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

1. a + b + c + d + e = 8
2. a² + b² + c² + d² + e² = 16

Hmm, let me think. Since we need to maximize e, maybe I should express the other variables in terms of e and then apply some inequality? Like Cauchy-Schwarz or something?

Let me start by letting the sum of a, b, c, d be something. Let me denote S = a + b + c + d. Then from the first equation, S + e = 8, so S = 8 - e.

Similarly, the sum of squares: a² + b² + c² + d² + e² = 16. Let me denote Q = a² + b² + c² + d². Then Q + e² = 16, so Q = 16 - e².

Now, I need to relate S and Q. For variables a, b, c, d, there's a relationship between their sum and sum of squares. Maybe using the Cauchy-Schwarz inequality? Or maybe the concept that for a fixed sum, the sum of squares is minimized when all variables are equal?

Wait, since I want to maximize e, perhaps I need to minimize Q? Because Q = 16 - e², so if I minimize Q, that would allow me to have a larger e²? Let me check.

If Q is minimized, then e² = 16 - Q would be maximized, so e would be maximized. So yes, to maximize e, we need to minimize the sum of squares of a, b, c, d given their sum is 8 - e.

So, how do I minimize Q given that a + b + c + d = 8 - e?

I remember that for real numbers, the sum of squares is minimized when all variables are equal. So, if I assume that a = b = c = d, then their sum is 4a = 8 - e, so a = (8 - e)/4. Then the sum of squares Q would be 4a² = 4 * [(8 - e)/4]^2 = [(8 - e)^2]/4.

Therefore, Q_min = [(8 - e)^2]/4. But we also have Q = 16 - e². Therefore, setting them equal:

[(8 - e)^2]/4 = 16 - e²

Let me solve this equation for e.

Multiply both sides by 4:

(8 - e)^2 = 64 - 4e²

Expanding the left side:

64 - 16e + e² = 64 - 4e²

Subtract 64 from both sides:

-16e + e² = -4e²

Bring all terms to one side:

-16e + e² + 4e² = 0 => 5e² - 16e = 0

Factor:

e(5e - 16) = 0

So, e = 0 or e = 16/5 = 3.2

Wait, but we are trying to maximize e. So 16/5 is 3.2, which is larger than 0, so maybe e = 16/5 is the answer?

But hold on, is this correct? Because I assumed that a = b = c = d to minimize Q. But is that valid? Let me think again.

Yes, for fixed sum, the sum of squares is minimized when all variables are equal. So if we have four variables with a fixed sum, their sum of squares is minimized when they are all equal. Therefore, that should give the minimal Q. Therefore, the equation [(8 - e)^2]/4 = 16 - e² is valid. Then solving gives e = 0 or 16/5. Therefore, since we want maximum e, e = 16/5. Therefore, the maximum value of e is 16/5.

But wait, let me verify this with another approach to make sure.

Another approach: Let me use Lagrange multipliers. Let me consider variables a, b, c, d, e with constraints.

But maybe that's more complicated. Alternatively, think of variables a, b, c, d as variables with sum S = 8 - e and sum of squares Q = 16 - e². To find the minimum Q given S, which we already did via equality case.

Alternatively, use Cauchy-Schwarz inequality. For vectors (1,1,1,1) and (a,b,c,d). The Cauchy-Schwarz inequality says that (a + b + c + d)^2 ≤ 4(a² + b² + c² + d²). So:

S² ≤ 4Q

So, (8 - e)^2 ≤ 4*(16 - e²)

Which is exactly the equation we had before. So solving:

(8 - e)^2 ≤ 4*(16 - e²)

Which again gives 64 - 16e + e² ≤ 64 - 4e² => -16e + e² ≤ -4e² => 5e² - 16e ≤ 0 => e(5e - 16) ≤ 0.

Which implies e ∈ [0, 16/5]. So maximum e is 16/5.

Therefore, this confirms that the maximum possible value of e is 16/5. So 16/5 is 3.2. But let me check if that actually works.

Let me plug e = 16/5 into S and Q.

S = 8 - e = 8 - 16/5 = 40/5 - 16/5 = 24/5. Then Q = 16 - e² = 16 - (256/25) = (400/25 - 256/25) = 144/25.

Now, if a = b = c = d = S/4 = (24/5)/4 = 6/5. Then Q should be 4*(6/5)^2 = 4*(36/25) = 144/25, which matches. Then the sum of squares is 144/25 + (16/5)^2 = 144/25 + 256/25 = 400/25 = 16. Which is correct. Also, the sum is 4*(6/5) + 16/5 = 24/5 + 16/5 = 40/5 = 8. Perfect.

Therefore, this works. So e = 16/5 is achievable. Therefore, the maximum value of e is 16/5.

But wait, is there a possibility that if we don't set a = b = c = d, but set some variables to be different, maybe we can get a higher e? For example, maybe making some variables larger and some smaller?

Wait, but according to Cauchy-Schwarz, the minimal Q is achieved when variables are equal. Therefore, if we have Q_min = [(8 - e)^2]/4. If we take Q to be larger, then e² = 16 - Q would be smaller, meaning e would be smaller. Therefore, in order to maximize e, we need Q to be as small as possible, which occurs when variables are equal. So this would indeed give the maximum e.

Alternatively, suppose we set three variables to zero, and let the other variable take the remaining sum. For example, suppose a = b = c = 0, then d + e = 8, and d² + e² = 16 - 0 - 0 - 0 = 16. Wait, let me check:

Wait, original equations:

a + b + c + d + e = 8

a² + b² + c² + d² + e² = 16

If a = b = c = 0, then d + e = 8, and d² + e² = 16. Let me solve for d and e.

Let me let d = 8 - e. Then:

(8 - e)^2 + e² = 16

64 - 16e + e² + e² = 16

2e² - 16e + 64 - 16 = 0

2e² - 16e + 48 = 0

Divide by 2:

e² - 8e + 24 = 0

Discriminant: 64 - 96 = -32 < 0. No real solutions. Therefore, this case is impossible. Therefore, if we set three variables to zero, there is no real solution. So that approach doesn't work.

Alternatively, suppose we set two variables to zero. Let me try a = b = 0, then c + d + e = 8, c² + d² + e² = 16. Now, trying to maximize e. Let me set c = d. Then 2c + e = 8 => c = (8 - e)/2. Then c² + c² + e² = 2c² + e² = 16. Substitute:

2*((8 - e)/2)^2 + e² = 16

2*( (64 - 16e + e²)/4 ) + e² = 16

(64 - 16e + e²)/2 + e² = 16

Multiply out:

32 - 8e + (e²)/2 + e² = 16

Combine terms:

32 - 8e + (3e²)/2 = 16

Subtract 16:

16 - 8e + (3e²)/2 = 0

Multiply by 2:

32 - 16e + 3e² = 0 => 3e² - 16e + 32 = 0

Discriminant: 256 - 384 = -128 < 0. No real solution again. So even if we set two variables to zero, there's no solution.

Alternatively, maybe set one variable to zero. Let me try a = 0, then b + c + d + e = 8, and b² + c² + d² + e² = 16. Trying to maximize e. Let me set b = c = d. Then 3b + e = 8 => b = (8 - e)/3. Then sum of squares: 3b² + e² = 3*[(8 - e)/3]^2 + e² = 3*(64 - 16e + e²)/9 + e² = (64 - 16e + e²)/3 + e² = (64 - 16e + e² + 3e²)/3 = (64 - 16e + 4e²)/3 = 16.

Multiply both sides by 3:

64 - 16e + 4e² = 48 => 4e² - 16e + 64 - 48 = 0 => 4e² - 16e + 16 = 0 => divide by 4: e² - 4e + 4 = 0 => (e - 2)^2 = 0 => e = 2.

So in this case, maximum e is 2, which is less than 16/5 (which is 3.2). Therefore, this approach gives a smaller e. Hence, the case where all four variables are equal gives a higher e.

Alternatively, what if we set variables to be unequal? For example, set three variables to some value and one different? Maybe that could lead to higher e? But according to our previous reasoning, the minimal Q is achieved when variables are equal, hence any deviation would make Q larger, which would require e to be smaller. Hence, the case where variables are equal gives the maximum e.

Alternatively, suppose that three variables are equal and one is different? Let me try. Suppose a = b = c = x, and d = y. Then:

Sum: 3x + y + e = 8 => 3x + y = 8 - e

Sum of squares: 3x² + y² + e² = 16 => 3x² + y² = 16 - e²

We need to minimize 3x² + y² given 3x + y = 8 - e. Let me use Lagrange multipliers again. Let me consider minimizing f(x, y) = 3x² + y² with constraint g(x, y) = 3x + y - (8 - e) = 0.

The Lagrangian is L = 3x² + y² - λ(3x + y - (8 - e)).

Taking partial derivatives:

∂L/∂x = 6x - 3λ = 0 => 6x = 3λ => λ = 2x

∂L/∂y = 2y - λ = 0 => 2y = λ => 2y = 2x => y = x.

Therefore, the minimum occurs when y = x. Hence, 3x + x = 8 - e => 4x = 8 - e => x = (8 - e)/4. Therefore, y = x, so this reduces to the case where all four variables are equal. Therefore, even if we assume three variables equal and one different, the minimum occurs when they are all equal. Therefore, this reinforces that the minimal Q is achieved when all variables are equal. Therefore, e = 16/5 is indeed the maximum value.

Another way: think of the problem as an optimization problem with constraints. We need to maximize e subject to:

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

a² + b² + c² + d² + e² = 16.

