Explanation:
Okay, so I need to find the probability that a randomly selected point (x, y) within the rectangle defined by 0 ≤ x ≤ 8 and 0 ≤ y ≤ 4 satisfies the condition x + y ≤ 4. Hmm, let me think about how to approach this.

First, probability problems like this usually involve finding the ratio of the favorable area to the total area. So, I should start by calculating the total area of the region where the point can be selected. The rectangle has length 8 along the x-axis and 4 along the y-axis, so the total area is 8 * 4. Let me write that down:

Total area = 8 * 4 = 32.

Now, the next part is figuring out the area of the region where x + y ≤ 4. This seems like a linear inequality, so the region defined by x + y ≤ 4 is a polygon, probably a triangle or a trapezoid. Let me visualize this.

If I consider the coordinate system with x from 0 to 8 and y from 0 to 4, the line x + y = 4 would intersect the axes at (4, 0) and (0, 4). Wait, but since our rectangle goes up to x=8 and y=4, the line x + y = 4 is entirely within the rectangle? Let me check. At x=0, y=4 is the upper limit, so the line goes from (0,4) to (4,0). Beyond x=4, if we go to x=8, y would have to be negative to satisfy x + y = 4, but since y can't be negative, the line only exists from (0,4) to (4,0) within the rectangle. Therefore, the region x + y ≤ 4 within the rectangle is a right triangle with vertices at (0,0), (0,4), and (4,0). Is that right?

Wait, let me confirm. If I take x and y both non-negative, then x + y ≤ 4 is indeed a triangle bounded by the x-axis, y-axis, and the line x + y = 4. So yes, that triangle has vertices at (0,0), (4,0), and (0,4). Therefore, the area of this triangle would be (base * height)/2. The base along the x-axis is 4 units, and the height along the y-axis is also 4 units. So area is (4 * 4)/2 = 8. Wait, but hold on, is that correct?

Wait, actually, the base and height for a right triangle are the two legs. Since the triangle is right-angled at (0,0), the base is from (0,0) to (4,0), which is 4 units, and the height is from (0,0) to (0,4), which is also 4 units. So area is indeed (4*4)/2 = 8. So the area where x + y ≤ 4 is 8?

But wait, let me think again. If the line is x + y = 4, then the region x + y ≤ 4 is all the points below that line. But since our rectangle extends beyond that line (up to x=8 and y=4), but the line only intersects the rectangle at (4,0) and (0,4). Therefore, the area under x + y ≤ 4 within the rectangle is exactly that triangle. Therefore, the area is 8. Therefore, the probability would be 8 divided by the total area 32, which is 8/32 = 1/4. So is the probability 1/4?

Wait, but let me make sure I didn't miss anything. Let me sketch mentally. The rectangle is 8 units wide and 4 units tall. The line x + y = 4 cuts through the rectangle from (0,4) to (4,0). So the area under that line is a triangle with area 8. But is there any other region? For example, if x is between 0 and 4, and y is between 0 and 4 - x. But if x is greater than 4, then 4 - x would be negative, so y would have to be less than a negative number, but since y is non-negative, there's no area there. Therefore, the entire region where x + y ≤ 4 is just that triangle. Therefore, area is 8. So probability is 8/32 = 1/4. So 1/4?

But wait, let me check with integration. Maybe integrating over the region. The area can be calculated as the integral from x=0 to x=4 of (4 - x) dx, since for each x between 0 and 4, y goes from 0 to 4 - x. Then integrating that:

Integral from 0 to 4 of (4 - x) dx = [4x - (x^2)/2] from 0 to 4 = (16 - 8) - 0 = 8. Yep, same result. So area is 8. Therefore, probability is 8/32 = 1/4. So is that the answer?

Wait, but let me think again. The problem says "a point (x,y) is randomly selected such that 0 ≤ x ≤ 8 and 0 ≤ y ≤ 4". So the total area is indeed 8*4=32. The area where x + y ≤ 4 is 8, so probability is 8/32=1/4. So 1/4? But I feel like maybe I need to check if there's another region?

Wait, suppose x is between 0 and 4, and y between 0 and 4 - x. But if x is between 0 and 4, then 4 - x is between 0 and 4. However, since y is allowed up to 4, but in this case, the upper limit for y is 4 - x. So that's correct. But if x is greater than 4, then 4 - x is negative, so there's no y that satisfies x + y ≤ 4 because y is non-negative. Therefore, the entire area is just that triangle. Therefore, 8/32=1/4. So probability is 1/4. But let me check with another approach.

Alternatively, maybe using geometry. The line x + y = 4 divides the rectangle into two regions. The area we want is the triangle, which is half of the square? Wait, no. Wait, if we consider the square from (0,0) to (4,4), then the area of the triangle would be half of that square, which is 8. But our rectangle is larger in x-direction. Wait, but in our case, the rectangle is 8x4, but the line x + y = 4 only intersects the rectangle at (0,4) and (4,0). So the area under the line is a triangle with area 8, as calculated. So 8 over 32 is 1/4. So that seems correct.

But maybe I made a mistake in assuming that the entire region is just that triangle. Let me think again. Suppose I pick a point where x is between 0 and 4, and y between 0 and 4. Then x + y ≤ 4 is the triangle. But if x is between 0 and 4, and y is between 0 and 4, but the rectangle is up to y=4. However, if x is between 0 and 4, and y is between 0 and 4, then the area where x + y ≤ 4 is the triangle. However, if x is between 4 and 8, then even if y is 0, x + y is already 4 or more. Therefore, there is no area in x between 4 and 8 that satisfies x + y ≤ 4. Therefore, the entire area is indeed the triangle with area 8. Therefore, probability is 8/32=1/4.

