Explanation:
Okay, so I need to find the maximum value of e given these two equations:

a + b + c + d + e = 8,

and

a² + b² + c² + d² + e² = 16.

Hmm, let me think. Since we need to maximize e, maybe I should try to minimize the other variables? Because if I can make a, b, c, d as small as possible, then e can be as large as possible. But how do I do that?

Wait, there are constraints here. The sum of the variables is fixed at 8, and the sum of their squares is fixed at 16. So if I want to maximize e, I need to adjust the other variables so that they take up as little "space" as possible in terms of their squares.

I remember something about the Cauchy-Schwarz inequality or maybe using the method of Lagrange multipliers. But since this is a problem likely intended for algebra, maybe there's a simpler way. Let me think.

Let me consider that if I fix e, then the sum of a + b + c + d = 8 - e, and the sum of their squares is 16 - e². So, maybe I can use the Cauchy-Schwarz inequality on the variables a, b, c, d.

The Cauchy-Schwarz inequality says that for any real numbers, (a₁² + a₂² + ... + aₙ²)(1² + 1² + ... + 1²) ≥ (a₁ + a₂ + ... + aₙ)².

So applying that to a, b, c, d:

(a² + b² + c² + d²)(1 + 1 + 1 + 1) ≥ (a + b + c + d)².

Which would be:

(16 - e²) * 4 ≥ (8 - e)².

Let me write that down:

4*(16 - e²) ≥ (8 - e)².

Let me compute both sides:

Left side: 4*(16 - e²) = 64 - 4e².

Right side: (8 - e)² = 64 - 16e + e².

So the inequality becomes:

64 - 4e² ≥ 64 - 16e + e².

Subtract 64 from both sides:

-4e² ≥ -16e + e².

Bring all terms to the left side:

-4e² + 16e - e² ≥ 0.

Combine like terms:

-5e² + 16e ≥ 0.

Multiply both sides by -1 (which reverses the inequality):

5e² - 16e ≤ 0.

Factor:

e*(5e - 16) ≤ 0.

So this inequality holds when e is between 0 and 16/5, inclusive. Wait, but we are trying to find the maximum value of e. So according to this, e can be at most 16/5? But let me check if this is correct.

Wait, but hold on. The original problem is to maximize e, so if the inequality gives that e ≤ 16/5, then 16/5 is the maximum? But let me verify if equality can be achieved. Because Cauchy-Schwarz gives equality when all the variables are equal. So in this case, for the variables a, b, c, d, equality occurs when a = b = c = d.

So if we set a = b = c = d, then let's check if that gives us e = 16/5.

Let me suppose that a = b = c = d = x. Then:

Sum: 4x + e = 8 => 4x = 8 - e => x = (8 - e)/4.

Sum of squares: 4x² + e² = 16.

Substituting x:

4*((8 - e)/4)² + e² = 16.

Compute that:

4*( (64 - 16e + e²)/16 ) + e² = 16.

Simplify:

(64 - 16e + e²)/4 + e² = 16.

Multiply through:

(64 - 16e + e²)/4 + e² = 16.

Multiply numerator:

16 - 4e + (e²)/4 + e² = 16.

Combine like terms:

16 - 4e + (5e²)/4 = 16.

Subtract 16 from both sides:

-4e + (5e²)/4 = 0.

Multiply both sides by 4 to eliminate denominator:

-16e + 5e² = 0.

Factor:

e*(-16 + 5e) = 0.

So solutions are e = 0 or e = 16/5.

So when e = 16/5, then x = (8 - 16/5)/4 = (40/5 - 16/5)/4 = (24/5)/4 = 6/5.

So then, the variables a, b, c, d are each 6/5, and e is 16/5. Let me check if the sum of squares is indeed 16:

Sum of squares: 4*(6/5)^2 + (16/5)^2 = 4*(36/25) + 256/25 = 144/25 + 256/25 = 400/25 = 16. Perfect, that works.

So that seems to satisfy both equations. Therefore, e can be as large as 16/5.

But wait, let me think again. The Cauchy-Schwarz gave us that e ≤ 16/5. But is there a possibility that even if the inequality is not tight, maybe there's another configuration where e is larger?

Wait, but the logic is that for the sum of squares of a, b, c, d to be minimized given their sum, they should be equal. Because variance is minimized when all variables are equal. Therefore, if we want to minimize the sum of squares of a, b, c, d given their sum, we set them equal. Therefore, if we set them equal, then we can get the minimal sum of squares, which would allow e to be as large as possible.

Therefore, if we have the minimal sum of squares for a, b, c, d, then the remaining value for e is maximized.

So that seems correct. Therefore, the maximum value of e is 16/5, which is 3.2.

But let me check another way. Let me try to use Lagrange multipliers. Maybe that will confirm the result.

Let me consider variables a, b, c, d, e. We need to maximize e subject to:

Constraint 1: a + b + c + d + e = 8,

Constraint 2: a² + b² + c² + d² + e² = 16.

Let me set up the Lagrangian:

L = e - λ(a + b + c + d + e - 8) - μ(a² + b² + c² + d² + e² - 16).

Wait, actually, since we are maximizing e, the Lagrangian should be e minus the constraints multiplied by Lagrange multipliers. But maybe I need to set up the problem as maximizing e with the constraints.

Alternatively, since we can consider the problem as maximizing e with the two constraints.

But maybe it's easier to use substitution. Let me consider that from the first equation, a + b + c + d = 8 - e. Let me denote S = 8 - e, and Q = 16 - e².

So, we have four variables a, b, c, d with sum S and sum of squares Q. To maximize e, we need to minimize Q given S? Wait, no. Wait, Q is fixed as 16 - e². Wait, but if we want to maximize e, then Q = 16 - e² must be as small as possible? Wait, no. Wait, if e is larger, then Q = 16 - e² is smaller. But Q is the sum of squares of a, b, c, d. So if Q is smaller, that would mean that the sum of squares of a, b, c, d is smaller. But given that their sum is fixed at S = 8 - e, then the minimal possible sum of squares occurs when all variables are equal. Therefore, if we have that minimal sum of squares, then Q must be equal to that minimal value. Therefore, if we set Q equal to the minimal sum of squares, then we can have equality.

Therefore, the minimal sum of squares for a, b, c, d given their sum S is (S²)/4, by Cauchy-Schwarz. Wait, because if you have four variables with sum S, then the minimal sum of squares is achieved when all variables are equal, so each is S/4, so sum of squares is 4*(S/4)^2 = S²/4.