Let me use Lagrange multipliers for this. Let me consider variables a, b, c, d, e. Let me set up the function to maximize e with constraints.

But since we have two constraints, we need to use Lagrange multipliers with two constraints. Wait, but maybe we can reduce the problem.

Alternatively, let me consider e as the variable to maximize. Let me fix e and then consider the problem for variables a, b, c, d. So, given e, find if there exist real numbers a, b, c, d such that:

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

a² + b² + c² + d² = 16 - e².

For real numbers to exist, the minimal value of the sum of squares given the sum must be less than or equal to 16 - e². As we discussed earlier, the minimal value of sum of squares is [(8 - e)^2]/4, so we need:

[(8 - e)^2]/4 ≤ 16 - e².

Which is the same equation as before, leading to e ≤ 16/5. Therefore, this gives the maximum e is 16/5. Therefore, the answer is 16/5.

Alternatively, thinking geometrically, the problem is equivalent to finding the maximum coordinate e such that the point (a, b, c, d, e) lies on the intersection of the hyperplane a + b + c + d + e = 8 and the sphere a² + b² + c² + d² + e² = 16. The maximum e would be the point on the sphere where the hyperplane touches the sphere at the highest e-coordinate. Which can be found by the method of Lagrange multipliers or by projecting the center of the sphere onto the hyperplane.

Wait, the sphere is centered at the origin with radius 4. The hyperplane is a + b + c + d + e = 8. The distance from the center (origin) to the hyperplane is |0 + 0 + 0 + 0 + 0 - 8| / sqrt(1² + 1² + 1² + 1² + 1²) ) = 8 / sqrt(5). Since the radius of the sphere is 4, and the distance from the center to the plane is 8/sqrt(5) ≈ 3.577, which is greater than the radius 4? Wait no, 8/sqrt(5) ≈ 3.577 < 4. Therefore, the hyperplane intersects the sphere. The maximum value of e on the intersection would be the point where you go from the center towards the hyperplane, then move along the direction of e-axis?

Wait, let me recall that the maximum value of e on the intersection of the hyperplane and the sphere can be found by parametrizing. Let me consider the vector from the origin to the point on the sphere. The hyperplane equation is a + b + c + d + e = 8. The maximum e occurs when the point is as far as possible in the e-direction, while still satisfying the hyperplane and sphere equations.

Alternatively, using the method of Lagrange multipliers. Let me try again. Let me maximize e subject to:

1. a + b + c + d + e = 8

2. a² + b² + c² + d² + e² = 16

Set up the Lagrangian:

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

Taking partial derivatives:

∂L/∂a = -λ - 2μa = 0 => λ = -2μa

Similarly for ∂L/∂b = -λ - 2μb = 0 => λ = -2μb

Same for c and d:

∂L/∂c = -λ - 2μc = 0 => λ = -2μc

∂L/∂d = -λ - 2μd = 0 => λ = -2μd

∂L/∂e = 1 - λ - 2μe = 0 => 1 - λ - 2μe = 0

Therefore, from the first four equations, we have:

-2μa = -2μb = -2μc = -2μd

Assuming μ ≠ 0, then a = b = c = d.

If μ = 0, then from ∂L/∂a = -λ = 0 => λ = 0, and ∂L/∂e = 1 - 0 - 0 = 1 ≠ 0, which is impossible. So μ ≠ 0, hence a = b = c = d.

Let me denote a = b = c = d = x. Then:

From the first constraint: 4x + e = 8 => x = (8 - e)/4

From the second constraint: 4x² + e² = 16 => 4*[(8 - e)/4]^2 + e² = 16

Which is the same equation as before. So:

[(8 - e)^2]/4 + e² = 16 => same steps as before, leading to e = 16/5.

Therefore, the maximum value of e is 16/5. So this method also confirms the same result.

Therefore, regardless of the method—Cauchy-Schwarz, Lagrange multipliers, or assuming equality in the variables—the maximum value of e is 16/5.

Just to make sure, let me check if there's another scenario where maybe e could be higher. Suppose that some variables are negative, allowing others to be larger? Wait, but in our previous case, when we set a = b = c = d = 6/5, which is positive. If we make some variables negative, perhaps we can have a higher e?

Let me try. Let me assume that three variables are negative, and one variable is positive. Wait, but how would that affect the sum?

Suppose I set three variables to some negative number and one variable to something else. Let me think. Suppose a = b = c = -k, and d is something. Then the sum is 3*(-k) + d + e = 8. The sum of squares is 3k² + d² + e² = 16.

But trying to maximize e. Let me try to set k as large as possible? Hmm, not sure.

Alternatively, perhaps to make the sum of a + b + c + d as small as possible, allowing e to be larger. Wait, but the sum of a + b + c + d is 8 - e. If I want e to be as large as possible, then 8 - e should be as small as possible, but the sum of squares of a, b, c, d is 16 - e². Therefore, for smaller 8 - e, the sum of squares would have to be larger? But how does that relate?

Wait, no. If I need to have a + b + c + d = 8 - e, and their sum of squares is 16 - e². For a fixed sum, the minimal sum of squares is achieved when variables are equal. If we want to have a smaller sum (i.e., 8 - e smaller), but then the required sum of squares is 16 - e². However, if 8 - e is smaller, but 16 - e² could be larger or smaller. For example, if e increases, 16 - e² decreases. So we need to balance between the sum and the sum of squares.

But our earlier method already took into account the minimal sum of squares for a given sum, hence allowing the maximum e. Therefore, if we don't take the minimal sum of squares (i.e., if variables are unequal), then the required sum of squares would be larger, which would require 16 - e² to be larger, hence e² to be smaller, leading to smaller e. Therefore, to maximize e, we need the minimal possible sum of squares for a given sum, which occurs when variables are equal.

Therefore, there's no way to have a higher e by making variables unequal. Therefore, 16/5 is indeed the maximum value.

Just to check with numbers, suppose e is 4. Then the sum of a + b + c + d = 8 - 4 = 4. The sum of squares would need to be 16 - 16 = 0. Therefore, a = b = c = d = 0, which satisfies the sum, but sum of squares is 0, which is okay. Wait, but if e = 4, then sum of squares is 0 + 16 = 16, which is correct. Wait, but that would mean that e can be 4?

Wait, hold on! Wait, if e = 4, then a + b + c + d = 4, and a² + b² + c² + d² = 16 - 16 = 0. Therefore, a = b = c = d = 0, which satisfies the sum of 4? No! Wait, 0 + 0 + 0 + 0 = 0 ≠ 4. Therefore, this is impossible. Therefore, my previous thought is wrong. There's a contradiction here. What's happening?

Wait, if e = 4, then sum a + b + c + d = 8 - e = 8 - 4 = 4. Sum of squares is 16 - e² = 16 - 16 = 0. Therefore, the sum of squares is zero implies all variables a, b, c, d are zero. But their sum is 0 ≠ 4. Contradiction. Therefore, e cannot be 4. Therefore, my previous thought was wrong.

Therefore, there must be a mistake here. Wait, how come?

Earlier, when I used Cauchy-Schwarz, we arrived at e ≤ 16/5. So 16/5 is 3.2. Let me check e = 3.2. Then 8 - e = 4.8. The sum of squares of a, b, c, d is 16 - (16/5)^2 = 16 - 256/25 = (400 - 256)/25 = 144/25. Which is 5.76. Then, if a = b = c = d = 4.8 / 4 = 1.2. Then sum of squares is 4*(1.2)^2 = 4*(1.44) = 5.76, which is correct. So that works.

But if I tried e = 4, then it doesn't work. Therefore, the maximum e is indeed 16/5. So why did I think there was a contradiction? Because I thought if we set e = 4, then the sum of a + b + c + d = 4 and the sum of squares is zero, which is impossible. Therefore, there is a maximum e where the minimal sum of squares equals 16 - e². Which is exactly what we solved earlier.

So, perhaps my confusion was because I thought e could be as high as 4, but that is not possible because of the sum of squares constraint. Therefore, the maximum e is 16/5.

Wait, let me check another value. Let's try e = 3. Let me compute 8 - e = 5. The sum of squares of a, b, c, d is 16 - 9 = 7. Then, if we set a = b = c = d = 5/4 = 1.25. Then the sum of squares is 4*(5/4)^2 = 4*(25/16) = 25/4 = 6.25, which is less than 7. Therefore, we have some slack. Therefore, we could vary the variables to make the sum of squares equal to 7. For example, set three variables to 1.25 and one variable to something else. Let me try.

Let me set three variables as 1.25 and the fourth variable as x. Then sum: 3*(1.25) + x = 5 => 3.75 + x = 5 => x = 1.25. Wait, same as before. Therefore, sum of squares is 4*(1.25)^2 = 6.25. But we need sum of squares to be 7. Therefore, we need to increase some variables. For example, let me set three variables to 1.25 and one variable to y. Then:

Sum: 3*1.25 + y = 5 => y = 5 - 3.75 = 1.25. Same as before. Therefore, if I want to increase the sum of squares, I need to make some variables larger and some smaller. For example, set two variables to some higher value and two variables to lower. Let me set two variables to a and two variables to b. Then:

2a + 2b = 5 => a + b = 2.5

And 2a² + 2b² = 7 => a² + b² = 3.5

Let me solve for a and b. Let me set b = 2.5 - a. Then:

a² + (2.5 - a)^2 = 3.5

Expand:

a² + 6.25 - 5a + a² = 3.5

2a² -5a + 6.25 - 3.5 = 0

2a² -5a + 2.75 = 0

Multiply by 4 to eliminate decimals:

8a² - 20a + 11 = 0

Discriminant: 400 - 352 = 48

Solutions: [20 ± sqrt(48)] / 16 = [20 ± 4*sqrt(3)] / 16 = [5 ± sqrt(3)] / 4 ≈ [5 ± 1.732]/4

So, a ≈ (5 + 1.732)/4 ≈ 6.732/4 ≈ 1.683, and b ≈ (5 - 1.732)/4 ≈ 3.268/4 ≈ 0.817

So, possible. Therefore, variables can be adjusted. Therefore, e = 3 is possible. However, that doesn't mean e can be higher than 16/5=3.2. So, e=3.2 is higher than 3, so that is better.

