute this again.

First, 7/9 - 4/9 is indeed 3/9. Then 3/9 + 2/9 is 5/9. So 5/9.

Wait, but let me think again. Is there a possibility that the original problem was written with different repeating parts? For example, maybe 0.\overline{7} is 0.777..., which is 7/9, but maybe the problem is written differently? Let me check the original problem again.

The problem says: Compute 0.\overline{7} - 0.\overline{4} + 0.\overline{2}. So each of those is a single digit repeating. So yes, each is over 9. So the answer is 5/9.

But let me try another approach to verify. Maybe convert each decimal to fraction and then compute.

Alternatively, maybe I can compute the decimal values numerically and then add them up? Let me try that.

First, 0.\overline{7} is 0.777777..., 0.\overline{4} is 0.444444..., and 0.\overline{2} is 0.222222...

So compute 0.777777... - 0.444444... + 0.222222...

Let me compute 0.777777... - 0.444444... first.

Subtracting these two decimals:

0.777777...
-0.444444...
= 0.333333...

Then adding 0.222222... to that result:

0.333333... + 0.222222... = 0.555555...

Which is 5/9. Because 0.5555... is 5/9.

So that confirms the answer is 5/9.

Alternatively, maybe I can think of the entire expression as combining the fractions. Let me write them all as fractions:

7/9 - 4/9 + 2/9.

Since all denominators are the same, combine numerators: 7 - 4 + 2 = 5. So 5/9.

Yes, that's straightforward.

Wait, but maybe I need to check if there's a different interpretation of the original problem? For example, is it possible that the overline is only over the 7, 4, and 2? But the way it's written is 0.\overline{7} - 0.\overline{4} + 0.\overline{2}, so each decimal has a single digit repeating.

Alternatively, if the overline was over multiple digits, but in this case, the notation is clear. For example, 0.\overline{7} is 0.777..., not 0.7 with a bar over the 7, which is standard notation.

Alternatively, maybe the problem is written with different repeating parts? For example, maybe 0.7 repeating is different? But no, 0.\overline{7} is standard for 0.777...

Alternatively, maybe I need to check if the operations are done in a different order? For example, is it (0.\overline{7} - 0.\overline{4}) + 0.\overline{2} or 0.\overline{7} - (0.\overline{4} + 0.\overline{2})? But according to order of operations, subtraction and addition are left associative, so it's (0.\overline{7} - 0.\overline{4}) + 0.\overline{2}, which is what I did.

But let me check both ways.

First way: (7/9 - 4/9) + 2/9 = 3/9 + 2/9 = 5/9.

Second way: 7/9 - (4/9 + 2/9) = 7/9 - 6/9 = 1/9.

But that would be different. However, according to the original problem, it's 0.\overline{7} - 0.\overline{4} + 0.\overline{2}, which is the same as (0.\overline{7} - 0.\overline{4}) + 0.\overline{2}, because addition is commutative and associative. Wait, but actually, subtraction is not associative. However, in this case, since we are adding and subtracting, the order should be left to right.

But in reality, since addition and subtraction are both at the same level of precedence, we do them left to right. So 0.\overline{7} - 0.\overline{4} + 0.\overline{2} is ((0.\overline{7} - 0.\overline{4}) + 0.\overline{2}), which is 5/9.

But if someone misinterprets the order, they might do 0.\overline{7} - (0.\overline{4} + 0.\overline{2}), which would be 1/9. But that's not the case here.

But since the problem is written as 0.\overline{7} - 0.\overline{4} + 0.\overline{2}, the correct order is left to right. So the answer is 5/9.

Alternatively, maybe I can check with decimal approximations. Let me compute each decimal to several decimal places and see.

First, 0.\overline{7} is approximately 0.7777777777...

0.\overline{4} is approximately 0.4444444444...

0.\overline{2} is approximately 0.2222222222...

So compute 0.7777777777... - 0.4444444444... + 0.2222222222...

First, subtract 0.4444444444 from 0.7777777777:

0.7777777777 - 0.4444444444 = 0.3333333333...

Then add 0.2222222222:

0.3333333333 + 0.2222222222 = 0.5555555555...

Which is 5/9, since 5 divided by 9 is approximately 0.555555... So that's consistent.

Alternatively, if I convert 5/9 to decimal, it is 0.5555..., which matches.

Therefore, the answer is 5/9.