Therefore, we have that Q ≥ S² / 4.

But Q is 16 - e², and S is 8 - e. Therefore:

16 - e² ≥ (8 - e)² / 4.

Which is exactly the same inequality as before. So solving this gives us the same result.

So, 16 - e² ≥ (64 - 16e + e²)/4.

Multiply both sides by 4:

64 - 4e² ≥ 64 - 16e + e².

Which simplifies to -5e² + 16e ≥ 0, leading to e(16 - 5e) ≥ 0. Wait, earlier I had 5e² - 16e ≤ 0, which is equivalent.

So, the solutions are e between 0 and 16/5. Therefore, maximum e is 16/5.

Therefore, the maximum value of e is 16/5.

But let me check if there's another approach. For example, using the concept of variance.

The variance of the variables a, b, c, d is minimized when they are equal, which again leads to the same conclusion.

Alternatively, think of the variables a, b, c, d as being as close to each other as possible to minimize their sum of squares.

Alternatively, think of the problem as an optimization problem where we want to maximize e, so we can consider that the other variables are all equal.

So, if I assume that a = b = c = d, then as before, we can solve for e. Which gives us e = 16/5.

Alternatively, suppose that three of them are equal and one is different? Would that give a higher e? Probably not, because making them unequal would increase the sum of squares.

For example, suppose that three variables are equal to x and one is y. Then:

Sum: 3x + y + e = 8,

Sum of squares: 3x² + y² + e² = 16.

But then, if we try to minimize the sum of squares 3x² + y² given that 3x + y = 8 - e.

But this would be more complicated, but perhaps we can check if the minimal sum of squares is achieved when all variables are equal.

In general, for fixed sum, the sum of squares is minimized when all variables are equal. Therefore, even if we have different numbers of variables, the minimal sum of squares occurs when they are all equal. Therefore, in this case, if we have four variables, the minimal sum of squares is achieved when they are all equal. Therefore, the previous approach is valid.

Therefore, the maximum value of e is 16/5.

But let me check with another example. Suppose that three variables are zero, and one variable is something. Let me try that.

Suppose a = b = c = 0, then d + e = 8, and d² + e² = 16.

So, d = 8 - e. Then:

(8 - e)² + e² = 16.

Expanding:

64 - 16e + e² + e² = 16.

So 2e² - 16e + 64 - 16 = 0.

Which is 2e² - 16e + 48 = 0.

Divide by 2: e² - 8e + 24 = 0.

Discriminant: 64 - 96 = -32 < 0. No real solutions.

So that's impossible. Therefore, if three variables are zero, there's no real solution.

What if two variables are zero? Let me try that. Let a = b = 0, then c + d + e = 8, and c² + d² + e² = 16.

Let me try to maximize e here. So set c + d = 8 - e, and c² + d² = 16 - e².

Again, for fixed sum, the minimal sum of squares occurs when c = d. So set c = d = (8 - e)/2. Then:

Sum of squares: 2*((8 - e)/2)^2 = 2*(64 - 16e + e²)/4 = (64 - 16e + e²)/2.

Set equal to 16 - e²:

(64 - 16e + e²)/2 = 16 - e².

Multiply both sides by 2:

64 - 16e + e² = 32 - 2e².

Bring all terms to left:

64 - 16e + e² - 32 + 2e² = 0 => 3e² - 16e + 32 = 0.

Discriminant: 256 - 384 = -128 < 0. Again, no real solution.

Hmm, so even if two variables are zero, there's no solution.

What if one variable is zero? Let me try. Let a = 0, then b + c + d + e = 8, and b² + c² + d² + e² = 16.

Again, trying to maximize e. So set b + c + d = 8 - e, and b² + c² + d² = 16 - e².

Again, minimal sum of squares for b, c, d occurs when they are equal. So set b = c = d = (8 - e)/3. Then sum of squares is 3*((8 - e)/3)^2 = 3*(64 - 16e + e²)/9 = (64 - 16e + e²)/3.

Set equal to 16 - e²:

(64 - 16e + e²)/3 = 16 - e².

Multiply both sides by 3:

64 - 16e + e² = 48 - 3e².

Bring all terms to left:

64 - 16e + e² - 48 + 3e² = 0 => 4e² - 16e + 16 = 0.

Divide by 4: e² - 4e + 4 = 0 => (e - 2)^2 = 0 => e = 2.

So in this case, e = 2. Which is less than 16/5 = 3.2. Therefore, this is worse.

So even if we set one variable to zero, the maximum e is 2, which is less than 16/5.

Therefore, the case where all four variables are equal gives a higher e.

Alternatively, if we set some variables to be different? For example, maybe set three variables equal and one different? But as I thought before, that would likely result in a higher sum of squares, which would require e to be smaller.

Alternatively, maybe set some variables to be negative? Wait, but if we allow variables to be negative, maybe we can have a larger e?

Wait, but the problem says that a, b, c, d, e are real numbers. So they can be negative.

But if we allow some variables to be negative, maybe we can have a larger e? Let me check.

Suppose that instead of making a, b, c, d equal, we make some of them negative. For example, suppose that three variables are negative and one is positive.

But how would that affect the sum of squares?

Wait, if we have some negative numbers, their squares are still positive. So maybe making some variables negative allows us to have a larger e?

Wait, but if we have negative numbers, then the sum a + b + c + d would be smaller, which would mean that e would have to be larger to compensate. Wait, but the sum a + b + c + d = 8 - e. If we make a + b + c + d smaller, then e would be larger. However, the sum of squares would be larger because of the negative numbers? Or maybe not?

Wait, for example, suppose that instead of having a, b, c, d all equal to 6/5, which is positive, we have some negative numbers. Let me try an example.

Suppose that three variables are equal to some value x, and one variable is y. Let me try to set up equations.

But this might get complicated. Let me think.

Alternatively, think of the problem as follows: given that the sum of a, b, c, d is S = 8 - e, and the sum of squares is Q = 16 - e².

We need to find the maximum e such that there exist real numbers a, b, c, d with sum S and sum of squares Q.

The condition for the existence of such numbers is that the minimal possible sum of squares for a given sum S is S² / 4. Therefore, we must have Q ≥ S² / 4.

Which gives 16 - e² ≥ (8 - e)² / 4. Which is the same inequality as before.