Therefore, all checks confirm that e=16/5 is possible, and higher values are not. Therefore, 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 first observe that we are working with five variables, but we are given only two constraints. Our goal is to **maximize $ e $** while keeping both equations valid.

---

### Step 1: Expressing Other Variables

Let us define:

- $ S = a + b + c + d $, so from (1):
  $$
  S + e = 8 \Rightarrow S = 8 - e
  $$

- $ Q = a^2 + b^2 + c^2 + d^2 $, so from (2):
  $$
  Q + e^2 = 16 \Rightarrow Q = 16 - e^2
  $$

We are now trying to find the **maximum value of $ e $** such that the sum $ S = 8 - e $ and the sum of squares $ Q = 16 - e^2 $ is **attainable**, i.e., there exist real numbers $ a, b, c, d $ satisfying:

$$
a + b + c + d = 8 - e, \quad a^2 + b^2 + c^2 + d^2 = 16 - e^2
$$

---

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

The sum of squares of real numbers can be bounded below using the **Cauchy-Schwarz inequality**. For any real numbers $ a, b, c, d $, we have:

$$
(a + b + c + d)^2 \leq 4(a^2 + b^2 + c^2 + d^2)
$$

Substituting the known values:

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

Expanding both sides:

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

Simplify:

$$
e^2 - 16e \leq -4e^2 \Rightarrow 5e^2 - 16e \leq 0
$$

Factor:

$$
e(5e - 16) \leq 0
$$

This inequality holds for:

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

Thus, the **maximum possible value** of $ e $ is $ \frac{16}{5} $.

---

### Step 3: Verifying the Solution

Let’s check whether $ e = \frac{16}{5} $ satisfies the original constraints:

- $ S = 8 - e = \frac{24}{5} $
- $ Q = 16 - e^2 = 16 - \left(\frac{256}{25}\right) = \frac{144}{25} $

If we set $ a = b = c = d = \frac{6}{5} $, then:

$$
a + b + c + d = 4 \cdot \frac{6}{5} = \frac{24}{5}, \quad a^2 + b^2 + c^2 + d^2 = 4 \cdot \left(\frac{6}{5}\right)^2 = \frac{144}{25}
$$

Then:

$$
a^2 + b^2 + c^2 + d^2 + e^2 = \frac{144}{25} + \left(\frac{16}{5}\right)^2 = \frac{144}{25} + \frac{256}{25} = \frac{400}{25} = 16
$$

And:

$$
a + b + c + d + e = \frac{24}{5} + \frac{16}{5} = \frac{40}{5} = 8
$$

Both constraints are satisfied. Therefore, $ e = \frac{16}{5} $ is indeed a **valid** solution, and since no higher $ e $ satisfies the inequality derived from Cauchy–Schwarz, it is the **maximum**.

---

### Final Answer

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

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

1. a + b + c + d + e = 8
2. a² + b² + c² + d² + e² = 16

Hmm, let me think. Since we need to maximize e, maybe I should express the other variables in terms of e and then apply some inequality? Like Cauchy-Schwarz or something?

Let me start by letting the sum of a, b, c, d be something. Let me denote S = a + b + c + d. Then from the first equation, S + e = 8, so S = 8 - e.

Similarly, the sum of squares: a² + b² + c² + d² + e² = 16. Let me denote Q = a² + b² + c² + d². Then Q + e² = 16, so Q = 16 - e².

Now, I need to relate S and Q. For variables a, b, c, d, there's a relationship between their sum and sum of squares. Maybe using the Cauchy-Schwarz inequality? Or maybe the concept that for a fixed sum, the sum of squares is minimized when all variables are equal?

Wait, since I want to maximize e, perhaps I need to minimize Q? Because Q = 16 - e², so if I minimize Q, that would allow me to have a larger e²? Let me check.

If Q is minimized, then e² = 16 - Q would be maximized, so e would be maximized. So yes, to maximize e, we need to minimize the sum of squares of a, b, c, d given their sum is 8 - e.

So, how do I minimize Q given that a + b + c + d = 8 - e?

I remember that for real numbers, the sum of squares is minimized when all variables are equal. So, if I assume that a = b = c = d, then their sum is 4a = 8 - e, so a = (8 - e)/4. Then the sum of squares Q would be 4a² = 4 * [(8 - e)/4]^2 = [(8 - e)^2]/4.

Therefore, Q_min = [(8 - e)^2]/4. But we also have Q = 16 - e². Therefore, setting them equal:

[(8 - e)^2]/4 = 16 - e²

Let me solve this equation for e.

Multiply both sides by 4:

(8 - e)^2 = 64 - 4e²

Expanding the left side:

64 - 16e + e² = 64 - 4e²

Subtract 64 from both sides:

-16e + e² = -4e²

Bring all terms to one side:

-16e + e² + 4e² = 0 => 5e² - 16e = 0

Factor:

e(5e - 16) = 0

So, e = 0 or e = 16/5 = 3.2

Wait, but we are trying to maximize e. So 16/5 is 3.2, which is larger than 0, so maybe e = 16/5 is the answer?

But hold on, is this correct? Because I assumed that a = b = c = d to minimize Q. But is that valid? Let me think again.

Yes, for fixed sum, the sum of squares is minimized when all variables are equal. So if we have four variables with a fixed sum, their sum of squares is minimized when they are all equal. Therefore, that should give the minimal Q. Therefore, the equation [(8 - e)^2]/4 = 16 - e² is valid. Then solving gives e = 0 or 16/5. Therefore, since we want maximum e, e = 16/5. Therefore, the maximum value of e is 16/5.

But wait, let me verify this with another approach to make sure.

Another approach: Let me use Lagrange multipliers. Let me consider variables a, b, c, d, e with constraints.

But maybe that's more complicated. Alternatively, think of variables a, b, c, d as variables with sum S = 8 - e and sum of squares Q = 16 - e². To find the minimum Q given S, which we already did via equality case.

Alternatively, use Cauchy-Schwarz inequality. For vectors (1,1,1,1) and (a,b,c,d). The Cauchy-Schwarz inequality says that (a + b + c + d)^2 ≤ 4(a² + b² + c² + d²). So:

S² ≤ 4Q

So, (8 - e)^2 ≤ 4*(16 - e²)

Which is exactly the equation we had before. So solving:

(8 - e)^2 ≤ 4*(16 - e²)

Which again gives 64 - 16e + e² ≤ 64 - 4e² => -16e + e² ≤ -4e² => 5e² - 16e ≤ 0 => e(5e - 16) ≤ 0.

Which implies e ∈ [0, 16/5]. So maximum e is 16/5.

Therefore, this confirms that the maximum possible value of e is 16/5. So 16/5 is 3.2. But let me check if that actually works.

Let me plug e = 16/5 into S and Q.

S = 8 - e = 8 - 16/5 = 40/5 - 16/5 = 24/5. Then Q = 16 - e² = 16 - (256/25) = (400/25 - 256/25) = 144/25.

Now, if a = b = c = d = S/4 = (24/5)/4 = 6/5. Then Q should be 4*(6/5)^2 = 4*(36/25) = 144/25, which matches. Then the sum of squares is 144/25 + (16/5)^2 = 144/25 + 256/25 = 400/25 = 16. Which is correct. Also, the sum is 4*(6/5) + 16/5 = 24/5 + 16/5 = 40/5 = 8. Perfect.

Therefore, this works. So e = 16/5 is achievable. Therefore, the maximum value of e is 16/5.

But wait, is there a possibility that if we don't set a = b = c = d, but set some variables to be different, maybe we can get a higher e? For example, maybe making some variables larger and some smaller?

Wait, but according to Cauchy-Schwarz, the minimal Q is achieved when variables are equal. Therefore, if we have Q_min = [(8 - e)^2]/4. If we take Q to be larger, then e² = 16 - Q would be smaller, meaning e would be smaller. Therefore, in order to maximize e, we need Q to be as small as possible, which occurs when variables are equal. So this would indeed give the maximum e.

Alternatively, suppose we set three variables to zero, and let the other variable take the remaining sum. For example, suppose a = b = c = 0, then d + e = 8, and d² + e² = 16 - 0 - 0 - 0 = 16. Wait, let me check:

Wait, original equations:

a + b + c + d + e = 8

a² + b² + c² + d² + e² = 16

If a = b = c = 0, then d + e = 8, and d² + e² = 16. Let me solve for d and e.

Let me let d = 8 - e. Then:

(8 - e)^2 + e² = 16

64 - 16e + e² + e² = 16

2e² - 16e + 64 - 16 = 0

2e² - 16e + 48 = 0

Divide by 2:

e² - 8e + 24 = 0

Discriminant: 64 - 96 = -32 < 0. No real solutions. Therefore, this case is impossible. Therefore, if we set three variables to zero, there is no real solution. So that approach doesn't work.