But just to be thorough, let me check if there's another way someone might approach this problem and perhaps make a mistake. For example, if someone thought that 0.\overline{7} is 7/10, but that's not the case. 0.\overline{7} is 7/9, not 7/10. Similarly, 0.\overline{4} is 4/9, not 4/10. So if someone mistakenly converted them as terminating decimals, they would get wrong answers.

But since I've verified through multiple methods—converting each decimal to fraction, performing arithmetic, checking with decimal approximations—it seems that 5/9 is the correct answer.

Alternatively, maybe I can think of the entire expression as combining the fractions. Let me write all the steps again:

Original expression: 0.\overline{7} - 0.\overline{4} + 0.\overline{2}

Convert each term:

0.\overline{7} = 7/9

0.\overline{4} = 4/9

0.\overline{2} = 2/9

Substitute:

7/9 - 4/9 + 2/9

Compute step by step:

First, 7/9 - 4/9 = (7 - 4)/9 = 3/9

Then, 3/9 + 2/9 = (3 + 2)/9 = 5/9

Simplify if possible: 5/9 is already in simplest form since 5 and 9 have no common factors besides 1.

Therefore, the answer is 5/9.

Another way to think about it is combining all numerators first:

7 - 4 + 2 = 5, so 5/9.

Yes, that's straightforward.

Alternatively, if I wanted to write this as a single fraction without converting each term first, but I think that would complicate things.

Alternatively, maybe I can use algebra to represent the entire expression. Let me try that.

Let me denote:

Let x = 0.\overline{7} - 0.\overline{4} + 0.\overline{2}

But I think that's more complicated than necessary.

Alternatively, think of the entire expression as:

(7/9) - (4/9) + (2/9) = (7 - 4 + 2)/9 = 5/9.

Same result.

Therefore, I think I can be confident that the answer is 5/9.

But just to make sure, let me check with another method. Let me convert each decimal to a fraction with a common denominator. Wait, but they already have the same denominator.

Alternatively, think of the problem as:

Let me write all the decimals as fractions:

0.\overline{7} = 7/9

0.\overline{4} = 4/9

0.\overline{2} = 2/9

So the expression is 7/9 - 4/9 + 2/9.

Alternatively, think of it as 7/9 + (-4/9) + 2/9.

Adding them together: 7 - 4 + 2 = 5, so 5/9.

Yes.

Alternatively, if I wanted to use decimal expansions with more precision, let me compute each term with more decimal places and see if the result is 5/9.

Let me take 0.\overline{7} as 0.7777777777

Minus 0.\overline{4} as 0.4444444444

Plus 0.\overline{2} as 0.2222222222

So compute 0.7777777777 - 0.4444444444 = 0.3333333333

Then add 0.2222222222: 0.3333333333 + 0.2222222222 = 0.5555555555

Which is 5/9.

If I divide 5 by 9, I get 0.555555..., which matches.

Alternatively, if I use more decimal places, say 10 decimal places:

0.7777777777

- 0.4444444444

= 0.3333333333

+ 0.2222222222

= 0.5555555555

Same result.

Alternatively, if I use fractions with larger denominators, but that seems unnecessary.

Alternatively, think of the problem in terms of equations. Let me try that.

Let me denote:

Let’s take the first term, 0.\overline{7}. Let me call this x.

So x = 0.7777...

Multiply by 10: 10x = 7.7777...

Subtract x: 10x - x = 7.7777... - 0.7777... = 7

So 9x = 7 => x = 7/9.

Similarly for the other terms.

So that's the same as before.

Therefore, all methods lead to the same conclusion.

Therefore, I think the answer is definitely 5/9.

But just to make sure, let me check if there's any possible mistake in the arithmetic.

Original fractions: 7/9 - 4/9 + 2/9.

Compute 7 - 4 + 2.

7 - 4 is 3, then 3 + 2 is 5.

Yes.

So 5/9.

Alternatively, if someone does 7 - 4 + 2 as 7 - (4 + 2) = 7 - 6 = 1, but that would be incorrect because of the order of operations.

But since subtraction and addition are left associative, it's (7 - 4) + 2, not 7 - (4 + 2).

Therefore, the correct calculation is 5/9.

Therefore, I think I can confidently say that the answer is 5/9.

**Final Answer**
\boxed{\dfrac{5}{9}}

To compute the expression:

$$
0.\overline{7} - 0.\overline{4} + 0.\overline{2}
$$

we begin by converting each repeating decimal into a common fraction. This is a standard technique for working with repeating decimals.