But wait, let me check with coordinates. Let me take some sample points. For example, (0,0): x + y = 0 ≤ 4, yes. (4,0): 4 + 0 = 4 ≤ 4, yes. (0,4): 0 + 4 = 4 ≤ 4, yes. (2, 2): 2 + 2 = 4 ≤ 4, yes. (3, 1): 3 + 1 = 4 ≤ 4, yes. (5, 0): 5 + 0 = 5 > 4, no. So that seems correct. So the area is indeed that triangle. Therefore, probability is 1/4.

But wait, I remember sometimes when you have inequalities, you have to be careful about the boundaries. But in this case, since the line is x + y = 4, and the region is x + y ≤ 4, which includes the boundary. But since we are dealing with areas, the boundary has zero area, so it doesn't affect the probability. Therefore, the area is indeed 8. Therefore, probability is 1/4.

But let me check again with another method. Let me think of the probability as the double integral over the region. The probability is the area where x + y ≤ 4 divided by total area. So:

Probability = (1 / 32) * ∫ (from x=0 to 4) ∫ (from y=0 to 4 - x) dy dx

Which is (1 / 32) * ∫ (from 0 to 4) (4 - x) dx = (1 / 32) * [4x - (x²)/2] from 0 to 4 = (1 / 32) * (16 - 8) = 8 / 32 = 1/4. Yep, same result. So that seems solid.

Alternatively, maybe using geometric probability. The region is a rectangle, and the condition is a linear inequality. The area is a triangle. So the answer is 1/4. Therefore, I think that is correct.

But wait, just to be thorough, let me think if there's another way someone might approach this problem and maybe get a different answer. For example, someone might think that since x can go up to 8 and y up to 4, maybe they consider the line x + y = 4 and think about the area in terms of some other shape. But no, because beyond x=4, the line x + y = 4 would require y to be negative, which is outside the rectangle. Therefore, the area is only the triangle. So I think that's correct.

Alternatively, someone might think that the region x + y ≤ 4 is a square or something else, but no. For example, if you have x and y both between 0 and 4, then x + y ≤ 4 is a triangle. But since our rectangle is larger in x, but the condition restricts y to be less than 4 - x, which for x beyond 4 is not possible. Therefore, the area is still the triangle. Therefore, 1/4.

Wait, but let me check with coordinates again. Suppose I have a rectangle from (0,0) to (8,4). The line x + y = 4 cuts through (0,4) and (4,0). So the area under the line is a triangle with vertices at (0,0), (0,4), and (4,0). So that triangle's area is ½ * base * height. The base is 4 (from (0,0) to (4,0)), the height is 4 (from (0,0) to (0,4)), so area is ½ * 4 * 4 = 8. So that's correct. Therefore, probability is 8 / 32 = 1/4. So yeah, that seems right.

Alternatively, if someone thought that the region is a square or something else, they might get confused, but I think this is solid. Therefore, I think the answer is 1/4.

But just to make sure, let me think of another approach. Let me consider the probability as the expected value or something else. Wait, maybe not necessary. Alternatively, think of it as uniform distribution over the rectangle. The probability that x + y ≤ 4 is equal to the area where x + y ≤ 4 divided by total area. Which we already calculated. So I think that's the way to go.

Alternatively, maybe using coordinates transformation. Let me set u = x, v = y. Then the condition is u + v ≤ 4. But I don't think that helps. Alternatively, maybe using probability distributions. The joint distribution of x and y is uniform over the rectangle. So the probability is the integral over the region x + y ≤ 4 of the joint PDF, which is 1/(8*4) = 1/32. Therefore, integrating over the region gives the area, which is 8, so 8*(1/32) = 1/4. Yep, same answer.

Therefore, I think I can be confident that the probability is 1/4. Therefore, the answer is \boxed{\dfrac{1}{4}}.

**Final Answer**
\boxed{\dfrac{1}{4}}

To determine the probability that a randomly selected point $(x, y)$ within the rectangle defined by $0 \le x \le 8$ and $0 \le y \le 4$ satisfies the condition $x + y \le 4$, we approach the problem using geometric probability.

---

### Step 1: Total Area of the Region

The region where the point $(x, y)$ can be selected is a rectangle with:

- Width: $8$ (from $x = 0$ to $x = 8$)
- Height: $4$ (from $y = 0$ to $y = 4$)

So, the **total area** of the region is:

$$
\text{Total Area} = 8 \times 4 = 32
$$

---

### Step 2: Area of the Region Satisfying $x + y \le 4$

We are interested in the area of the region where $x + y \le 4$, within the rectangle.

The inequality $x + y \le 4$ represents the region **below** the line $x + y = 4$, which is a straight line that intersects the axes at:

- $(0, 4)$ when $x = 0$
- $(4, 0)$ when $y = 0$

This line forms a **right triangle** with the coordinate axes, bounded by:

- The x-axis from $(0, 0)$ to $(4, 0)$
- The y-axis from $(0, 0)$ to $(0, 4)$
- The line $x + y = 4$ from $(0, 4)$ to $(4, 0)$

This triangle lies entirely within the rectangle since the rectangle extends to $x = 8$ and $y = 4$, and the triangle is fully contained within that.