Therefore, regardless of the values of a, b, c, d, this inequality must hold. Therefore, the maximum e is 16/5.

Therefore, even if we allow negative numbers, the inequality still holds, and the maximum e is 16/5.

Therefore, the answer is 16/5.

But let me check with another approach. Let me consider variables a, b, c, d. Let me think of them as vectors.

The problem is equivalent to: given that the vector (a, b, c, d) has a fixed sum and fixed norm, find the maximum value of e.

But since e is related to the sum and the norm, maybe we can think of it as follows.

Let me denote vector v = (a, b, c, d). Then, the sum of v is 8 - e, and the norm squared is 16 - e².

The Cauchy-Schwarz inequality tells us that the norm squared of v is at least (sum of v)^2 / 4. Therefore, 16 - e² ≥ (8 - e)^2 / 4. Which again gives the same inequality.

Therefore, regardless of the approach, we get that e ≤ 16/5.

Therefore, the maximum value of e is 16/5.

But just to be thorough, let me check if there's a case where e is larger than 16/5. Suppose e = 4. Then, the sum of a + b + c + d = 8 - 4 = 4. The sum of squares of a, b, c, d would be 16 - 16 = 0. Therefore, all of a, b, c, d must be zero. But then their sum is 0, not 4. Contradiction.

If e = 3.5, which is 7/2. Then, sum of a + b + c + d = 8 - 3.5 = 4.5. Sum of squares is 16 - (3.5)^2 = 16 - 12.25 = 3.75.

Now, check if there exist real numbers a, b, c, d with sum 4.5 and sum of squares 3.75.

The minimal sum of squares for sum 4.5 is (4.5)^2 / 4 = 20.25 / 4 = 5.0625. But 3.75 < 5.0625. Therefore, impossible. Therefore, e cannot be 3.5.

So, e = 16/5 = 3.2. Let me check that.

Sum of a + b + c + d = 8 - 16/5 = (40 - 16)/5 = 24/5 = 4.8.

Sum of squares is 16 - (16/5)^2 = 16 - 256/25 = (400 - 256)/25 = 144/25 = 5.76.

The minimal sum of squares for sum 4.8 is (4.8)^2 / 4 = 23.04 / 4 = 5.76. Which matches. Therefore, equality holds, so it is possible.

Therefore, e = 16/5 is indeed attainable.

Therefore, the maximum value of e is 16/5.

So, after all these checks and approaches, I think the answer is 16/5.

**Final Answer**
The maximum value of $ e $ is \boxed{\dfrac{16}{5}}.

To determine the **maximum value of $ e $** given the constraints:

$$
a + b + c + d + e = 8 \quad \text{(1)}
$$
$$
a^2 + b^2 + c^2 + d^2 + e^2 = 16 \quad \text{(2)}
$$

we aim to **maximize $ e $** under these conditions.

---

### Step 1: Express the Remaining Variables

From equation (1), we can isolate the sum of the other variables:

$$
a + b + c + d = 8 - e \quad \text{(3)}
$$

From equation (2), we isolate the sum of their squares:

$$
a^2 + b^2 + c^2 + d^2 = 16 - e^2 \quad \text{(4)}
$$

Now, we are left with the problem of finding the **maximum possible value of $ e $** such that there exist real numbers $ a, b, c, d $ satisfying both (3) and (4).

---

### Step 2: Apply the Cauchy-Schwarz Inequality

We apply the **Cauchy-Schwarz inequality** to the variables $ a, b, c, d $:

$$
(a^2 + b^2 + c^2 + d^2)(1^2 + 1^2 + 1^2 + 1^2) \geq (a + b + c + d)^2
$$

Substituting from (3) and (4):

$$
(16 - e^2)(4) \geq (8 - e)^2
$$

Simplify both sides:

- Left-hand side: $ 4(16 - e^2) = 64 - 4e^2 $
- Right-hand side: $ (8 - e)^2 = 64 - 16e + e^2 $

So the inequality becomes:

$$
64 - 4e^2 \geq 64 - 16e + e^2
$$

Subtract 64 from both sides:

$$
-4e^2 \geq -16e + e^2
$$

Bring all terms to one side:

$$
-5e^2 + 16e \geq 0
$$

Factor:

$$
e(-5e + 16) \geq 0
$$

This inequality holds when:

$$
0 \leq e \leq \frac{16}{5}
$$

---

### Step 3: Check for Equality Case

The maximum value of $ e $ is $ \frac{16}{5} $, and we must verify that this value is **achievable**.

Let’s suppose $ e = \frac{16}{5} $. Then:

- From (3): $ a + b + c + d = 8 - \frac{16}{5} = \frac{24}{5} $
- From (4): $ a^2 + b^2 + c^2 + d^2 = 16 - \left(\frac{16}{5}\right)^2 = 16 - \frac{256}{25} = \frac{144}{25} $

Now, to minimize the sum of squares of $ a, b, c, d $ for a fixed sum, the variables should be **equal**. So set:

$$
a = b = c = d = \frac{24}{5} \div 4 = \frac{6}{5}
$$

Then:

$$
a^2 + b^2 + c^2 + d^2 = 4 \cdot \left(\frac{6}{5}\right)^2 = 4 \cdot \frac{36}{25} = \frac{144}{25}
$$

Which matches the required sum of squares. Therefore, the configuration is valid.

---

### Final Conclusion

The maximum value of $ e $ that satisfies both constraints is:

$$
\boxed{\dfrac{16}{5}}
$$

[Assistant]
Explanation:
Okay, so I need to find the maximum value of e given these two equations:

a + b + c + d + e = 8,

and

a² + b² + c² + d² + e² = 16.

Hmm, let me think. Since we need to maximize e, maybe I should try to minimize the other variables? Because if I can make a, b, c, d as small as possible, then e can be as large as possible. But how do I do that?

Wait, there are constraints here. The sum of the variables is fixed at 8, and the sum of their squares is fixed at 16. So if I want to maximize e, I need to adjust the other variables so that they take up as little "space" as possible in terms of their squares.

I remember something about the Cauchy-Schwarz inequality or maybe using the method of Lagrange multipliers. But since this is a problem likely intended for algebra, maybe there's a simpler way. Let me think.

Let me consider that if I fix e, then the sum of a + b + c + d = 8 - e, and the sum of their squares is 16 - e². So, maybe I can use the Cauchy-Schwarz inequality on the variables a, b, c, d.