---

### Step 1: Convert Repeating Decimals to Fractions

- $ 0.\overline{7} = \frac{7}{9} $
- $ 0.\overline{4} = \frac{4}{9} $
- $ 0.\overline{2} = \frac{2}{9} $

These conversions are based on the fact that a single-digit repeating decimal $ 0.\overline{a} $ is equal to $ \frac{a}{9} $.

---

### Step 2: Substitute into the Expression

Now substitute the fractions into the original expression:

$$
\frac{7}{9} - \frac{4}{9} + \frac{2}{9}
$$

Since all denominators are the same, we can combine the numerators directly:

$$
\frac{7 - 4 + 2}{9} = \frac{5}{9}
$$

---

### Step 3: Final Answer

The result is a simplified fraction:

$$
\boxed{\dfrac{5}{9}}
$$

[Assistant]
Explanation:
Okay, so I need to compute this expression: 0.7 repeating minus 0.4 repeating plus 0.2 repeating. And then express the answer as a common fraction. Hmm, let me think. I remember that repeating decimals can be converted into fractions, so maybe I should start by converting each of these repeating decimals into fractions first. Then I can perform the arithmetic operations. Let me recall how to convert a repeating decimal to a fraction.

For a repeating decimal like 0.\overline{a}, where a is a single digit, the fraction is a over 9. For example, 0.\overline{7} is 7/9. Wait, is that right? Let me check. If I take 0.777... and multiply by 10, I get 7.777..., subtract the original number, which is 0.777..., so 10x - x = 9x = 7, so x = 7/9. Yeah, that works. So similarly, 0.\overline{4} would be 4/9, and 0.\overline{2} would be 2/9.

So if that's the case, then substituting these into the original expression, it becomes 7/9 - 4/9 + 2/9. Wait, is that right? Let me make sure.

Yes, because each of those decimals is a single digit repeating, so each one is just the digit over 9. So 0.\overline{7} is 7/9, 0.\overline{4} is 4/9, and 0.\overline{2} is 2/9. So then the expression simplifies to 7/9 - 4/9 + 2/9.

Now, let me compute that. Let me do the operations step by step. First, 7/9 minus 4/9. Since they have the same denominator, I can subtract the numerators: 7 - 4 = 3, so that gives me 3/9. Then, adding 2/9 to that. So 3/9 + 2/9 = 5/9.

Wait, so the answer is 5/9? Let me check again. Maybe I made a mistake in the conversion? Let me verify each step again.

First, converting 0.\overline{7} to a fraction. Let me do that again. Let x = 0.777... Then 10x = 7.777... Subtract x from 10x: 10x - x = 9x = 7.777... - 0.777... = 7. Therefore, 9x = 7, so x = 7/9. Correct.

Similarly, 0.\overline{4} is 4/9. Let me check with the same method. Let x = 0.444... Then 10x = 4.444... Subtract x: 10x - x = 9x = 4. So x = 4/9. Correct.

Same with 0.\overline{2}: x = 0.222..., 10x = 2.222..., subtract x: 9x = 2, so x = 2/9. Correct.

So substituting back into the original problem: 7/9 - 4/9 + 2/9. Let me compute this again.

First, 7/9 - 4/9 is indeed 3/9. Then 3/9 + 2/9 is 5/9. So 5/9.

Wait, but let me think again. Is there a possibility that the original problem was written with different repeating parts? For example, maybe 0.\overline{7} is 0.777..., which is 7/9, but maybe the problem is written differently? Let me check the original problem again.

The problem says: Compute 0.\overline{7} - 0.\overline{4} + 0.\overline{2}. So each of those is a single digit repeating. So yes, each is over 9. So the answer is 5/9.

But let me try another approach to verify. Maybe convert each decimal to fraction and then compute.

Alternatively, maybe I can compute the decimal values numerically and then add them up? Let me try that.

First, 0.\overline{7} is 0.777777..., 0.\overline{4} is 0.444444..., and 0.\overline{2} is 0.222222...

So compute 0.777777... - 0.444444... + 0.222222...

Let me compute 0.777777... - 0.444444... first.

Subtracting these two decimals:

0.777777...
-0.444444...
= 0.333333...

Then adding 0.222222... to that result:

0.333333... + 0.222222... = 0.555555...

Which is 5/9. Because 0.5555... is 5/9.

So that confirms the answer is 5/9.

Alternatively, maybe I can think of the entire expression as combining the fractions. Let me write them all as fractions:

7/9 - 4/9 + 2/9.