Alternatively, suppose we set two variables to zero. Let me try a = b = 0, then c + d + e = 8, c² + d² + e² = 16. Now, trying to maximize e. Let me set c = d. Then 2c + e = 8 => c = (8 - e)/2. Then c² + c² + e² = 2c² + e² = 16. Substitute:

2*((8 - e)/2)^2 + e² = 16

2*( (64 - 16e + e²)/4 ) + e² = 16

(64 - 16e + e²)/2 + e² = 16

Multiply out:

32 - 8e + (e²)/2 + e² = 16

Combine terms:

32 - 8e + (3e²)/2 = 16

Subtract 16:

16 - 8e + (3e²)/2 = 0

Multiply by 2:

32 - 16e + 3e² = 0 => 3e² - 16e + 32 = 0

Discriminant: 256 - 384 = -128 < 0. No real solution again. So even if we set two variables to zero, there's no solution.

Alternatively, maybe set one variable to zero. Let me try a = 0, then b + c + d + e = 8, and b² + c² + d² + e² = 16. Trying to maximize e. Let me set b = c = d. Then 3b + e = 8 => b = (8 - e)/3. Then sum of squares: 3b² + e² = 3*[(8 - e)/3]^2 + e² = 3*(64 - 16e + e²)/9 + e² = (64 - 16e + e²)/3 + e² = (64 - 16e + e² + 3e²)/3 = (64 - 16e + 4e²)/3 = 16.

Multiply both sides by 3:

64 - 16e + 4e² = 48 => 4e² - 16e + 64 - 48 = 0 => 4e² - 16e + 16 = 0 => divide by 4: e² - 4e + 4 = 0 => (e - 2)^2 = 0 => e = 2.

So in this case, maximum e is 2, which is less than 16/5 (which is 3.2). Therefore, this approach gives a smaller e. Hence, the case where all four variables are equal gives a higher e.

Alternatively, what if we set variables to be unequal? For example, set three variables to some value and one different? Maybe that could lead to higher e? But according to our previous reasoning, the minimal Q is achieved when variables are equal, hence any deviation would make Q larger, which would require e to be smaller. Hence, the case where variables are equal gives the maximum e.

Alternatively, suppose that three variables are equal and one is different? Let me try. Suppose a = b = c = x, and d = y. Then:

Sum: 3x + y + e = 8 => 3x + y = 8 - e

Sum of squares: 3x² + y² + e² = 16 => 3x² + y² = 16 - e²

We need to minimize 3x² + y² given 3x + y = 8 - e. Let me use Lagrange multipliers again. Let me consider minimizing f(x, y) = 3x² + y² with constraint g(x, y) = 3x + y - (8 - e) = 0.

The Lagrangian is L = 3x² + y² - λ(3x + y - (8 - e)).

Taking partial derivatives:

∂L/∂x = 6x - 3λ = 0 => 6x = 3λ => λ = 2x

∂L/∂y = 2y - λ = 0 => 2y = λ => 2y = 2x => y = x.

Therefore, the minimum occurs when y = x. Hence, 3x + x = 8 - e => 4x = 8 - e => x = (8 - e)/4. Therefore, y = x, so this reduces to the case where all four variables are equal. Therefore, even if we assume three variables equal and one different, the minimum occurs when they are all equal. Therefore, this reinforces that the minimal Q is achieved when all variables are equal. Therefore, e = 16/5 is indeed the maximum value.

Another way: think of the problem as an optimization problem with constraints. We need to maximize e subject to:

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

a² + b² + c² + d² + e² = 16.

Let me use Lagrange multipliers for this. Let me consider variables a, b, c, d, e. Let me set up the function to maximize e with constraints.

But since we have two constraints, we need to use Lagrange multipliers with two constraints. Wait, but maybe we can reduce the problem.

Alternatively, let me consider e as the variable to maximize. Let me fix e and then consider the problem for variables a, b, c, d. So, given e, find if there exist real numbers a, b, c, d such that:

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

a² + b² + c² + d² = 16 - e².

For real numbers to exist, the minimal value of the sum of squares given the sum must be less than or equal to 16 - e². As we discussed earlier, the minimal value of sum of squares is [(8 - e)^2]/4, so we need:

[(8 - e)^2]/4 ≤ 16 - e².

Which is the same equation as before, leading to e ≤ 16/5. Therefore, this gives the maximum e is 16/5. Therefore, the answer is 16/5.

Alternatively, thinking geometrically, the problem is equivalent to finding the maximum coordinate e such that the point (a, b, c, d, e) lies on the intersection of the hyperplane a + b + c + d + e = 8 and the sphere a² + b² + c² + d² + e² = 16. The maximum e would be the point on the sphere where the hyperplane touches the sphere at the highest e-coordinate. Which can be found by the method of Lagrange multipliers or by projecting the center of the sphere onto the hyperplane.

Wait, the sphere is centered at the origin with radius 4. The hyperplane is a + b + c + d + e = 8. The distance from the center (origin) to the hyperplane is |0 + 0 + 0 + 0 + 0 - 8| / sqrt(1² + 1² + 1² + 1² + 1²) ) = 8 / sqrt(5). Since the radius of the sphere is 4, and the distance from the center to the plane is 8/sqrt(5) ≈ 3.577, which is greater than the radius 4? Wait no, 8/sqrt(5) ≈ 3.577 < 4. Therefore, the hyperplane intersects the sphere. The maximum value of e on the intersection would be the point where you go from the center towards the hyperplane, then move along the direction of e-axis?

Wait, let me recall that the maximum value of e on the intersection of the hyperplane and the sphere can be found by parametrizing. Let me consider the vector from the origin to the point on the sphere. The hyperplane equation is a + b + c + d + e = 8. The maximum e occurs when the point is as far as possible in the e-direction, while still satisfying the hyperplane and sphere equations.

Alternatively, using the method of Lagrange multipliers. Let me try again. Let me maximize e subject to:

1. a + b + c + d + e = 8

2. a² + b² + c² + d² + e² = 16

Set up the Lagrangian:

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

Taking partial derivatives:

∂L/∂a = -λ - 2μa = 0 => λ = -2μa

Similarly for ∂L/∂b = -λ - 2μb = 0 => λ = -2μb

Same for c and d:

∂L/∂c = -λ - 2μc = 0 => λ = -2μc

∂L/∂d = -λ - 2μd = 0 => λ = -2μd

∂L/∂e = 1 - λ - 2μe = 0 => 1 - λ - 2μe = 0

Therefore, from the first four equations, we have:

-2μa = -2μb = -2μc = -2μd

Assuming μ ≠ 0, then a = b = c = d.

If μ = 0, then from ∂L/∂a = -λ = 0 => λ = 0, and ∂L/∂e = 1 - 0 - 0 = 1 ≠ 0, which is impossible. So μ ≠ 0, hence a = b = c = d.

Let me denote a = b = c = d = x. Then:

From the first constraint: 4x + e = 8 => x = (8 - e)/4

From the second constraint: 4x² + e² = 16 => 4*[(8 - e)/4]^2 + e² = 16

Which is the same equation as before. So:

[(8 - e)^2]/4 + e² = 16 => same steps as before, leading to e = 16/5.

Therefore, the maximum value of e is 16/5. So this method also confirms the same result.

Therefore, regardless of the method—Cauchy-Schwarz, Lagrange multipliers, or assuming equality in the variables—the maximum value of e is 16/5.

Just to make sure, let me check if there's another scenario where maybe e could be higher. Suppose that some variables are negative, allowing others to be larger? Wait, but in our previous case, when we set a = b = c = d = 6/5, which is positive. If we make some variables negative, perhaps we can have a higher e?

Let me try. Let me assume that three variables are negative, and one variable is positive. Wait, but how would that affect the sum?

Suppose I set three variables to some negative number and one variable to something else. Let me think. Suppose a = b = c = -k, and d is something. Then the sum is 3*(-k) + d + e = 8. The sum of squares is 3k² + d² + e² = 16.

But trying to maximize e. Let me try to set k as large as possible? Hmm, not sure.

Alternatively, perhaps to make the sum of a + b + c + d as small as possible, allowing e to be larger. Wait, but the sum of a + b + c + d is 8 - e. If I want e to be as large as possible, then 8 - e should be as small as possible, but the sum of squares of a, b, c, d is 16 - e². Therefore, for smaller 8 - e, the sum of squares would have to be larger? But how does that relate?

Wait, no. If I need to have a + b + c + d = 8 - e, and their sum of squares is 16 - e². For a fixed sum, the minimal sum of squares is achieved when variables are equal. If we want to have a smaller sum (i.e., 8 - e smaller), but then the required sum of squares is 16 - e². However, if 8 - e is smaller, but 16 - e² could be larger or smaller. For example, if e increases, 16 - e² decreases. So we need to balance between the sum and the sum of squares.

But our earlier method already took into account the minimal sum of squares for a given sum, hence allowing the maximum e. Therefore, if we don't take the minimal sum of squares (i.e., if variables are unequal), then the required sum of squares would be larger, which would require 16 - e² to be larger, hence e² to be smaller, leading to smaller e. Therefore, to maximize e, we need the minimal possible sum of squares for a given sum, which occurs when variables are equal.

Therefore, there's no way to have a higher e by making variables unequal. Therefore, 16/5 is indeed the maximum value.