The Cauchy-Schwarz inequality says that for any real numbers, (a₁² + a₂² + ... + aₙ²)(1² + 1² + ... + 1²) ≥ (a₁ + a₂ + ... + aₙ)².

So applying that to a, b, c, d:

(a² + b² + c² + d²)(1 + 1 + 1 + 1) ≥ (a + b + c + d)².

Which would be:

(16 - e²) * 4 ≥ (8 - e)².

Let me write that down:

4*(16 - e²) ≥ (8 - e)².

Let me compute both sides:

Left side: 4*(16 - e²) = 64 - 4e².

Right side: (8 - e)² = 64 - 16e + e².

So the inequality becomes:

64 - 4e² ≥ 64 - 16e + e².

Subtract 64 from both sides:

-4e² ≥ -16e + e².

Bring all terms to the left side:

-4e² + 16e - e² ≥ 0.

Combine like terms:

-5e² + 16e ≥ 0.

Multiply both sides by -1 (which reverses the inequality):

5e² - 16e ≤ 0.

Factor:

e*(5e - 16) ≤ 0.

So this inequality holds when e is between 0 and 16/5, inclusive. Wait, but we are trying to find the maximum value of e. So according to this, e can be at most 16/5? But let me check if this is correct.

Wait, but hold on. The original problem is to maximize e, so if the inequality gives that e ≤ 16/5, then 16/5 is the maximum? But let me verify if equality can be achieved. Because Cauchy-Schwarz gives equality when all the variables are equal. So in this case, for the variables a, b, c, d, equality occurs when a = b = c = d.

So if we set a = b = c = d, then let's check if that gives us e = 16/5.

Let me suppose that a = b = c = d = x. Then:

Sum: 4x + e = 8 => 4x = 8 - e => x = (8 - e)/4.

Sum of squares: 4x² + e² = 16.

Substituting x:

4*((8 - e)/4)² + e² = 16.

Compute that:

4*( (64 - 16e + e²)/16 ) + e² = 16.

Simplify:

(64 - 16e + e²)/4 + e² = 16.

Multiply through:

(64 - 16e + e²)/4 + e² = 16.

Multiply numerator:

16 - 4e + (e²)/4 + e² = 16.

Combine like terms:

16 - 4e + (5e²)/4 = 16.

Subtract 16 from both sides:

-4e + (5e²)/4 = 0.

Multiply both sides by 4 to eliminate denominator:

-16e + 5e² = 0.

Factor:

e*(-16 + 5e) = 0.

So solutions are e = 0 or e = 16/5.

So when e = 16/5, then x = (8 - 16/5)/4 = (40/5 - 16/5)/4 = (24/5)/4 = 6/5.

So then, the variables a, b, c, d are each 6/5, and e is 16/5. Let me check if the sum of squares is indeed 16:

Sum of squares: 4*(6/5)^2 + (16/5)^2 = 4*(36/25) + 256/25 = 144/25 + 256/25 = 400/25 = 16. Perfect, that works.

So that seems to satisfy both equations. Therefore, e can be as large as 16/5.

But wait, let me think again. The Cauchy-Schwarz gave us that e ≤ 16/5. But is there a possibility that even if the inequality is not tight, maybe there's another configuration where e is larger?

Wait, but the logic is that for the sum of squares of a, b, c, d to be minimized given their sum, they should be equal. Because variance is minimized when all variables are equal. Therefore, if we want to minimize the sum of squares of a, b, c, d given their sum, we set them equal. Therefore, if we set them equal, then we can get the minimal sum of squares, which would allow e to be as large as possible.

Therefore, if we have the minimal sum of squares for a, b, c, d, then the remaining value for e is maximized.

So that seems correct. Therefore, the maximum value of e is 16/5, which is 3.2.

But let me check another way. Let me try to use Lagrange multipliers. Maybe that will confirm the result.

Let me consider variables a, b, c, d, e. We need to maximize e subject to:

Constraint 1: a + b + c + d + e = 8,

Constraint 2: a² + b² + c² + d² + e² = 16.

Let me set up the Lagrangian:

L = e - λ(a + b + c + d + e - 8) - μ(a² + b² + c² + d² + e² - 16).

Wait, actually, since we are maximizing e, the Lagrangian should be e minus the constraints multiplied by Lagrange multipliers. But maybe I need to set up the problem as maximizing e with the constraints.

Alternatively, since we can consider the problem as maximizing e with the two constraints.

But maybe it's easier to use substitution. Let me consider that from the first equation, a + b + c + d = 8 - e. Let me denote S = 8 - e, and Q = 16 - e².

So, we have four variables a, b, c, d with sum S and sum of squares Q. To maximize e, we need to minimize Q given S? Wait, no. Wait, Q is fixed as 16 - e². Wait, but if we want to maximize e, then Q = 16 - e² must be as small as possible? Wait, no. Wait, if e is larger, then Q = 16 - e² is smaller. But Q is the sum of squares of a, b, c, d. So if Q is smaller, that would mean that the sum of squares of a, b, c, d is smaller. But given that their sum is fixed at S = 8 - e, then the minimal possible sum of squares occurs when all variables are equal. Therefore, if we have that minimal sum of squares, then Q must be equal to that minimal value. Therefore, if we set Q equal to the minimal sum of squares, then we can have equality.

Therefore, the minimal sum of squares for a, b, c, d given their sum S is (S²)/4, by Cauchy-Schwarz. Wait, because if you have four variables with sum S, then the minimal sum of squares is achieved when all variables are equal, so each is S/4, so sum of squares is 4*(S/4)^2 = S²/4.

Therefore, we have that Q ≥ S² / 4.

But Q is 16 - e², and S is 8 - e. Therefore:

16 - e² ≥ (8 - e)² / 4.

Which is exactly the same inequality as before. So solving this gives us the same result.

So, 16 - e² ≥ (64 - 16e + e²)/4.

Multiply both sides by 4:

64 - 4e² ≥ 64 - 16e + e².

Which simplifies to -5e² + 16e ≥ 0, leading to e(16 - 5e) ≥ 0. Wait, earlier I had 5e² - 16e ≤ 0, which is equivalent.

So, the solutions are e between 0 and 16/5. Therefore, maximum e is 16/5.

Therefore, the maximum value of e is 16/5.

But let me check if there's another approach. For example, using the concept of variance.