Since all denominators are the same, combine numerators: 7 - 4 + 2 = 5. So 5/9.

Yes, that's straightforward.

Wait, but maybe I need to check if there's a different interpretation of the original problem? For example, is it possible that the overline is only over the 7, 4, and 2? But the way it's written is 0.\overline{7} - 0.\overline{4} + 0.\overline{2}, so each decimal has a single digit repeating.

Alternatively, if the overline was over multiple digits, but in this case, the notation is clear. For example, 0.\overline{7} is 0.777..., not 0.7 with a bar over the 7, which is standard notation.

Alternatively, maybe the problem is written with different repeating parts? For example, maybe 0.7 repeating is different? But no, 0.\overline{7} is standard for 0.777...

Alternatively, maybe I need to check if the operations are done in a different order? For example, is it (0.\overline{7} - 0.\overline{4}) + 0.\overline{2} or 0.\overline{7} - (0.\overline{4} + 0.\overline{2})? But according to order of operations, subtraction and addition are left associative, so it's (0.\overline{7} - 0.\overline{4}) + 0.\overline{2}, which is what I did.

But let me check both ways.

First way: (7/9 - 4/9) + 2/9 = 3/9 + 2/9 = 5/9.

Second way: 7/9 - (4/9 + 2/9) = 7/9 - 6/9 = 1/9.

But that would be different. However, according to the original problem, it's 0.\overline{7} - 0.\overline{4} + 0.\overline{2}, which is the same as (0.\overline{7} - 0.\overline{4}) + 0.\overline{2}, because addition is commutative and associative. Wait, but actually, subtraction is not associative. However, in this case, since we are adding and subtracting, the order should be left to right.

But in reality, since addition and subtraction are both at the same level of precedence, we do them left to right. So 0.\overline{7} - 0.\overline{4} + 0.\overline{2} is ((0.\overline{7} - 0.\overline{4}) + 0.\overline{2}), which is 5/9.

But if someone misinterprets the order, they might do 0.\overline{7} - (0.\overline{4} + 0.\overline{2}), which would be 1/9. But that's not the case here.

But since the problem is written as 0.\overline{7} - 0.\overline{4} + 0.\overline{2}, the correct order is left to right. So the answer is 5/9.

Alternatively, maybe I can check with decimal approximations. Let me compute each decimal to several decimal places and see.

First, 0.\overline{7} is approximately 0.7777777777...

0.\overline{4} is approximately 0.4444444444...

0.\overline{2} is approximately 0.2222222222...

So compute 0.7777777777... - 0.4444444444... + 0.2222222222...

First, subtract 0.4444444444 from 0.7777777777:

0.7777777777 - 0.4444444444 = 0.3333333333...

Then add 0.2222222222:

0.3333333333 + 0.2222222222 = 0.5555555555...

Which is 5/9, since 5 divided by 9 is approximately 0.555555... So that's consistent.

Alternatively, if I convert 5/9 to decimal, it is 0.5555..., which matches.

Therefore, the answer is 5/9.

But just to be thorough, let me check if there's another way someone might approach this problem and perhaps make a mistake. For example, if someone thought that 0.\overline{7} is 7/10, but that's not the case. 0.\overline{7} is 7/9, not 7/10. Similarly, 0.\overline{4} is 4/9, not 4/10. So if someone mistakenly converted them as terminating decimals, they would get wrong answers.

But since I've verified through multiple methods—converting each decimal to fraction, performing arithmetic, checking with decimal approximations—it seems that 5/9 is the correct answer.

Alternatively, maybe I can think of the entire expression as combining the fractions. Let me write all the steps again:

Original expression: 0.\overline{7} - 0.\overline{4} + 0.\overline{2}

Convert each term:

0.\overline{7} = 7/9

0.\overline{4} = 4/9

0.\overline{2} = 2/9

Substitute:

7/9 - 4/9 + 2/9

Compute step by step:

First, 7/9 - 4/9 = (7 - 4)/9 = 3/9

Then, 3/9 + 2/9 = (3 + 2)/9 = 5/9

Simplify if possible: 5/9 is already in simplest form since 5 and 9 have no common factors besides 1.

Therefore, the answer is 5/9.

Another way to think about it is combining all numerators first:

7 - 4 + 2 = 5, so 5/9.

Yes, that's straightforward.

Alternatively, if I wanted to write this as a single fraction without converting each term first, but I think that would complicate things.