The area of this triangle is:

$$
\text{Area of Triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4 \times 4 = 8
$$

---

### Step 3: Compute the Probability

The probability that a randomly selected point $(x, y)$ satisfies $x + y \le 4$ is the ratio of the favorable area to the total area:

$$
\text{Probability} = \frac{\text{Area of Triangle}}{\text{Total Area}} = \frac{8}{32} = \frac{1}{4}
$$

---

### Final Answer

$$
\boxed{\dfrac{1}{4}}
$$

[Assistant]
Explanation:
Okay, so I need to find the probability that a randomly selected point (x, y) within the rectangle defined by 0 ≤ x ≤ 8 and 0 ≤ y ≤ 4 satisfies the condition x + y ≤ 4. Hmm, let me think about how to approach this.

First, probability problems like this usually involve finding the ratio of the favorable area to the total area. So, I should start by calculating the total area of the region where the point can be selected. The rectangle has length 8 along the x-axis and 4 along the y-axis, so the total area is 8 * 4. Let me write that down:

Total area = 8 * 4 = 32.

Now, the next part is figuring out the area of the region where x + y ≤ 4. This seems like a linear inequality, so the region defined by x + y ≤ 4 is a polygon, probably a triangle or a trapezoid. Let me visualize this.

If I consider the coordinate system with x from 0 to 8 and y from 0 to 4, the line x + y = 4 would intersect the axes at (4, 0) and (0, 4). Wait, but since our rectangle goes up to x=8 and y=4, the line x + y = 4 is entirely within the rectangle? Let me check. At x=0, y=4 is the upper limit, so the line goes from (0,4) to (4,0). Beyond x=4, if we go to x=8, y would have to be negative to satisfy x + y = 4, but since y can't be negative, the line only exists from (0,4) to (4,0) within the rectangle. Therefore, the region x + y ≤ 4 within the rectangle is a right triangle with vertices at (0,0), (0,4), and (4,0). Is that right?

Wait, let me confirm. If I take x and y both non-negative, then x + y ≤ 4 is indeed a triangle bounded by the x-axis, y-axis, and the line x + y = 4. So yes, that triangle has vertices at (0,0), (4,0), and (0,4). Therefore, the area of this triangle would be (base * height)/2. The base along the x-axis is 4 units, and the height along the y-axis is also 4 units. So area is (4 * 4)/2 = 8. Wait, but hold on, is that correct?

Wait, actually, the base and height for a right triangle are the two legs. Since the triangle is right-angled at (0,0), the base is from (0,0) to (4,0), which is 4 units, and the height is from (0,0) to (0,4), which is also 4 units. So area is indeed (4*4)/2 = 8. So the area where x + y ≤ 4 is 8?

But wait, let me think again. If the line is x + y = 4, then the region x + y ≤ 4 is all the points below that line. But since our rectangle extends beyond that line (up to x=8 and y=4), but the line only intersects the rectangle at (4,0) and (0,4). Therefore, the area under x + y ≤ 4 within the rectangle is exactly that triangle. Therefore, the area is 8. Therefore, the probability would be 8 divided by the total area 32, which is 8/32 = 1/4. So is the probability 1/4?

Wait, but let me make sure I didn't miss anything. Let me sketch mentally. The rectangle is 8 units wide and 4 units tall. The line x + y = 4 cuts through the rectangle from (0,4) to (4,0). So the area under that line is a triangle with area 8. But is there any other region? For example, if x is between 0 and 4, and y is between 0 and 4 - x. But if x is greater than 4, then 4 - x would be negative, so y would have to be less than a negative number, but since y is non-negative, there's no area there. Therefore, the entire region where x + y ≤ 4 is just that triangle. Therefore, area is 8. So probability is 8/32 = 1/4. So 1/4?

But wait, let me check with integration. Maybe integrating over the region. The area can be calculated as the integral from x=0 to x=4 of (4 - x) dx, since for each x between 0 and 4, y goes from 0 to 4 - x. Then integrating that:

Integral from 0 to 4 of (4 - x) dx = [4x - (x^2)/2] from 0 to 4 = (16 - 8) - 0 = 8. Yep, same result. So area is 8. Therefore, probability is 8/32 = 1/4. So is that the answer?

Wait, but let me think again. The problem says "a point (x,y) is randomly selected such that 0 ≤ x ≤ 8 and 0 ≤ y ≤ 4". So the total area is indeed 8*4=32. The area where x + y ≤ 4 is 8, so probability is 8/32=1/4. So 1/4? But I feel like maybe I need to check if there's another region?

Wait, suppose x is between 0 and 4, and y between 0 and 4 - x. But if x is between 0 and 4, then 4 - x is between 0 and 4. However, since y is allowed up to 4, but in this case, the upper limit for y is 4 - x. So that's correct. But if x is greater than 4, then 4 - x is negative, so there's no y that satisfies x + y ≤ 4 because y is non-negative. Therefore, the entire area is just that triangle. Therefore, 8/32=1/4. So probability is 1/4. But let me check with another approach.

Alternatively, maybe using geometry. The line x + y = 4 divides the rectangle into two regions. The area we want is the triangle, which is half of the square? Wait, no. Wait, if we consider the square from (0,0) to (4,4), then the area of the triangle would be half of that square, which is 8. But our rectangle is larger in x-direction. Wait, but in our case, the rectangle is 8x4, but the line x + y = 4 only intersects the rectangle at (0,4) and (4,0). So the area under the line is a triangle with area 8, as calculated. So 8 over 32 is 1/4. So that seems correct.

But maybe I made a mistake in assuming that the entire region is just that triangle. Let me think again. Suppose I pick a point where x is between 0 and 4, and y between 0 and 4. Then x + y ≤ 4 is the triangle. But if x is between 0 and 4, and y is between 0 and 4, but the rectangle is up to y=4. However, if x is between 0 and 4, and y is between 0 and 4, then the area where x + y ≤ 4 is the triangle. However, if x is between 4 and 8, then even if y is 0, x + y is already 4 or more. Therefore, there is no area in x between 4 and 8 that satisfies x + y ≤ 4. Therefore, the entire area is indeed the triangle with area 8. Therefore, probability is 8/32=1/4.