The variance of the variables a, b, c, d is minimized when they are equal, which again leads to the same conclusion.

Alternatively, think of the variables a, b, c, d as being as close to each other as possible to minimize their sum of squares.

Alternatively, think of the problem as an optimization problem where we want to maximize e, so we can consider that the other variables are all equal.

So, if I assume that a = b = c = d, then as before, we can solve for e. Which gives us e = 16/5.

Alternatively, suppose that three of them are equal and one is different? Would that give a higher e? Probably not, because making them unequal would increase the sum of squares.

For example, suppose that three variables are equal to x and one is y. Then:

Sum: 3x + y + e = 8,

Sum of squares: 3x² + y² + e² = 16.

But then, if we try to minimize the sum of squares 3x² + y² given that 3x + y = 8 - e.

But this would be more complicated, but perhaps we can check if the minimal sum of squares is achieved when all variables are equal.

In general, for fixed sum, the sum of squares is minimized when all variables are equal. Therefore, even if we have different numbers of variables, the minimal sum of squares occurs when they are all equal. Therefore, in this case, if we have four variables, the minimal sum of squares is achieved when they are all equal. Therefore, the previous approach is valid.

Therefore, the maximum value of e is 16/5.

But let me check with another example. Suppose that three variables are zero, and one variable is something. Let me try that.

Suppose a = b = c = 0, then d + e = 8, and d² + e² = 16.

So, d = 8 - e. Then:

(8 - e)² + e² = 16.

Expanding:

64 - 16e + e² + e² = 16.

So 2e² - 16e + 64 - 16 = 0.

Which is 2e² - 16e + 48 = 0.

Divide by 2: e² - 8e + 24 = 0.

Discriminant: 64 - 96 = -32 < 0. No real solutions.

So that's impossible. Therefore, if three variables are zero, there's no real solution.

What if two variables are zero? Let me try that. Let a = b = 0, then c + d + e = 8, and c² + d² + e² = 16.

Let me try to maximize e here. So set c + d = 8 - e, and c² + d² = 16 - e².

Again, for fixed sum, the minimal sum of squares occurs when c = d. So set c = d = (8 - e)/2. Then:

Sum of squares: 2*((8 - e)/2)^2 = 2*(64 - 16e + e²)/4 = (64 - 16e + e²)/2.

Set equal to 16 - e²:

(64 - 16e + e²)/2 = 16 - e².

Multiply both sides by 2:

64 - 16e + e² = 32 - 2e².

Bring all terms to left:

64 - 16e + e² - 32 + 2e² = 0 => 3e² - 16e + 32 = 0.

Discriminant: 256 - 384 = -128 < 0. Again, no real solution.

Hmm, so even if two variables are zero, there's no solution.

What if one variable is zero? Let me try. Let a = 0, then b + c + d + e = 8, and b² + c² + d² + e² = 16.

Again, trying to maximize e. So set b + c + d = 8 - e, and b² + c² + d² = 16 - e².

Again, minimal sum of squares for b, c, d occurs when they are equal. So set b = c = d = (8 - e)/3. Then sum of squares is 3*((8 - e)/3)^2 = 3*(64 - 16e + e²)/9 = (64 - 16e + e²)/3.

Set equal to 16 - e²:

(64 - 16e + e²)/3 = 16 - e².

Multiply both sides by 3:

64 - 16e + e² = 48 - 3e².

Bring all terms to left:

64 - 16e + e² - 48 + 3e² = 0 => 4e² - 16e + 16 = 0.

Divide by 4: e² - 4e + 4 = 0 => (e - 2)^2 = 0 => e = 2.

So in this case, e = 2. Which is less than 16/5 = 3.2. Therefore, this is worse.

So even if we set one variable to zero, the maximum e is 2, which is less than 16/5.

Therefore, the case where all four variables are equal gives a higher e.

Alternatively, if we set some variables to be different? For example, maybe set three variables equal and one different? But as I thought before, that would likely result in a higher sum of squares, which would require e to be smaller.

Alternatively, maybe set some variables to be negative? Wait, but if we allow variables to be negative, maybe we can have a larger e?

Wait, but the problem says that a, b, c, d, e are real numbers. So they can be negative.

But if we allow some variables to be negative, maybe we can have a larger e? Let me check.

Suppose that instead of making a, b, c, d equal, we make some of them negative. For example, suppose that three variables are negative and one is positive.

But how would that affect the sum of squares?

Wait, if we have some negative numbers, their squares are still positive. So maybe making some variables negative allows us to have a larger e?

Wait, but if we have negative numbers, then the sum a + b + c + d would be smaller, which would mean that e would have to be larger to compensate. Wait, but the sum a + b + c + d = 8 - e. If we make a + b + c + d smaller, then e would be larger. However, the sum of squares would be larger because of the negative numbers? Or maybe not?

Wait, for example, suppose that instead of having a, b, c, d all equal to 6/5, which is positive, we have some negative numbers. Let me try an example.

Suppose that three variables are equal to some value x, and one variable is y. Let me try to set up equations.

But this might get complicated. Let me think.

Alternatively, think of the problem as follows: given that the sum of a, b, c, d is S = 8 - e, and the sum of squares is Q = 16 - e².

We need to find the maximum e such that there exist real numbers a, b, c, d with sum S and sum of squares Q.

The condition for the existence of such numbers is that the minimal possible sum of squares for a given sum S is S² / 4. Therefore, we must have Q ≥ S² / 4.

Which gives 16 - e² ≥ (8 - e)² / 4. Which is the same inequality as before.

Therefore, regardless of the values of a, b, c, d, this inequality must hold. Therefore, the maximum e is 16/5.

Therefore, even if we allow negative numbers, the inequality still holds, and the maximum e is 16/5.

Therefore, the answer is 16/5.

But let me check with another approach. Let me consider variables a, b, c, d. Let me think of them as vectors.

The problem is equivalent to: given that the vector (a, b, c, d) has a fixed sum and fixed norm, find the maximum value of e.

But since e is related to the sum and the norm, maybe we can think of it as follows.

Let me denote vector v = (a, b, c, d). Then, the sum of v is 8 - e, and the norm squared is 16 - e².

The Cauchy-Schwarz inequality tells us that the norm squared of v is at least (sum of v)^2 / 4. Therefore, 16 - e² ≥ (8 - e)^2 / 4. Which again gives the same inequality.

Therefore, regardless of the approach, we get that e ≤ 16/5.