Just to check with numbers, suppose e is 4. Then the sum of a + b + c + d = 8 - 4 = 4. The sum of squares would need to be 16 - 16 = 0. Therefore, a = b = c = d = 0, which satisfies the sum, but sum of squares is 0, which is okay. Wait, but if e = 4, then sum of squares is 0 + 16 = 16, which is correct. Wait, but that would mean that e can be 4?

Wait, hold on! Wait, if e = 4, then a + b + c + d = 4, and a² + b² + c² + d² = 16 - 16 = 0. Therefore, a = b = c = d = 0, which satisfies the sum of 4? No! Wait, 0 + 0 + 0 + 0 = 0 ≠ 4. Therefore, this is impossible. Therefore, my previous thought is wrong. There's a contradiction here. What's happening?

Wait, if e = 4, then sum a + b + c + d = 8 - e = 8 - 4 = 4. Sum of squares is 16 - e² = 16 - 16 = 0. Therefore, the sum of squares is zero implies all variables a, b, c, d are zero. But their sum is 0 ≠ 4. Contradiction. Therefore, e cannot be 4. Therefore, my previous thought was wrong.

Therefore, there must be a mistake here. Wait, how come?

Earlier, when I used Cauchy-Schwarz, we arrived at e ≤ 16/5. So 16/5 is 3.2. Let me check e = 3.2. Then 8 - e = 4.8. The sum of squares of a, b, c, d is 16 - (16/5)^2 = 16 - 256/25 = (400 - 256)/25 = 144/25. Which is 5.76. Then, if a = b = c = d = 4.8 / 4 = 1.2. Then sum of squares is 4*(1.2)^2 = 4*(1.44) = 5.76, which is correct. So that works.

But if I tried e = 4, then it doesn't work. Therefore, the maximum e is indeed 16/5. So why did I think there was a contradiction? Because I thought if we set e = 4, then the sum of a + b + c + d = 4 and the sum of squares is zero, which is impossible. Therefore, there is a maximum e where the minimal sum of squares equals 16 - e². Which is exactly what we solved earlier.

So, perhaps my confusion was because I thought e could be as high as 4, but that is not possible because of the sum of squares constraint. Therefore, the maximum e is 16/5.

Wait, let me check another value. Let's try e = 3. Let me compute 8 - e = 5. The sum of squares of a, b, c, d is 16 - 9 = 7. Then, if we set a = b = c = d = 5/4 = 1.25. Then the sum of squares is 4*(5/4)^2 = 4*(25/16) = 25/4 = 6.25, which is less than 7. Therefore, we have some slack. Therefore, we could vary the variables to make the sum of squares equal to 7. For example, set three variables to 1.25 and one variable to something else. Let me try.

Let me set three variables as 1.25 and the fourth variable as x. Then sum: 3*(1.25) + x = 5 => 3.75 + x = 5 => x = 1.25. Wait, same as before. Therefore, sum of squares is 4*(1.25)^2 = 6.25. But we need sum of squares to be 7. Therefore, we need to increase some variables. For example, let me set three variables to 1.25 and one variable to y. Then:

Sum: 3*1.25 + y = 5 => y = 5 - 3.75 = 1.25. Same as before. Therefore, if I want to increase the sum of squares, I need to make some variables larger and some smaller. For example, set two variables to some higher value and two variables to lower. Let me set two variables to a and two variables to b. Then:

2a + 2b = 5 => a + b = 2.5

And 2a² + 2b² = 7 => a² + b² = 3.5

Let me solve for a and b. Let me set b = 2.5 - a. Then:

a² + (2.5 - a)^2 = 3.5

Expand:

a² + 6.25 - 5a + a² = 3.5

2a² -5a + 6.25 - 3.5 = 0

2a² -5a + 2.75 = 0

Multiply by 4 to eliminate decimals:

8a² - 20a + 11 = 0

Discriminant: 400 - 352 = 48

Solutions: [20 ± sqrt(48)] / 16 = [20 ± 4*sqrt(3)] / 16 = [5 ± sqrt(3)] / 4 ≈ [5 ± 1.732]/4

So, a ≈ (5 + 1.732)/4 ≈ 6.732/4 ≈ 1.683, and b ≈ (5 - 1.732)/4 ≈ 3.268/4 ≈ 0.817

So, possible. Therefore, variables can be adjusted. Therefore, e = 3 is possible. However, that doesn't mean e can be higher than 16/5=3.2. So, e=3.2 is higher than 3, so that is better.

Therefore, all checks confirm that e=16/5 is possible, and higher values are not. Therefore, 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 first observe that we are working with five variables, but we are given only two constraints. Our goal is to **maximize $ e $** while keeping both equations valid.

---

### Step 1: Expressing Other Variables

Let us define:

- $ S = a + b + c + d $, so from (1):
  $$
  S + e = 8 \Rightarrow S = 8 - e
  $$

- $ Q = a^2 + b^2 + c^2 + d^2 $, so from (2):
  $$
  Q + e^2 = 16 \Rightarrow Q = 16 - e^2
  $$

We are now trying to find the **maximum value of $ e $** such that the sum $ S = 8 - e $ and the sum of squares $ Q = 16 - e^2 $ is **attainable**, i.e., there exist real numbers $ a, b, c, d $ satisfying:

$$
a + b + c + d = 8 - e, \quad a^2 + b^2 + c^2 + d^2 = 16 - e^2
$$

---

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

The sum of squares of real numbers can be bounded below using the **Cauchy-Schwarz inequality**. For any real numbers $ a, b, c, d $, we have:

$$
(a + b + c + d)^2 \leq 4(a^2 + b^2 + c^2 + d^2)
$$

Substituting the known values:

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

Expanding both sides:

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

Simplify:

$$
e^2 - 16e \leq -4e^2 \Rightarrow 5e^2 - 16e \leq 0
$$

Factor:

$$
e(5e - 16) \leq 0
$$

This inequality holds for:

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

Thus, the **maximum possible value** of $ e $ is $ \frac{16}{5} $.

---

### Step 3: Verifying the Solution

Let’s check whether $ e = \frac{16}{5} $ satisfies the original constraints:

- $ S = 8 - e = \frac{24}{5} $
- $ Q = 16 - e^2 = 16 - \left(\frac{256}{25}\right) = \frac{144}{25} $

If we set $ a = b = c = d = \frac{6}{5} $, then:

$$
a + b + c + d = 4 \cdot \frac{6}{5} = \frac{24}{5}, \quad a^2 + b^2 + c^2 + d^2 = 4 \cdot \left(\frac{6}{5}\right)^2 = \frac{144}{25}
$$

Then:

$$
a^2 + b^2 + c^2 + d^2 + e^2 = \frac{144}{25} + \left(\frac{16}{5}\right)^2 = \frac{144}{25} + \frac{256}{25} = \frac{400}{25} = 16
$$

And:

$$
a + b + c + d + e = \frac{24}{5} + \frac{16}{5} = \frac{40}{5} = 8
$$

Both constraints are satisfied. Therefore, $ e = \frac{16}{5} $ is indeed a **valid** solution, and since no higher $ e $ satisfies the inequality derived from Cauchy–Schwarz, it is the **maximum**.

---

### Final Answer

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

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.

So, we have three people: Amy, Ben, and Chris. The average of their ages is 6. That means if I add their ages together and divide by 3, I get 6. So, the total sum of their ages now is 6 multiplied by 3, which is 18. Let me write that down:

Amy's age + Ben's age + Chris's age = 18.

Now, the next part says: Four years ago, Chris was the same age as Amy is now. Hmm, okay. Let me parse that. Four years ago, Chris's age was equal to Amy's current age. Let me denote their current ages as variables. Let me assign variables:

Let A = Amy's current age

B = Ben's current age

C = Chris's current age

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

Now, the second piece of information: 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 another equation: C - 4 = A. Which can be rewritten as A = C - 4.

Okay, so we have two equations so far:

1. A + B + C = 18

2. A = C - 4

Now, the third piece of information: 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. Amy's age at that time will be A + 4. The problem says that 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)

Our unknowns are A, B, C. So we can solve this system of equations. Let me try substituting equations 2 into equations 1 and 3 to reduce the number of variables.

From equation 2, since A = C - 4, we can substitute A in equation 1:

(C - 4) + B + C = 18

Simplify that:

C - 4 + B + C = 18 → 2C + B - 4 = 18 → 2C + B = 22 → equation 1a: 2C + B = 22

Now, equation 3: B + 4 = (3/5)(A + 4). But A is C - 4, so substitute that:

B + 4 = (3/5)( (C - 4) + 4 ) → simplify inside the parentheses:

(C - 4 + 4) = C. So equation 3 becomes:

B + 4 = (3/5)C

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

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

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

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

Let me write equation 3a again:

B = (3/5)C - 4

Now substitute this expression for B into equation 1a:

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

Let me compute that:

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

Combine like terms. Let me convert 2C to fifths to combine with 3/5 C:

2C = 10/5 C

So:

10/5 C + 3/5 C - 4 = 22 → (13/5)C - 4 = 22

Now, add 4 to both sides:

(13/5)C = 26

Multiply both sides by 5:

13C = 130

Divide both sides by 13:

C = 10

So, Chris is currently 10 years old. Let me check if this makes sense.

First, let's check equation 2: A = C - 4. So A = 10 - 4 = 6. So Amy is 6 now.

Then, equation 1: A + B + C = 18 → 6 + B + 10 = 18 → B = 18 - 16 = 2. So Ben is 2 years old?