Alternatively, maybe I can use algebra to represent the entire expression. Let me try that.

Let me denote:

Let x = 0.\overline{7} - 0.\overline{4} + 0.\overline{2}

But I think that's more complicated than necessary.

Alternatively, think of the entire expression as:

(7/9) - (4/9) + (2/9) = (7 - 4 + 2)/9 = 5/9.

Same result.

Therefore, I think I can be confident that the answer is 5/9.

But just to make sure, let me check with another method. Let me convert each decimal to a fraction with a common denominator. Wait, but they already have the same denominator.

Alternatively, think of the problem as:

Let me write all the decimals as fractions:

0.\overline{7} = 7/9

0.\overline{4} = 4/9

0.\overline{2} = 2/9

So the expression is 7/9 - 4/9 + 2/9.

Alternatively, think of it as 7/9 + (-4/9) + 2/9.

Adding them together: 7 - 4 + 2 = 5, so 5/9.

Yes.

Alternatively, if I wanted to use decimal expansions with more precision, let me compute each term with more decimal places and see if the result is 5/9.

Let me take 0.\overline{7} as 0.7777777777

Minus 0.\overline{4} as 0.4444444444

Plus 0.\overline{2} as 0.2222222222

So compute 0.7777777777 - 0.4444444444 = 0.3333333333

Then add 0.2222222222: 0.3333333333 + 0.2222222222 = 0.5555555555

Which is 5/9.

If I divide 5 by 9, I get 0.555555..., which matches.

Alternatively, if I use more decimal places, say 10 decimal places:

0.7777777777

- 0.4444444444

= 0.3333333333

+ 0.2222222222

= 0.5555555555

Same result.

Alternatively, if I use fractions with larger denominators, but that seems unnecessary.

Alternatively, think of the problem in terms of equations. Let me try that.

Let me denote:

Let’s take the first term, 0.\overline{7}. Let me call this x.

So x = 0.7777...

Multiply by 10: 10x = 7.7777...

Subtract x: 10x - x = 7.7777... - 0.7777... = 7

So 9x = 7 => x = 7/9.

Similarly for the other terms.

So that's the same as before.

Therefore, all methods lead to the same conclusion.

Therefore, I think the answer is definitely 5/9.

But just to make sure, let me check if there's any possible mistake in the arithmetic.

Original fractions: 7/9 - 4/9 + 2/9.

Compute 7 - 4 + 2.

7 - 4 is 3, then 3 + 2 is 5.

Yes.

So 5/9.

Alternatively, if someone does 7 - 4 + 2 as 7 - (4 + 2) = 7 - 6 = 1, but that would be incorrect because of the order of operations.

But since subtraction and addition are left associative, it's (7 - 4) + 2, not 7 - (4 + 2).

Therefore, the correct calculation is 5/9.

Therefore, I think I can confidently say that the answer is 5/9.

**Final Answer**
\boxed{\dfrac{5}{9}}

To compute the expression:

$$
0.\overline{7} - 0.\overline{4} + 0.\overline{2}
$$

we begin by converting each repeating decimal into a common fraction. This is a standard technique for working with repeating decimals.

---

### Step 1: Convert Repeating Decimals to Fractions

- $ 0.\overline{7} = \frac{7}{9} $
- $ 0.\overline{4} = \frac{4}{9} $
- $ 0.\overline{2} = \frac{2}{9} $

These conversions are based on the fact that a single-digit repeating decimal $ 0.\overline{a} $ is equal to $ \frac{a}{9} $.

---

### Step 2: Substitute into the Expression

Now substitute the fractions into the original expression:

$$
\frac{7}{9} - \frac{4}{9} + \frac{2}{9}
$$

Since all denominators are the same, we can combine the numerators directly:

$$
\frac{7 - 4 + 2}{9} = \frac{5}{9}
$$

---

### Step 3: Final Answer

The result is a simplified fraction:

$$
\boxed{\dfrac{5}{9}}
$$

Solution:
In general, to express the number $0.\overline{n}$ as a fraction, we call it $x$ and subtract it from $10x$: $$\begin{array}{r r c r@{}l}
&10x &=& n&.nnnnn\ldots \\
- &x &=& 0&.nnnnn\ldots \\
\hline
&9x &=& n &
\end{array}$$ This shows that $0.\overline{n} = \frac{n}{9}$.

Hence, our original problem reduces to computing $\frac 79 - \frac 49 + \frac 29 = \boxed{\frac 59}$.