Therefore, the maximum value of e is 16/5.

But just to be thorough, let me check if there's a case where e is larger than 16/5. Suppose e = 4. Then, the sum of a + b + c + d = 8 - 4 = 4. The sum of squares of a, b, c, d would be 16 - 16 = 0. Therefore, all of a, b, c, d must be zero. But then their sum is 0, not 4. Contradiction.

If e = 3.5, which is 7/2. Then, sum of a + b + c + d = 8 - 3.5 = 4.5. Sum of squares is 16 - (3.5)^2 = 16 - 12.25 = 3.75.

Now, check if there exist real numbers a, b, c, d with sum 4.5 and sum of squares 3.75.

The minimal sum of squares for sum 4.5 is (4.5)^2 / 4 = 20.25 / 4 = 5.0625. But 3.75 < 5.0625. Therefore, impossible. Therefore, e cannot be 3.5.

So, e = 16/5 = 3.2. Let me check that.

Sum of a + b + c + d = 8 - 16/5 = (40 - 16)/5 = 24/5 = 4.8.

Sum of squares is 16 - (16/5)^2 = 16 - 256/25 = (400 - 256)/25 = 144/25 = 5.76.

The minimal sum of squares for sum 4.8 is (4.8)^2 / 4 = 23.04 / 4 = 5.76. Which matches. Therefore, equality holds, so it is possible.

Therefore, e = 16/5 is indeed attainable.

Therefore, the maximum value of e is 16/5.

So, after all these checks and approaches, I think the answer is 16/5.

**Final Answer**
The maximum value of $ e $ is \boxed{\dfrac{16}{5}}.

To determine the **maximum value of $ e $** given the constraints:

$$
a + b + c + d + e = 8 \quad \text{(1)}
$$
$$
a^2 + b^2 + c^2 + d^2 + e^2 = 16 \quad \text{(2)}
$$

we aim to **maximize $ e $** under these conditions.

---

### Step 1: Express the Remaining Variables

From equation (1), we can isolate the sum of the other variables:

$$
a + b + c + d = 8 - e \quad \text{(3)}
$$

From equation (2), we isolate the sum of their squares:

$$
a^2 + b^2 + c^2 + d^2 = 16 - e^2 \quad \text{(4)}
$$

Now, we are left with the problem of finding the **maximum possible value of $ e $** such that there exist real numbers $ a, b, c, d $ satisfying both (3) and (4).

---

### Step 2: Apply the Cauchy-Schwarz Inequality

We apply the **Cauchy-Schwarz inequality** to the variables $ a, b, c, d $:

$$
(a^2 + b^2 + c^2 + d^2)(1^2 + 1^2 + 1^2 + 1^2) \geq (a + b + c + d)^2
$$

Substituting from (3) and (4):

$$
(16 - e^2)(4) \geq (8 - e)^2
$$

Simplify both sides:

- Left-hand side: $ 4(16 - e^2) = 64 - 4e^2 $
- Right-hand side: $ (8 - e)^2 = 64 - 16e + e^2 $

So the inequality becomes:

$$
64 - 4e^2 \geq 64 - 16e + e^2
$$

Subtract 64 from both sides:

$$
-4e^2 \geq -16e + e^2
$$

Bring all terms to one side:

$$
-5e^2 + 16e \geq 0
$$

Factor:

$$
e(-5e + 16) \geq 0
$$

This inequality holds when:

$$
0 \leq e \leq \frac{16}{5}
$$

---

### Step 3: Check for Equality Case

The maximum value of $ e $ is $ \frac{16}{5} $, and we must verify that this value is **achievable**.

Let’s suppose $ e = \frac{16}{5} $. Then:

- From (3): $ a + b + c + d = 8 - \frac{16}{5} = \frac{24}{5} $
- From (4): $ a^2 + b^2 + c^2 + d^2 = 16 - \left(\frac{16}{5}\right)^2 = 16 - \frac{256}{25} = \frac{144}{25} $

Now, to minimize the sum of squares of $ a, b, c, d $ for a fixed sum, the variables should be **equal**. So set:

$$
a = b = c = d = \frac{24}{5} \div 4 = \frac{6}{5}
$$

Then:

$$
a^2 + b^2 + c^2 + d^2 = 4 \cdot \left(\frac{6}{5}\right)^2 = 4 \cdot \frac{36}{25} = \frac{144}{25}
$$

Which matches the required sum of squares. Therefore, the configuration is valid.

---

### Final Conclusion

The maximum value of $ e $ that satisfies both constraints is:

$$
\boxed{\dfrac{16}{5}}
$$

Solution:
By Cauchy-Schwarz,
\[(a^2 + b^2 + c^2 + d^2)(1 + 1 + 1 + 1) \ge (a + b + c + d)^2.\]Thus, $(16 - e^2)(4) \ge (8 - e)^2.$  This simplifies to $16e - 5e^2 \ge 0,$ or $e(16 - 5e) \ge 0.$  Therefore, $e \le \frac{16}{5}.$

Equality occurs when $a = b = c = d = \frac{6}{5}$ and $e = \frac{16}{5},$ so the maximum value of $e$ is $\boxed{\frac{16}{5}}.$


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 assign variables to their current ages. Let me denote:

Let A = Amy's current age

B = Ben's current age

C = Chris's current age

So, the first equation is A + B + C = 18.

Now, the second part: Four years ago, Chris was the same age as Amy is now. 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's the second equation: C - 4 = A. Which can be rewritten as A = C - 4.

Third part: "In four years, Ben's age will be 3/5 of Amy's age at that time." Let me think. In four years, Ben's age will be B + 4. At that same time, Amy's age will be A + 4. And according to the problem, Ben's age then will be 3/5 of Amy's age then. So:

B + 4 = (3/5)(A + 4)

So that's the third equation.

So, now we have three equations:

1) A + B + C = 18

2) A = C - 4

3) B + 4 = (3/5)(A + 4)

So, we can solve these equations step by step.

First, since we have equation 2, which relates A and C, maybe we can substitute A in terms of C into the other equations.

So, let me substitute A = C - 4 into equation 1 and equation 3.

Starting with equation 1:

A + B + C = 18

Substitute A with C - 4:

(C - 4) + B + C = 18

Combine like terms:

2C + B - 4 = 18

Then, 2C + B = 18 + 4 = 22

So, equation 1 becomes: 2C + B = 22. Let me note that as equation 1a.