Wait, that seems very young. Let me check the third equation.

Third equation: In four years, Ben's age will be 3/5 of Amy's age at that time.

Ben's age in four years: 2 + 4 = 6

Amy's age in four years: 6 + 4 = 10

Is 6 equal to 3/5 of 10? 3/5 of 10 is 6. Yes, that works. So that checks out.

Wait, so even though Ben is 2 now, which seems young, but mathematically, it's correct. Let me check the first part again: Four years ago, Chris was the same age as Amy is now. Chris is currently 10, so four years ago he was 6, which is Amy's current age. That's correct. So Amy is 6, Ben is 2, Chris is 10. Their average age is (6 + 2 + 10)/3 = 18/3 = 6. Correct.

So even though Ben is 2, which seems very young, the problem doesn't state any constraints on their ages besides the given equations, so it's possible. So the answer is Chris is 10 years old now.

But let me just make sure I didn't make any calculation errors. Let me go through the equations again.

From equation 3a: B + 4 = (3/5)C. If C=10, then (3/5)*10 = 6, so B + 4 = 6 → B = 2. Correct.

Equation 1a: 2C + B = 22. 2*10 + 2 = 22. 20 + 2 = 22. Correct.

So all equations are satisfied. Therefore, Chris is 10 years old.

But let me think again, is there a possible mistake here? For example, maybe interpreting the problem differently? Let me check the problem statement again:

"Four years ago, Chris was the same age as Amy is now."

So, Chris's age four years ago equals Amy's current age. So, Chris is currently 4 years older than Amy. Which is exactly what we have: Chris is 10, Amy is 6. 10 - 6 = 4. Correct.

"In four years, Ben's age will be 3/5 of Amy's age at that time."

Ben's age in four years is 2 + 4 = 6. Amy's age in four years is 6 + 4 = 10. 6 is 3/5 of 10. Correct.

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

But just to be thorough, let me check if there are other possible solutions. Suppose I made a mistake in setting up equations. Let me check:

First equation: average is 6, so total is 18. That's correct.

Second equation: Four years ago, Chris was same age as Amy now. So Chris's age then: C - 4 = Amy's age now: A. So A = C - 4. 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 all equations are set up correctly. Then substitution steps:

From A = C - 4, substituted into first equation, got 2C + B = 22. Then from third equation, B = (3/5)C - 4. Then substituted into 2C + B = 22, getting 2C + (3/5 C - 4) = 22. Then 13/5 C = 26, so C = 10. Correct.

Therefore, answer is 10. It seems correct.

But let me think again about Ben's age. If Ben is 2 now, that means in four years he's 6, and Amy is 10. 6 is 3/5 of 10. So that's right.

Alternatively, if someone thought that "In four years, Ben's age will be 3/5 of Amy's age at that time" might mean something else, but no, it's straightforward.

Alternatively, maybe there's a trick with the wording? For example, maybe "Chris was the same age as Amy is now" refers to something else? But no, it's clearly saying four years ago Chris's age equals Amy's current age. So that's straightforward.

Alternatively, maybe the problem is in the translation? But assuming the problem is stated in English correctly, the equations are set up properly.

So, I think the answer is 10. Let me check once again with actual ages:

Amy: 6

Ben: 2

Chris: 10

Average: (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.

So, everything checks out. Therefore, Chris is 10 years old now.

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

To determine Chris's current age, we approach the problem using the given conditions and set up algebraic equations to solve for the unknowns.

---

### Step 1: Define Variables

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

From the problem, the **average of their ages is 6**, which gives us the following equation:
$$
\frac{A + B + C}{3} = 6 \Rightarrow A + B + C = 18 \tag{1}
$$

---

### Step 2: Translate Second Condition

> "Four years ago, Chris was the same age as Amy is now."

This translates to:
$$
C - 4 = A \Rightarrow A = C - 4 \tag{2}
$$

---

### Step 3: Translate Third Condition

> "In four years, Ben's age will be $ \frac{3}{5} $ of Amy's age at that time."

In four years:
- Ben's age: $ B + 4 $
- Amy's age: $ A + 4 $

So we get:
$$
B + 4 = \frac{3}{5}(A + 4) \tag{3}
$$

---

### Step 4: Substitute and Solve

Using equation (2):
$$
A = C - 4
$$
Substitute into equation (3):
$$
B + 4 = \frac{3}{5}((C - 4) + 4) = \frac{3}{5}C
\Rightarrow B = \frac{3}{5}C - 4 \tag{3a}
$$

Now substitute equations (2) and (3a) into equation (1):
$$
A + B + C = 18 \Rightarrow (C - 4) + \left(\frac{3}{5}C - 4\right) + C = 18
$$
Simplify:
$$
C - 4 + \frac{3}{5}C - 4 + C = 18
\Rightarrow 2C + \frac{3}{5}C - 8 = 18
$$
Combine like terms:
$$
\left(2 + \frac{3}{5}\right)C = 26 \Rightarrow \frac{13}{5}C = 26
$$
Solve for $ C $:
$$
C = \frac{26 \cdot 5}{13} = 10
$$

---

### Step 5: Verify the Solution

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

Check average:
$$
\frac{A + B + C}{3} = \frac{6 + 2 + 10}{3} = \frac{18}{3} = 6 \quad \text{✓}
$$

Check condition about four years ago:
- Chris’s age 4 years ago: $ 10 - 4 = 6 $ = Amy’s current age $ A = 6 $ ✓

Check future condition:
- In 4 years: Ben = $ 2 + 4 = 6 $, Amy = $ 6 + 4 = 10 $
- Is 6 equal to $ \frac{3}{5} \times 10 $? $ \frac{3}{5} \times 10 = 6 $ ✓

---

### 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.

So, we have three people: Amy, Ben, and Chris. The average of their ages is 6. That means if I add their ages together and divide by 3, I get 6. So, the total sum of their ages now is 6 multiplied by 3, which is 18. Let me write that down:

Amy's age + Ben's age + Chris's age = 18.

Now, the next part says: Four years ago, Chris was the same age as Amy is now. Hmm, okay. Let me parse that. Four years ago, Chris's age was equal to Amy's current age. Let me denote their current ages as variables. Let me assign variables:

Let A = Amy's current age

B = Ben's current age

C = Chris's current age

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

Now, the second piece of information: 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 another equation: C - 4 = A. Which can be rewritten as A = C - 4.

Okay, so we have two equations so far:

1. A + B + C = 18

2. A = C - 4

Now, the third piece of information: 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. Amy's age at that time will be A + 4. The problem says that 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)

Our unknowns are A, B, C. So we can solve this system of equations. Let me try substituting equations 2 into equations 1 and 3 to reduce the number of variables.

From equation 2, since A = C - 4, we can substitute A in equation 1:

(C - 4) + B + C = 18

Simplify that:

C - 4 + B + C = 18 → 2C + B - 4 = 18 → 2C + B = 22 → equation 1a: 2C + B = 22

Now, equation 3: B + 4 = (3/5)(A + 4). But A is C - 4, so substitute that:

B + 4 = (3/5)( (C - 4) + 4 ) → simplify inside the parentheses:

(C - 4 + 4) = C. So equation 3 becomes:

B + 4 = (3/5)C

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

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

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

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

Let me write equation 3a again:

B = (3/5)C - 4

Now substitute this expression for B into equation 1a:

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

Let me compute that:

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

Combine like terms. Let me convert 2C to fifths to combine with 3/5 C:

2C = 10/5 C

So:

10/5 C + 3/5 C - 4 = 22 → (13/5)C - 4 = 22

Now, add 4 to both sides:

(13/5)C = 26

Multiply both sides by 5:

13C = 130

Divide both sides by 13:

C = 10

So, Chris is currently 10 years old. Let me check if this makes sense.

First, let's check equation 2: A = C - 4. So A = 10 - 4 = 6. So Amy is 6 now.

Then, equation 1: A + B + C = 18 → 6 + B + 10 = 18 → B = 18 - 16 = 2. So Ben is 2 years old?

Wait, that seems very young. Let me check the third equation.

Third equation: In four years, Ben's age will be 3/5 of Amy's age at that time.

Ben's age in four years: 2 + 4 = 6

Amy's age in four years: 6 + 4 = 10

Is 6 equal to 3/5 of 10? 3/5 of 10 is 6. Yes, that works. So that checks out.

Wait, so even though Ben is 2 now, which seems young, but mathematically, it's correct. Let me check the first part again: Four years ago, Chris was the same age as Amy is now. Chris is currently 10, so four years ago he was 6, which is Amy's current age. That's correct. So Amy is 6, Ben is 2, Chris is 10. Their average age is (6 + 2 + 10)/3 = 18/3 = 6. Correct.

So even though Ben is 2, which seems very young, the problem doesn't state any constraints on their ages besides the given equations, so it's possible. So the answer is Chris is 10 years old now.

But let me just make sure I didn't make any calculation errors. Let me go through the equations again.

From equation 3a: B + 4 = (3/5)C. If C=10, then (3/5)*10 = 6, so B + 4 = 6 → B = 2. Correct.

Equation 1a: 2C + B = 22. 2*10 + 2 = 22. 20 + 2 = 22. Correct.

So all equations are satisfied. Therefore, Chris is 10 years old.

But let me think again, is there a possible mistake here? For example, maybe interpreting the problem differently? Let me check the problem statement again:

"Four years ago, Chris was the same age as Amy is now."