But wait, let me check with coordinates. Let me take some sample points. For example, (0,0): x + y = 0 ≤ 4, yes. (4,0): 4 + 0 = 4 ≤ 4, yes. (0,4): 0 + 4 = 4 ≤ 4, yes. (2, 2): 2 + 2 = 4 ≤ 4, yes. (3, 1): 3 + 1 = 4 ≤ 4, yes. (5, 0): 5 + 0 = 5 > 4, no. So that seems correct. So the area is indeed that triangle. Therefore, probability is 1/4.

But wait, I remember sometimes when you have inequalities, you have to be careful about the boundaries. But in this case, since the line is x + y = 4, and the region is x + y ≤ 4, which includes the boundary. But since we are dealing with areas, the boundary has zero area, so it doesn't affect the probability. Therefore, the area is indeed 8. Therefore, probability is 1/4.

But let me check again with another method. Let me think of the probability as the double integral over the region. The probability is the area where x + y ≤ 4 divided by total area. So:

Probability = (1 / 32) * ∫ (from x=0 to 4) ∫ (from y=0 to 4 - x) dy dx

Which is (1 / 32) * ∫ (from 0 to 4) (4 - x) dx = (1 / 32) * [4x - (x²)/2] from 0 to 4 = (1 / 32) * (16 - 8) = 8 / 32 = 1/4. Yep, same result. So that seems solid.

Alternatively, maybe using geometric probability. The region is a rectangle, and the condition is a linear inequality. The area is a triangle. So the answer is 1/4. Therefore, I think that is correct.

But wait, just to be thorough, let me think if there's another way someone might approach this problem and maybe get a different answer. For example, someone might think that since x can go up to 8 and y up to 4, maybe they consider the line x + y = 4 and think about the area in terms of some other shape. But no, because beyond x=4, the line x + y = 4 would require y to be negative, which is outside the rectangle. Therefore, the area is only the triangle. So I think that's correct.

Alternatively, someone might think that the region x + y ≤ 4 is a square or something else, but no. For example, if you have x and y both between 0 and 4, then x + y ≤ 4 is a triangle. But since our rectangle is larger in x, but the condition restricts y to be less than 4 - x, which for x beyond 4 is not possible. Therefore, the area is still the triangle. Therefore, 1/4.

Wait, but let me check with coordinates again. Suppose I have a rectangle from (0,0) to (8,4). The line x + y = 4 cuts through (0,4) and (4,0). So the area under the line is a triangle with vertices at (0,0), (0,4), and (4,0). So that triangle's area is ½ * base * height. The base is 4 (from (0,0) to (4,0)), the height is 4 (from (0,0) to (0,4)), so area is ½ * 4 * 4 = 8. So that's correct. Therefore, probability is 8 / 32 = 1/4. So yeah, that seems right.

Alternatively, if someone thought that the region is a square or something else, they might get confused, but I think this is solid. Therefore, I think the answer is 1/4.

But just to make sure, let me think of another approach. Let me consider the probability as the expected value or something else. Wait, maybe not necessary. Alternatively, think of it as uniform distribution over the rectangle. The probability that x + y ≤ 4 is equal to the area where x + y ≤ 4 divided by total area. Which we already calculated. So I think that's the way to go.

Alternatively, maybe using coordinates transformation. Let me set u = x, v = y. Then the condition is u + v ≤ 4. But I don't think that helps. Alternatively, maybe using probability distributions. The joint distribution of x and y is uniform over the rectangle. So the probability is the integral over the region x + y ≤ 4 of the joint PDF, which is 1/(8*4) = 1/32. Therefore, integrating over the region gives the area, which is 8, so 8*(1/32) = 1/4. Yep, same answer.

Therefore, I think I can be confident that the probability is 1/4. Therefore, the answer is \boxed{\dfrac{1}{4}}.

**Final Answer**
\boxed{\dfrac{1}{4}}

To determine the probability that a randomly selected point $(x, y)$ within the rectangle defined by $0 \le x \le 8$ and $0 \le y \le 4$ satisfies the condition $x + y \le 4$, we approach the problem using geometric probability.

---

### Step 1: Total Area of the Region

The region where the point $(x, y)$ can be selected is a rectangle with:

- Width: $8$ (from $x = 0$ to $x = 8$)
- Height: $4$ (from $y = 0$ to $y = 4$)

So, the **total area** of the region is:

$$
\text{Total Area} = 8 \times 4 = 32
$$

---

### Step 2: Area of the Region Satisfying $x + y \le 4$

We are interested in the area of the region where $x + y \le 4$, within the rectangle.

The inequality $x + y \le 4$ represents the region **below** the line $x + y = 4$, which is a straight line that intersects the axes at:

- $(0, 4)$ when $x = 0$
- $(4, 0)$ when $y = 0$

This line forms a **right triangle** with the coordinate axes, bounded by:

- The x-axis from $(0, 0)$ to $(4, 0)$
- The y-axis from $(0, 0)$ to $(0, 4)$
- The line $x + y = 4$ from $(0, 4)$ to $(4, 0)$

This triangle lies entirely within the rectangle since the rectangle extends to $x = 8$ and $y = 4$, and the triangle is fully contained within that.