Now, equation 3:

B + 4 = (3/5)(A + 4)

Again, substitute A with C - 4:

B + 4 = (3/5)( (C - 4) + 4 )

Simplify inside the parentheses:

(C - 4) + 4 = C. So, that simplifies equation 3 to:

B + 4 = (3/5)(C)

Therefore, equation 3 becomes: B + 4 = (3/5)C. Let me write that as equation 3a.

So now, we have equation 1a: 2C + B = 22

and equation 3a: B + 4 = (3/5)C

So, we can solve these two equations for B and C.

Let me solve equation 3a for B:

From equation 3a: B = (3/5)C - 4

Now, substitute this expression for B into equation 1a:

2C + [ (3/5)C - 4 ] = 22

Let me compute that:

First, combine the terms with C:

2C + (3/5)C - 4 = 22

Convert 2C to fifths to combine with 3/5 C:

2C is 10/5 C, so:

10/5 C + 3/5 C = 13/5 C

Therefore:

13/5 C - 4 = 22

Add 4 to both sides:

13/5 C = 22 + 4 = 26

So, 13/5 C = 26

Multiply both sides by 5:

13C = 130

Divide both sides by 13:

C = 130 / 13 = 10

So, Chris is currently 10 years old.

Wait, let me check if that makes sense. Let me verify the answer.

If Chris is 10, then from equation 2: A = C - 4 = 10 - 4 = 6. So Amy is 6.

Then, from equation 3a: B = (3/5)C - 4 = (3/5)*10 - 4 = 6 - 4 = 2. So Ben is 2.

Now check the first equation: A + B + C = 6 + 2 + 10 = 18. Which matches the total. Good.

Now check the second condition: Four years ago, Chris was the same age as Amy is now. Four years ago, Chris was 10 - 4 = 6, which is Amy's current age. Correct.

Third condition: 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 conditions are satisfied. Therefore, Chris is 10 years old now.

Wait, but let me think again. The problem says "the average of Amy's, Ben's, and Chris's ages is 6." So average is 6, so total is 18. Which we have. So 6 + 2 + 10 is 18. Correct.

But let me check if there's another way to approach this, just to make sure.

Alternatively, maybe I can use substitution differently. But I think the way I did it is correct.

Alternatively, let me think if there's any possible mistake in the equations.

First equation: A + B + C = 18. Correct.

Second equation: Four years ago, Chris was same age as Amy is now. So Chris's age four years ago: C - 4. Amy's current age: A. Therefore, C - 4 = A. Correct.

Third equation: In four years, Ben's age will be 3/5 of Amy's age at that time. So Ben's age in four years: B + 4. Amy's age in four years: A + 4. Therefore, B + 4 = (3/5)(A + 4). Correct.

So equations are correct. Then solving them gives C = 10. So answer is 10. Seems correct.

But let me check if maybe the problem says "In four years, Ben's age will be 3/5 of Amy's age at that time." So, maybe I need to check if the wording is correct. For example, is it Ben's age in four years is 3/5 of Amy's age in four years? Yes, that's how I interpreted it. So B + 4 = (3/5)(A + 4). Which is correct.

Alternatively, if it was "Ben's age will be 3/5 of Amy's current age," but no, the problem says "at that time," meaning when Ben's age is being considered, which is in four years. So Amy's age at that time is A + 4. So the equation is correct.

Therefore, I think the answer is 10. So Chris is 10 years old now.

But let me check again with the numbers:

Amy is 6, Ben is 2, Chris is 10.

Average age: (6 + 2 + 10)/3 = 18/3 = 6. Correct.

Four years ago, Chris was 10 - 4 = 6, which is Amy's current age. Correct.

In four years, Ben will be 2 + 4 = 6, Amy will be 6 + 4 = 10. 6 is 3/5 of 10. Correct.

All conditions are satisfied. Therefore, the answer is 10. So Chris is 10 years old now.

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

To determine Chris's current age, we begin by translating the problem into a system of equations based on the given information.

---

### Step 1: Define Variables

Let:
- $ A $ = Amy's current age
- $ B $ = Ben's current age
- $ C $ = Chris's current age

---

### Step 2: Translate the Problem into Equations

1. **Average Age Condition**
   The average of their ages is 6, so the total of their ages is:
   $$
   A + B + C = 18 \tag{1}
   $$

2. **Chris's Age Four Years Ago**
   Four years ago, Chris was the same age as Amy is now:
   $$
   C - 4 = A \Rightarrow A = C - 4 \tag{2}
   $$

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

---

### Step 3: Solve the System of Equations

**Substitute Equation (2) into Equation (1):**

$$
(C - 4) + B + C = 18 \Rightarrow 2C + B - 4 = 18 \Rightarrow 2C + B = 22 \tag{4}
$$

**Substitute Equation (2) into Equation (3):**

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

**Substitute Equation (5) into Equation (4):**

$$
2C + \left( \frac{3}{5}C - 4 \right) = 22 \Rightarrow \frac{13}{5}C - 4 = 22
$$

$$
\frac{13}{5}C = 26 \Rightarrow C = \frac{26 \times 5}{13} = 10
$$

---

### Step 4: Find Other Ages

- From Equation (2): $ A = C - 4 = 10 - 4 = 6 $
- From Equation (5): $ B = \frac{3}{5}C - 4 = \frac{3}{5}(10) - 4 = 6 - 4 = 2 $

---

### Step 5: Verify the Solution

- Total age: $ A + B + C = 6 + 2 + 10 = 18 $ ✅
- Four years ago, Chris was $ 10 - 4 = 6 $, which is Amy's current age ✅
- In four years, Ben will be $ 2 + 4 = 6 $, and Amy will be $ 6 + 4 = 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 assign variables to their current ages. Let me denote:

Let A = Amy's current age

B = Ben's current age

C = Chris's current age

So, the first equation is A + B + C = 18.

Now, the second part: Four years ago, Chris was the same age as Amy is now. 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's the second equation: C - 4 = A. Which can be rewritten as A = C - 4.

Third part: "In four years, Ben's age will be 3/5 of Amy's age at that time." Let me think. In four years, Ben's age will be B + 4. At that same time, Amy's age will be A + 4. And according to the problem, Ben's age then will be 3/5 of Amy's age then. So:

B + 4 = (3/5)(A + 4)

So that's the third equation.