So, Chris's age four years ago equals Amy's current age. So, Chris is currently 4 years older than Amy. Which is exactly what we have: Chris is 10, Amy is 6. 10 - 6 = 4. Correct.

"In four years, Ben's age will be 3/5 of Amy's age at that time."

Ben's age in four years is 2 + 4 = 6. Amy's age in four years is 6 + 4 = 10. 6 is 3/5 of 10. Correct.

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

But just to be thorough, let me check if there are other possible solutions. Suppose I made a mistake in setting up equations. Let me check:

First equation: average is 6, so total is 18. That's correct.

Second equation: Four years ago, Chris was same age as Amy now. So Chris's age then: C - 4 = Amy's age now: A. So A = C - 4. 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 all equations are set up correctly. Then substitution steps:

From A = C - 4, substituted into first equation, got 2C + B = 22. Then from third equation, B = (3/5)C - 4. Then substituted into 2C + B = 22, getting 2C + (3/5 C - 4) = 22. Then 13/5 C = 26, so C = 10. Correct.

Therefore, answer is 10. It seems correct.

But let me think again about Ben's age. If Ben is 2 now, that means in four years he's 6, and Amy is 10. 6 is 3/5 of 10. So that's right.

Alternatively, if someone thought that "In four years, Ben's age will be 3/5 of Amy's age at that time" might mean something else, but no, it's straightforward.

Alternatively, maybe there's a trick with the wording? For example, maybe "Chris was the same age as Amy is now" refers to something else? But no, it's clearly saying four years ago Chris's age equals Amy's current age. So that's straightforward.

Alternatively, maybe the problem is in the translation? But assuming the problem is stated in English correctly, the equations are set up properly.

So, I think the answer is 10. Let me check once again with actual ages:

Amy: 6

Ben: 2

Chris: 10

Average: (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.

So, everything checks out. Therefore, Chris is 10 years old now.

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

To determine Chris's current age, we approach the problem using the given conditions and set up algebraic equations to solve for the unknowns.

---

### Step 1: Define Variables

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

From the problem, the **average of their ages is 6**, which gives us the following equation:
$$
\frac{A + B + C}{3} = 6 \Rightarrow A + B + C = 18 \tag{1}
$$

---

### Step 2: Translate Second Condition

> "Four years ago, Chris was the same age as Amy is now."

This translates to:
$$
C - 4 = A \Rightarrow A = C - 4 \tag{2}
$$

---

### Step 3: Translate Third Condition

> "In four years, Ben's age will be $ \frac{3}{5} $ of Amy's age at that time."

In four years:
- Ben's age: $ B + 4 $
- Amy's age: $ A + 4 $

So we get:
$$
B + 4 = \frac{3}{5}(A + 4) \tag{3}
$$

---

### Step 4: Substitute and Solve

Using equation (2):
$$
A = C - 4
$$
Substitute into equation (3):
$$
B + 4 = \frac{3}{5}((C - 4) + 4) = \frac{3}{5}C
\Rightarrow B = \frac{3}{5}C - 4 \tag{3a}
$$

Now substitute equations (2) and (3a) into equation (1):
$$
A + B + C = 18 \Rightarrow (C - 4) + \left(\frac{3}{5}C - 4\right) + C = 18
$$
Simplify:
$$
C - 4 + \frac{3}{5}C - 4 + C = 18
\Rightarrow 2C + \frac{3}{5}C - 8 = 18
$$
Combine like terms:
$$
\left(2 + \frac{3}{5}\right)C = 26 \Rightarrow \frac{13}{5}C = 26
$$
Solve for $ C $:
$$
C = \frac{26 \cdot 5}{13} = 10
$$

---

### Step 5: Verify the Solution

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

Check average:
$$
\frac{A + B + C}{3} = \frac{6 + 2 + 10}{3} = \frac{18}{3} = 6 \quad \text{✓}
$$

Check condition about four years ago:
- Chris’s age 4 years ago: $ 10 - 4 = 6 $ = Amy’s current age $ A = 6 $ ✓

Check future condition:
- In 4 years: Ben = $ 2 + 4 = 6 $, Amy = $ 6 + 4 = 10 $
- Is 6 equal to $ \frac{3}{5} \times 10 $? $ \frac{3}{5} \times 10 = 6 $ ✓

---

### 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. Hmm, let me think. First, I remember that repeating decimals can be converted into fractions. Maybe I should convert each of these repeating decimals into fractions first and then perform the operations? That seems like a plan.

Alright, starting with 0.7 repeating. Let me recall how to convert a repeating decimal to a fraction. For a decimal like 0.\overline{a}, where a is a single digit, the fraction is a over 9. So 0.\overline{7} should be 7/9? Let me check that. If I take 7 divided by 9, I get 0.7777..., right? Yeah, that seems correct. So 0.\overline{7} is 7/9.

Next, 0.\overline{4}. Similarly, that should be 4/9. Let me verify. 4 divided by 9 is 0.4444..., yes. So that's 4/9.

Then 0.\overline{2}. Following the same pattern, that would be 2/9. 2 divided by 9 is 0.2222... Correct. So that's 2/9.

So substituting these into the original expression, it becomes:

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

Wait, so all the denominators are the same. That makes things easier. Let me compute the numerators:

7 - 4 is 3, and then 3 + 2 is 5. So the result is 5/9?

Wait, is that right? Let me double-check my steps.

First, converting each decimal to fraction:

0.777... = 7/9

0.444... = 4/9

0.222... = 2/9

So substituting:

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

So, combining them:

7/9 - 4/9 = 3/9. Then 3/9 + 2/9 = 5/9. Yes, that seems straightforward. So the answer is 5/9?

Wait, but let me think again. Maybe I made a mistake in the order of operations? The original problem is 0.\overline{7} - 0.\overline{4} + 0.\overline{2}. So it's subtraction first, then addition. So yes, 7/9 - 4/9 is 3/9, then adding 2/9 gives 5/9. So 5/9 is the answer? Let me check with another method to be sure.

Alternatively, I can compute the decimal values first and then perform the operations. Let me try that approach.

First, 0.\overline{7} is 0.7777..., 0.\overline{4} is 0.4444..., and 0.\overline{2} is 0.2222...

So compute 0.7777... - 0.4444... + 0.2222...

Let me compute 0.7777... - 0.4444... first. Let's subtract these two decimals:

0.7777...
-0.4444...
= ?

Subtracting digit by digit:

0.7777... - 0.4444... = 0.3333... which is 0.\overline{3}. Then adding 0.2222... to that.

So 0.3333... + 0.2222... = 0.5555..., which is 0.\overline{5}. And 0.\overline{5} is equal to 5/9. So that also gives 5/9. Therefore, both methods give the same result. So that confirms that the answer is indeed 5/9.

Wait, but let me think again. Sometimes when dealing with repeating decimals, there can be some nuances, but in this case, since all the decimals are single-digit repeating, converting them to fractions is straightforward. So I think my answer is correct. But just to be thorough, let me do the algebraic conversion again for each decimal.

Starting with 0.\overline{7}. Let me use algebra to convert it to a fraction. Let x = 0.7777...

Multiply both sides by 10: 10x = 7.7777...

Subtract original equation: 10x - x = 7.7777... - 0.7777...

So 9x = 7 => x = 7/9. Yep, that works. Same for 0.\overline{4}. Let me do that:

x = 0.4444...

10x = 4.4444...

10x - x = 4.4444... - 0.4444... => 9x = 4 => x = 4/9. And for 0.\overline{2}:

x = 0.2222...

10x = 2.2222...

Subtract: 9x = 2 => x = 2/9. So all conversions are correct.

Therefore, substituting back into the original problem:

7/9 - 4/9 + 2/9. Let me compute this again:

First, 7/9 - 4/9 = (7 - 4)/9 = 3/9 = 1/3. Then 1/3 + 2/9. To add these, common denominator is 9. So 1/3 is 3/9, so 3/9 + 2/9 = 5/9. Yep, same result. Therefore, 5/9 is the answer.

Alternatively, maybe I can think of the entire expression as combining all three fractions first. Let me write them all over 9:

7/9 - 4/9 + 2/9 = (7 - 4 + 2)/9 = (5)/9. Yep, same answer. So all methods lead to 5/9.

But maybe there's a trick here? Wait, the problem says "Compute: 0.\overline{7} - 0.\overline{4} + 0.\overline{2}". So maybe someone might misinterpret the order of operations? For example, if someone does 0.\overline{7} - (0.\overline{4} + 0.\overline{2}), but no, the original expression is written as 0.\overline{7} - 0.\overline{4} + 0.\overline{2}, so according to order of operations, it's left to right. So subtraction first, then addition. But even if someone did it differently, let me check:

If someone thought it was 0.\overline{7} - (0.\overline{4} + 0.\overline{2}), then that would be 7/9 - (4/9 + 2/9) = 7/9 - 6/9 = 1/9. But that's different. But according to standard order of operations, addition and subtraction have the same precedence and are evaluated left to right. So the original expression is (0.\overline{7} - 0.\overline{4}) + 0.\overline{2}, which is (7/9 - 4/9) + 2/9 = 3/9 + 2/9 = 5/9. So that's correct.

Alternatively, if there were parentheses, it would be different, but as written, left to right.

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

But let me check with decimal approximation. Let's compute each decimal to several decimal places and perform the operations.