The area of this triangle is:

$$
\text{Area of Triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4 \times 4 = 8
$$

---

### Step 3: Compute the Probability

The probability that a randomly selected point $(x, y)$ satisfies $x + y \le 4$ is the ratio of the favorable area to the total area:

$$
\text{Probability} = \frac{\text{Area of Triangle}}{\text{Total Area}} = \frac{8}{32} = \frac{1}{4}
$$

---

### Final Answer

$$
\boxed{\dfrac{1}{4}}
$$

Solution:
Rewrite $x+y\leq 4$ as $y\leq 4-x$.  This inequality is satisfied by the points on and under the line $y=4-x$.  Sketching this line along with the $4\times 8$ rectangle determined by the inequalities $0\leq x\leq 8$ and $0\leq y\leq 4$, we find that the points satisfying $x+y\leq 4$ are those in the shaded triangle (see figure).  The  area of the triangle is $\frac{1}{2}(4)(4)=8$ square units, and the area of the rectangle is $(4)(8)=32$ square units, so the probability that a randomly selected point would fall in the shaded triangle is $\boxed{\frac{1}{4}}$.

[asy] import graph; size(200); defaultpen(linewidth(0.7)+fontsize(10));

dotfactor=4;

real f(real x) { return 4-x; }

pair A=(0,4), B=(8,4), C=(8,0), D=(0,0); pair[] dots={A,B,C,D};

fill(A--(4,0)--D--cycle,gray(0.7)); draw(A--B--C);

xaxis(xmin=-3,xmax=9,Ticks(" ",1.0, begin=false, end=false, NoZero, Size=3), Arrows(4), above=true);

yaxis(ymin=-1,ymax=5,Ticks(" ",1.0,begin=false, end=false, NoZero, Size=3), Arrows(4), above=true);

draw(graph(f,-0.8,4.5),Arrows(4)); label("$x+y=4$",(-2.2,5.2));[/asy]


Explanation:
Okay, so I need to find the sum of all multiples of 7 between 100 and 200. Hmm, let me think about how to approach this. First, I remember that when you want to find the sum of a series of numbers, especially an arithmetic sequence, there's a formula for that. But first, I need to figure out which numbers between 100 and 200 are multiples of 7.

Let me start by recalling that a multiple of 7 is any number that can be expressed as 7 times some integer. So, I need to find all numbers in the range from 100 to 200 that fit this description.

First, maybe I should find the smallest multiple of 7 that is greater than or equal to 100. How do I do that? Well, I can divide 100 by 7 and then round up to the nearest whole number. Let me try that.

Calculating 100 divided by 7... 7 times 14 is 98, right? 7*14=98. Then 100-98=2, so 100 divided by 7 is 14 with a remainder of 2. That means the next multiple of 7 after 98 is 98 + 7 = 105. So, 105 is the first multiple of 7 that's greater than or equal to 100.

Now, I need to check if 105 is less than 200. Yes, it is. So, the multiples of 7 between 100 and 200 start at 105. Now, I need to find the last multiple of 7 that is less than or equal to 200. Let me do a similar calculation.

Divide 200 by 7. Let me see, 7*28 is 196, because 7*25=175, 7*28=175 + 7*3=175+21=196. Then 200 - 196 = 4, so 200 divided by 7 is 28 with a remainder of 4. Therefore, the last multiple of 7 before 200 is 196.

So, the multiples of 7 between 100 and 200 are 105, 112, 119, ..., up to 196. Now, I need to figure out how many terms are in this sequence. Once I have the number of terms, I can use the formula for the sum of an arithmetic sequence.

The formula for the sum of an arithmetic sequence is S = n/2 * (a1 + an), where n is the number of terms, a1 is the first term, and an is the last term.

So, first, let me find n. The sequence starts at 105 and ends at 196, with a common difference of 7.

To find the number of terms, I can use the formula for the nth term of an arithmetic sequence: an = a1 + (n - 1)d. Here, an is 196, a1 is 105, d is 7.

Plugging in the values: 196 = 105 + (n - 1)*7. Let me solve for n.

Subtract 105 from both sides: 196 - 105 = (n - 1)*7.

Calculating 196 - 105: 196 - 100 is 96, minus 5 more is 91. So, 91 = (n - 1)*7.

Divide both sides by 7: 91 / 7 = n - 1.

Calculating 91 divided by 7: 7*13 is 91. So, 13 = n - 1.

Therefore, n = 13 + 1 = 14.

Wait, so there are 14 terms? Let me check that. Let me list them out to make sure.

Starting at 105:

105, 112, 119, 126, 133, 140, 147, 154, 161, 168, 175, 182, 189, 196.

Let me count them.

1. 105
2. 112
3. 119
4. 126
5. 133
6. 140
7. 147
8. 154
9. 161
10. 168
11. 175
12. 182
13. 189
14. 196

Yes, that's 14 terms. Okay, so n is 14.

Now, using the sum formula: S = n/2 * (a1 + an).

So, plugging in the numbers: S = 14/2 * (105 + 196).

Calculating 14 divided by 2 is 7. Then, 105 + 196. Let me add those.

105 + 196: 100 + 196 is 296, plus 5 is 301.

So, S = 7 * 301.

Now, calculating 7 * 301. Let me break that down.

7 * 300 = 2100, and 7 * 1 = 7. So, 2100 + 7 = 2107.

Therefore, the sum of all multiples of 7 between 100 and 200 is 2107.

Wait, let me verify this with another method to make sure I didn't make a mistake.

Another way to approach this is to find how many multiples of 7 there are between 100 and 200, then use the formula for the sum.