So, now we have three equations:

1) A + B + C = 18

2) A = C - 4

3) B + 4 = (3/5)(A + 4)

So, we can solve these equations step by step.

First, since we have equation 2, which relates A and C, maybe we can substitute A in terms of C into the other equations.

So, let me substitute A = C - 4 into equation 1 and equation 3.

Starting with equation 1:

A + B + C = 18

Substitute A with C - 4:

(C - 4) + B + C = 18

Combine like terms:

2C + B - 4 = 18

Then, 2C + B = 18 + 4 = 22

So, equation 1 becomes: 2C + B = 22. Let me note that as equation 1a.

Now, equation 3:

B + 4 = (3/5)(A + 4)

Again, substitute A with C - 4:

B + 4 = (3/5)( (C - 4) + 4 )

Simplify inside the parentheses:

(C - 4) + 4 = C. So, that simplifies equation 3 to:

B + 4 = (3/5)(C)

Therefore, equation 3 becomes: B + 4 = (3/5)C. Let me write that as equation 3a.

So now, we have equation 1a: 2C + B = 22

and equation 3a: B + 4 = (3/5)C

So, we can solve these two equations for B and C.

Let me solve equation 3a for B:

From equation 3a: B = (3/5)C - 4

Now, substitute this expression for B into equation 1a:

2C + [ (3/5)C - 4 ] = 22

Let me compute that:

First, combine the terms with C:

2C + (3/5)C - 4 = 22

Convert 2C to fifths to combine with 3/5 C:

2C is 10/5 C, so:

10/5 C + 3/5 C = 13/5 C

Therefore:

13/5 C - 4 = 22

Add 4 to both sides:

13/5 C = 22 + 4 = 26

So, 13/5 C = 26

Multiply both sides by 5:

13C = 130

Divide both sides by 13:

C = 130 / 13 = 10

So, Chris is currently 10 years old.

Wait, let me check if that makes sense. Let me verify the answer.

If Chris is 10, then from equation 2: A = C - 4 = 10 - 4 = 6. So Amy is 6.

Then, from equation 3a: B = (3/5)C - 4 = (3/5)*10 - 4 = 6 - 4 = 2. So Ben is 2.

Now check the first equation: A + B + C = 6 + 2 + 10 = 18. Which matches the total. Good.

Now check the second condition: Four years ago, Chris was the same age as Amy is now. Four years ago, Chris was 10 - 4 = 6, which is Amy's current age. Correct.

Third condition: 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 conditions are satisfied. Therefore, Chris is 10 years old now.

Wait, but let me think again. The problem says "the average of Amy's, Ben's, and Chris's ages is 6." So average is 6, so total is 18. Which we have. So 6 + 2 + 10 is 18. Correct.

But let me check if there's another way to approach this, just to make sure.

Alternatively, maybe I can use substitution differently. But I think the way I did it is correct.

Alternatively, let me think if there's any possible mistake in the equations.

First equation: A + B + C = 18. Correct.

Second equation: Four years ago, Chris was same age as Amy is now. So Chris's age four years ago: C - 4. Amy's current age: A. Therefore, C - 4 = A. Correct.

Third equation: In four years, Ben's age will be 3/5 of Amy's age at that time. So Ben's age in four years: B + 4. Amy's age in four years: A + 4. Therefore, B + 4 = (3/5)(A + 4). Correct.

So equations are correct. Then solving them gives C = 10. So answer is 10. Seems correct.

But let me check if maybe the problem says "In four years, Ben's age will be 3/5 of Amy's age at that time." So, maybe I need to check if the wording is correct. For example, is it Ben's age in four years is 3/5 of Amy's age in four years? Yes, that's how I interpreted it. So B + 4 = (3/5)(A + 4). Which is correct.

Alternatively, if it was "Ben's age will be 3/5 of Amy's current age," but no, the problem says "at that time," meaning when Ben's age is being considered, which is in four years. So Amy's age at that time is A + 4. So the equation is correct.

Therefore, I think the answer is 10. So Chris is 10 years old now.

But let me check again with the numbers:

Amy is 6, Ben is 2, Chris is 10.

Average age: (6 + 2 + 10)/3 = 18/3 = 6. Correct.

Four years ago, Chris was 10 - 4 = 6, which is Amy's current age. Correct.

In four years, Ben will be 2 + 4 = 6, Amy will be 6 + 4 = 10. 6 is 3/5 of 10. Correct.

All conditions are satisfied. Therefore, the answer is 10. So Chris is 10 years old now.

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

To determine Chris's current age, we begin by translating the problem into a system of equations based on the given information.

---

### Step 1: Define Variables

Let:
- $ A $ = Amy's current age
- $ B $ = Ben's current age
- $ C $ = Chris's current age

---

### Step 2: Translate the Problem into Equations

1. **Average Age Condition**
   The average of their ages is 6, so the total of their ages is:
   $$
   A + B + C = 18 \tag{1}
   $$

2. **Chris's Age Four Years Ago**
   Four years ago, Chris was the same age as Amy is now:
   $$
   C - 4 = A \Rightarrow A = C - 4 \tag{2}
   $$

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

---

### Step 3: Solve the System of Equations

**Substitute Equation (2) into Equation (1):**

$$
(C - 4) + B + C = 18 \Rightarrow 2C + B - 4 = 18 \Rightarrow 2C + B = 22 \tag{4}
$$

**Substitute Equation (2) into Equation (3):**

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

**Substitute Equation (5) into Equation (4):**

$$
2C + \left( \frac{3}{5}C - 4 \right) = 22 \Rightarrow \frac{13}{5}C - 4 = 22
$$

$$
\frac{13}{5}C = 26 \Rightarrow C = \frac{26 \times 5}{13} = 10
$$

---

### Step 4: Find Other Ages

- From Equation (2): $ A = C - 4 = 10 - 4 = 6 $
- From Equation (5): $ B = \frac{3}{5}C - 4 = \frac{3}{5}(10) - 4 = 6 - 4 = 2 $

---

### Step 5: Verify the Solution

- Total age: $ A + B + C = 6 + 2 + 10 = 18 $ ✅
- Four years ago, Chris was $ 10 - 4 = 6 $, which is Amy's current age ✅
- In four years, Ben will be $ 2 + 4 = 6 $, and Amy will be $ 6 + 4 = 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 $\frac 79 - \frac 49 + \frac 29 = \boxed{\frac 59}$.