Compute 0.\overline{7} ≈ 0.7777777777

0.\overline{4} ≈ 0.4444444444

0.\overline{2} ≈ 0.2222222222

Compute 0.7777777777 - 0.4444444444 = 0.3333333333

Then add 0.2222222222: 0.3333333333 + 0.2222222222 = 0.5555555555

Which is approximately 0.555555..., which is 5/9, since 5 divided by 9 is approximately 0.555555...

Therefore, decimal approximation also confirms that the answer is 5/9. So, all methods converge to the same answer. Therefore, I can be confident that 5/9 is the correct answer.

But just to be 100% thorough, let me try another approach. Let me write all three decimals as fractions with denominator 9:

0.\overline{7} = 7/9

0.\overline{4} = 4/9

0.\overline{2} = 2/9

Therefore, the expression is 7/9 - 4/9 + 2/9. Let me compute this as fractions:

First, subtract 4/9 from 7/9: 7/9 - 4/9 = 3/9. Then, adding 2/9: 3/9 + 2/9 = 5/9. Yes.

Alternatively, combining all terms:

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

Another way: think of the entire expression as 7/9 + 2/9 - 4/9. Which is (7 + 2)/9 - 4/9 = 9/9 - 4/9 = 1 - 4/9 = 5/9. Wait, that's another way to see it. Let me check:

If I rearrange the terms: 7/9 + 2/9 - 4/9. Combine the first two: 9/9, which is 1. Then subtract 4/9: 1 - 4/9 = 5/9. Yep, same answer. So that works.

Therefore, regardless of how I approach it, the answer is consistently 5/9. Therefore, I think that's solid.

I can't see any mistakes in my reasoning. All methods confirm the same result, decimal approximation, algebraic conversion, fraction arithmetic. Therefore, I think the answer is 5/9.

**Final Answer**
The result of the expression is \boxed{\dfrac{5}{9}}.

To compute the expression:

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

we begin by converting the repeating decimals into fractions. Each of these repeating decimals has a single repeating digit, and we use the standard conversion method for such numbers:

---

### 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} $

---

### Step 2: Substitute and Simplify

Now substitute the fractional equivalents into the expression:

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

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

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

---

### Final Answer

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

[Assistant]
Explanation:
Okay, so I need to compute this expression: 0.7 repeating minus 0.4 repeating plus 0.2 repeating. Hmm, let me think. First, I remember that repeating decimals can be converted into fractions. Maybe I should convert each of these repeating decimals into fractions first and then perform the operations? That seems like a plan.

Alright, starting with 0.7 repeating. Let me recall how to convert a repeating decimal to a fraction. For a decimal like 0.\overline{a}, where a is a single digit, the fraction is a over 9. So 0.\overline{7} should be 7/9? Let me check that. If I take 7 divided by 9, I get 0.7777..., right? Yeah, that seems correct. So 0.\overline{7} is 7/9.

Next, 0.\overline{4}. Similarly, that should be 4/9. Let me verify. 4 divided by 9 is 0.4444..., yes. So that's 4/9.

Then 0.\overline{2}. Following the same pattern, that would be 2/9. 2 divided by 9 is 0.2222... Correct. So that's 2/9.

So substituting these into the original expression, it becomes:

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

Wait, so all the denominators are the same. That makes things easier. Let me compute the numerators:

7 - 4 is 3, and then 3 + 2 is 5. So the result is 5/9?

Wait, is that right? Let me double-check my steps.

First, converting each decimal to fraction:

0.777... = 7/9

0.444... = 4/9

0.222... = 2/9

So substituting:

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

So, combining them:

7/9 - 4/9 = 3/9. Then 3/9 + 2/9 = 5/9. Yes, that seems straightforward. So the answer is 5/9?

Wait, but let me think again. Maybe I made a mistake in the order of operations? The original problem is 0.\overline{7} - 0.\overline{4} + 0.\overline{2}. So it's subtraction first, then addition. So yes, 7/9 - 4/9 is 3/9, then adding 2/9 gives 5/9. So 5/9 is the answer? Let me check with another method to be sure.

Alternatively, I can compute the decimal values first and then perform the operations. Let me try that approach.

First, 0.\overline{7} is 0.7777..., 0.\overline{4} is 0.4444..., and 0.\overline{2} is 0.2222...

So compute 0.7777... - 0.4444... + 0.2222...

Let me compute 0.7777... - 0.4444... first. Let's subtract these two decimals:

0.7777...
-0.4444...
= ?

Subtracting digit by digit:

0.7777... - 0.4444... = 0.3333... which is 0.\overline{3}. Then adding 0.2222... to that.

So 0.3333... + 0.2222... = 0.5555..., which is 0.\overline{5}. And 0.\overline{5} is equal to 5/9. So that also gives 5/9. Therefore, both methods give the same result. So that confirms that the answer is indeed 5/9.

Wait, but let me think again. Sometimes when dealing with repeating decimals, there can be some nuances, but in this case, since all the decimals are single-digit repeating, converting them to fractions is straightforward. So I think my answer is correct. But just to be thorough, let me do the algebraic conversion again for each decimal.

Starting with 0.\overline{7}. Let me use algebra to convert it to a fraction. Let x = 0.7777...

Multiply both sides by 10: 10x = 7.7777...

Subtract original equation: 10x - x = 7.7777... - 0.7777...

So 9x = 7 => x = 7/9. Yep, that works. Same for 0.\overline{4}. Let me do that:

x = 0.4444...

10x = 4.4444...

10x - x = 4.4444... - 0.4444... => 9x = 4 => x = 4/9. And for 0.\overline{2}:

x = 0.2222...

10x = 2.2222...

Subtract: 9x = 2 => x = 2/9. So all conversions are correct.

Therefore, substituting back into the original problem:

7/9 - 4/9 + 2/9. Let me compute this again:

First, 7/9 - 4/9 = (7 - 4)/9 = 3/9 = 1/3. Then 1/3 + 2/9. To add these, common denominator is 9. So 1/3 is 3/9, so 3/9 + 2/9 = 5/9. Yep, same result. Therefore, 5/9 is the answer.

Alternatively, maybe I can think of the entire expression as combining all three fractions first. Let me write them all over 9:

7/9 - 4/9 + 2/9 = (7 - 4 + 2)/9 = (5)/9. Yep, same answer. So all methods lead to 5/9.

But maybe there's a trick here? Wait, the problem says "Compute: 0.\overline{7} - 0.\overline{4} + 0.\overline{2}". So maybe someone might misinterpret the order of operations? For example, if someone does 0.\overline{7} - (0.\overline{4} + 0.\overline{2}), but no, the original expression is written as 0.\overline{7} - 0.\overline{4} + 0.\overline{2}, so according to order of operations, it's left to right. So subtraction first, then addition. But even if someone did it differently, let me check:

If someone thought it was 0.\overline{7} - (0.\overline{4} + 0.\overline{2}), then that would be 7/9 - (4/9 + 2/9) = 7/9 - 6/9 = 1/9. But that's different. But according to standard order of operations, addition and subtraction have the same precedence and are evaluated left to right. So the original expression is (0.\overline{7} - 0.\overline{4}) + 0.\overline{2}, which is (7/9 - 4/9) + 2/9 = 3/9 + 2/9 = 5/9. So that's correct.

Alternatively, if there were parentheses, it would be different, but as written, left to right.

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

But let me check with decimal approximation. Let's compute each decimal to several decimal places and perform the operations.

Compute 0.\overline{7} ≈ 0.7777777777

0.\overline{4} ≈ 0.4444444444

0.\overline{2} ≈ 0.2222222222

Compute 0.7777777777 - 0.4444444444 = 0.3333333333

Then add 0.2222222222: 0.3333333333 + 0.2222222222 = 0.5555555555

Which is approximately 0.555555..., which is 5/9, since 5 divided by 9 is approximately 0.555555...

Therefore, decimal approximation also confirms that the answer is 5/9. So, all methods converge to the same answer. Therefore, I can be confident that 5/9 is the correct answer.

But just to be 100% thorough, let me try another approach. Let me write all three decimals as fractions with denominator 9:

0.\overline{7} = 7/9

0.\overline{4} = 4/9

0.\overline{2} = 2/9

Therefore, the expression is 7/9 - 4/9 + 2/9. Let me compute this as fractions:

First, subtract 4/9 from 7/9: 7/9 - 4/9 = 3/9. Then, adding 2/9: 3/9 + 2/9 = 5/9. Yes.

Alternatively, combining all terms:

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

Another way: think of the entire expression as 7/9 + 2/9 - 4/9. Which is (7 + 2)/9 - 4/9 = 9/9 - 4/9 = 1 - 4/9 = 5/9. Wait, that's another way to see it. Let me check:

If I rearrange the terms: 7/9 + 2/9 - 4/9. Combine the first two: 9/9, which is 1. Then subtract 4/9: 1 - 4/9 = 5/9. Yep, same answer. So that works.

Therefore, regardless of how I approach it, the answer is consistently 5/9. Therefore, I think that's solid.

I can't see any mistakes in my reasoning. All methods confirm the same result, decimal approximation, algebraic conversion, fraction arithmetic. Therefore, I think the answer is 5/9.

**Final Answer**
The result of the expression is \boxed{\dfrac{5}{9}}.

To compute the expression:

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

we begin by converting the repeating decimals into fractions. Each of these repeating decimals has a single repeating digit, and we use the standard conversion method for such numbers:

---

### 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} $

---

### Step 2: Substitute and Simplify

Now substitute the fractional equivalents into the expression:

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

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

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

---

### Final Answer

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

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 $\frac 79 - \frac 49 + \frac 29 = \boxed{\frac 59}$.