Alternatively, I can think of the multiples of 7 as 7 times integers. So, if I can find the integers k such that 100 ≤ 7k ≤ 200, then the sum would be 7 times the sum of those integers.

So, solving for k:

Lower bound: 100 ≤ 7k ⇒ k ≥ 100/7 ≈ 14.2857. Since k must be an integer, the smallest integer k is 15.

Upper bound: 7k ≤ 200 ⇒ k ≤ 200/7 ≈ 28.5714. So, the largest integer k is 28.

Therefore, k ranges from 15 to 28 inclusive.

Now, the number of terms here is 28 - 15 + 1 = 14. Which matches the previous count.

So, the sum is 7*(sum from k=15 to k=28 of k).

So, let me compute the sum of integers from 15 to 28.

The formula for the sum from 1 to n is n(n + 1)/2. Therefore, the sum from 15 to 28 is equal to the sum from 1 to 28 minus the sum from 1 to 14.

Calculating sum from 1 to 28: 28*29/2 = (28/2)*29 = 14*29. Let me compute that. 14*29: 10*29=290, 4*29=116, so 290 + 116 = 406.

Sum from 1 to 14: 14*15/2 = 7*15 = 105.

Therefore, the sum from 15 to 28 is 406 - 105 = 301.

Then, multiplying by 7 gives 7*301 = 2107.

Same result as before. So that seems to confirm that the answer is 2107.

Alternatively, maybe I can check with another approach. Let me try adding the numbers step by step, but that might take longer.

But just to be thorough, let me add the numbers I listed earlier:

105, 112, 119, 126, 133, 140, 147, 154, 161, 168, 175, 182, 189, 196.

Let me pair them from the start and end to make addition easier.

First pair: 105 + 196 = 301

Second pair: 112 + 189 = 301

Third pair: 119 + 182 = 301

Fourth pair: 126 + 175 = 301

Fifth pair: 133 + 168 = 301

Sixth pair: 140 + 161 = 301

Seventh pair: 147 + 154 = 301

Wait, how many pairs is that? Let me check. There are 14 numbers, so 7 pairs. Each pair adds up to 301. Therefore, total sum is 7*301 = 2107.

Same result again. So that's three different methods giving me the same answer. That gives me confidence that 2107 is correct.

But just to be absolutely sure, let me check one of the pairings. For example, 105 + 196: 100 + 196 is 296, plus 5 is 301. Correct.

Another pair: 112 + 189. 112 + 189: 112 + 180 = 292, plus 9 is 301. Correct.

Another pair: 119 + 182. 119 + 182: 100 + 182 = 282, 19 + 0 = 19, so 282 + 19 = 301. Correct.

Another pair: 126 + 175. 126 + 175: 126 + 175. Let me do 126 + 175. 126 + 175 = 301. Yes, because 126 + 175 = 126 + 175. 126 + 175 = 301.

Similarly, 133 + 168: 133 + 168. 133 + 168 = 301.

140 + 161: 140 + 161 = 301.

147 + 154: 147 + 154. 147 + 154 = 301.

So all pairs add up to 301. Therefore, 7 pairs give 7*301 = 2107.

Therefore, I can be confident that the answer is 2107.

But just to make sure I didn't make any calculation errors in the earlier steps, let me check the number of terms again.

Starting from 105, which is 7*15, up to 196, which is 7*28. So, the number of terms is 28 - 15 + 1 = 14. Correct.

Alternatively, if I think of the multiples of 7 between 100 and 200, inclusive, how many are there?

But wait, the original question says "between 100 and 200". Depending on interpretation, sometimes "between" can be exclusive, meaning not including 100 and 200. But in this case, since 100 is not a multiple of 7, and 200 is not a multiple of 7, the numbers between 100 and 200 would still include 105 up to 196. So, even if it's exclusive, since 100 and 200 aren't multiples of 7, the count remains the same.

But just to check, if the question had said "between 100 and 200 inclusive", then we would have to check if 100 or 200 are multiples of 7. But since they aren't, the answer remains the same.

Therefore, the sum is indeed 2107.

Another way to think about it: if I use the formula for the sum of an arithmetic series, which is S = n/2 * (2a + (n - 1)d). Let me try that formula as well.

Here, a = 105, d = 7, n = 14.

So, S = 14/2 * [2*105 + (14 - 1)*7]

Calculates to 7 * [210 + 13*7]

Calculating 13*7: 91.

So, 210 + 91 = 301.

Then, 7 * 301 = 2107.

Same result.

Therefore, regardless of the method, the answer is consistent.

I think that's thorough enough. I can't see any mistakes in my reasoning. All methods converge to 2107.

**Final Answer**
The sum of all multiples of 7 between 100 and 200 is \boxed{2107}.

To find the **sum of all multiples of 7 between 100 and 200**, we can approach the problem using the properties of arithmetic sequences.

---

### Step 1: Identify the First and Last Multiples of 7 in the Range

We are looking for all **multiples of 7** that lie **between 100 and 200** (inclusive). That is, we want all numbers of the form $ 7k $ such that:

$$
100 \leq 7k \leq 200
$$

Solving for $ k $:

- $ k \geq \frac{100}{7} \approx 14.2857 $ → So the smallest integer $ k $ is **15**
- $ k \leq \frac{200}{7} \approx 28.5714 $ → So the largest integer $ k $ is **28**

Thus, the values of $ k $ range from **15 to 28**, inclusive.

---

### Step 2: Determine the Number of Terms

The number of integers from 15 to 28 inclusive is:

$$
n = 28 - 15 + 1 = 14
$$

So, there are **14 terms** in the sequence.

---

### Step 3: Use the Arithmetic Series Sum Formula

The sequence of multiples of 7 is:

$$
7 \times 15, 7 \times 16, \dots, 7 \times 28
$$

This is an arithmetic sequence with:

- First term $ a = 7 \times 15 = 105 $
- Last term $ l = 7 \times 28 = 196 $
- Number of terms $ n = 14 $

The sum $ S $ of an arithmetic sequence is given by:

$$
S = \frac{n}{2} \times (a + l)
$$

Substitute the values:

$$
S = \frac{14}{2} \times (105 + 196) = 7 \times 301 = 2107
$$

---

### Final Answer

$$
\boxed{2107}
$$

[Assistant]
Explanation:
Okay, so I need to find the sum of all multiples of 7 between 100 and 200. Hmm, let me think about how to approach this. First, I remember that when you want to find the sum of a series of numbers, especially an arithmetic sequence, there's a formula for that. But first, I need to figure out which numbers between 100 and 200 are multiples of 7.

Let me start by recalling that a multiple of 7 is any number that can be expressed as 7 times some integer. So, I need to find all numbers in the range from 100 to 200 that fit this description.

First, maybe I should find the smallest multiple of 7 that is greater than or equal to 100. How do I do that? Well, I can divide 100 by 7 and then round up to the nearest whole number. Let me try that.

Calculating 100 divided by 7... 7 times 14 is 98, right? 7*14=98. Then 100-98=2, so 100 divided by 7 is 14 with a remainder of 2. That means the next multiple of 7 after 98 is 98 + 7 = 105. So, 105 is the first multiple of 7 that's greater than or equal to 100.

Now, I need to check if 105 is less than 200. Yes, it is. So, the multiples of 7 between 100 and 200 start at 105. Now, I need to find the last multiple of 7 that is less than or equal to 200. Let me do a similar calculation.

Divide 200 by 7. Let me see, 7*28 is 196, because 7*25=175, 7*28=175 + 7*3=175+21=196. Then 200 - 196 = 4, so 200 divided by 7 is 28 with a remainder of 4. Therefore, the last multiple of 7 before 200 is 196.

So, the multiples of 7 between 100 and 200 are 105, 112, 119, ..., up to 196. Now, I need to figure out how many terms are in this sequence. Once I have the number of terms, I can use the formula for the sum of an arithmetic sequence.

The formula for the sum of an arithmetic sequence is S = n/2 * (a1 + an), where n is the number of terms, a1 is the first term, and an is the last term.

So, first, let me find n. The sequence starts at 105 and ends at 196, with a common difference of 7.

To find the number of terms, I can use the formula for the nth term of an arithmetic sequence: an = a1 + (n - 1)d. Here, an is 196, a1 is 105, d is 7.

Plugging in the values: 196 = 105 + (n - 1)*7. Let me solve for n.

Subtract 105 from both sides: 196 - 105 = (n - 1)*7.

Calculating 196 - 105: 196 - 100 is 96, minus 5 more is 91. So, 91 = (n - 1)*7.

Divide both sides by 7: 91 / 7 = n - 1.

Calculating 91 divided by 7: 7*13 is 91. So, 13 = n - 1.

Therefore, n = 13 + 1 = 14.

Wait, so there are 14 terms? Let me check that. Let me list them out to make sure.

Starting at 105:

105, 112, 119, 126, 133, 140, 147, 154, 161, 168, 175, 182, 189, 196.

Let me count them.

1. 105
2. 112
3. 119
4. 126
5. 133
6. 140
7. 147
8. 154
9. 161
10. 168
11. 175
12. 182
13. 189
14. 196

Yes, that's 14 terms. Okay, so n is 14.

Now, using the sum formula: S = n/2 * (a1 + an).

So, plugging in the numbers: S = 14/2 * (105 + 196).

Calculating 14 divided by 2 is 7. Then, 105 + 196. Let me add those.

105 + 196: 100 + 196 is 296, plus 5 is 301.

So, S = 7 * 301.

Now, calculating 7 * 301. Let me break that down.

7 * 300 = 2100, and 7 * 1 = 7. So, 2100 + 7 = 2107.

Therefore, the sum of all multiples of 7 between 100 and 200 is 2107.

Wait, let me verify this with another method to make sure I didn't make a mistake.

Another way to approach this is to find how many multiples of 7 there are between 100 and 200, then use the formula for the sum.

Alternatively, I can think of the multiples of 7 as 7 times integers. So, if I can find the integers k such that 100 ≤ 7k ≤ 200, then the sum would be 7 times the sum of those integers.

So, solving for k:

Lower bound: 100 ≤ 7k ⇒ k ≥ 100/7 ≈ 14.2857. Since k must be an integer, the smallest integer k is 15.

Upper bound: 7k ≤ 200 ⇒ k ≤ 200/7 ≈ 28.5714. So, the largest integer k is 28.

Therefore, k ranges from 15 to 28 inclusive.

Now, the number of terms here is 28 - 15 + 1 = 14. Which matches the previous count.

So, the sum is 7*(sum from k=15 to k=28 of k).

So, let me compute the sum of integers from 15 to 28.

The formula for the sum from 1 to n is n(n + 1)/2. Therefore, the sum from 15 to 28 is equal to the sum from 1 to 28 minus the sum from 1 to 14.

Calculating sum from 1 to 28: 28*29/2 = (28/2)*29 = 14*29. Let me compute that. 14*29: 10*29=290, 4*29=116, so 290 + 116 = 406.

Sum from 1 to 14: 14*15/2 = 7*15 = 105.

Therefore, the sum from 15 to 28 is 406 - 105 = 301.

Then, multiplying by 7 gives 7*301 = 2107.

Same result as before. So that seems to confirm that the answer is 2107.

Alternatively, maybe I can check with another approach. Let me try adding the numbers step by step, but that might take longer.

But just to be thorough, let me add the numbers I listed earlier:

105, 112, 119, 126, 133, 140, 147, 154, 161, 168, 175, 182, 189, 196.

Let me pair them from the start and end to make addition easier.

First pair: 105 + 196 = 301

Second pair: 112 + 189 = 301

Third pair: 119 + 182 = 301

Fourth pair: 126 + 175 = 301

Fifth pair: 133 + 168 = 301

Sixth pair: 140 + 161 = 301

Seventh pair: 147 + 154 = 301

Wait, how many pairs is that? Let me check. There are 14 numbers, so 7 pairs. Each pair adds up to 301. Therefore, total sum is 7*301 = 2107.

Same result again. So that's three different methods giving me the same answer. That gives me confidence that 2107 is correct.

But just to be absolutely sure, let me check one of the pairings. For example, 105 + 196: 100 + 196 is 296, plus 5 is 301. Correct.

Another pair: 112 + 189. 112 + 189: 112 + 180 = 292, plus 9 is 301. Correct.

Another pair: 119 + 182. 119 + 182: 100 + 182 = 282, 19 + 0 = 19, so 282 + 19 = 301. Correct.

Another pair: 126 + 175. 126 + 175: 126 + 175. Let me do 126 + 175. 126 + 175 = 301. Yes, because 126 + 175 = 126 + 175. 126 + 175 = 301.

Similarly, 133 + 168: 133 + 168. 133 + 168 = 301.

140 + 161: 140 + 161 = 301.

147 + 154: 147 + 154. 147 + 154 = 301.

So all pairs add up to 301. Therefore, 7 pairs give 7*301 = 2107.

Therefore, I can be confident that the answer is 2107.

But just to make sure I didn't make any calculation errors in the earlier steps, let me check the number of terms again.

Starting from 105, which is 7*15, up to 196, which is 7*28. So, the number of terms is 28 - 15 + 1 = 14. Correct.

Alternatively, if I think of the multiples of 7 between 100 and 200, inclusive, how many are there?

But wait, the original question says "between 100 and 200". Depending on interpretation, sometimes "between" can be exclusive, meaning not including 100 and 200. But in this case, since 100 is not a multiple of 7, and 200 is not a multiple of 7, the numbers between 100 and 200 would still include 105 up to 196. So, even if it's exclusive, since 100 and 200 aren't multiples of 7, the count remains the same.

But just to check, if the question had said "between 100 and 200 inclusive", then we would have to check if 100 or 200 are multiples of 7. But since they aren't, the answer remains the same.

Therefore, the sum is indeed 2107.

Another way to think about it: if I use the formula for the sum of an arithmetic series, which is S = n/2 * (2a + (n - 1)d). Let me try that formula as well.

Here, a = 105, d = 7, n = 14.

So, S = 14/2 * [2*105 + (14 - 1)*7]

Calculates to 7 * [210 + 13*7]

Calculating 13*7: 91.

So, 210 + 91 = 301.

Then, 7 * 301 = 2107.

Same result.

Therefore, regardless of the method, the answer is consistent.

I think that's thorough enough. I can't see any mistakes in my reasoning. All methods converge to 2107.

**Final Answer**
The sum of all multiples of 7 between 100 and 200 is \boxed{2107}.

To find the **sum of all multiples of 7 between 100 and 200**, we can approach the problem using the properties of arithmetic sequences.

---

### Step 1: Identify the First and Last Multiples of 7 in the Range

We are looking for all **multiples of 7** that lie **between 100 and 200** (inclusive). That is, we want all numbers of the form $ 7k $ such that:

$$
100 \leq 7k \leq 200
$$

Solving for $ k $:

- $ k \geq \frac{100}{7} \approx 14.2857 $ → So the smallest integer $ k $ is **15**
- $ k \leq \frac{200}{7} \approx 28.5714 $ → So the largest integer $ k $ is **28**

Thus, the values of $ k $ range from **15 to 28**, inclusive.

---

### Step 2: Determine the Number of Terms

The number of integers from 15 to 28 inclusive is:

$$
n = 28 - 15 + 1 = 14
$$

So, there are **14 terms** in the sequence.

---

### Step 3: Use the Arithmetic Series Sum Formula

The sequence of multiples of 7 is:

$$
7 \times 15, 7 \times 16, \dots, 7 \times 28
$$

This is an arithmetic sequence with:

- First term $ a = 7 \times 15 = 105 $
- Last term $ l = 7 \times 28 = 196 $
- Number of terms $ n = 14 $

The sum $ S $ of an arithmetic sequence is given by:

$$
S = \frac{n}{2} \times (a + l)
$$

Substitute the values:

$$
S = \frac{14}{2} \times (105 + 196) = 7 \times 301 = 2107
$$

---

### Final Answer

$$
\boxed{2107}
$$

Solution:
The smallest multiple of 7 between 100 and 200 is 105, and the largest multiple is 196.  Thus, we want to find the sum of the arithmetic series $105 + 112 + \dots + 196$.

The $n^{\text{th}}$ term in this arithmetic sequence is $105 + 7(n - 1) = 7n + 98$.  If $7n + 98 = 196$, then $n = 14$, so the number of terms in this sequence is 14.

The sum of an arithmetic series is equal to the average of the first and last term, multiplied by the number of terms, so the sum is $(105 + 196)/2 \cdot 14 = \boxed{2107}$.


Explanation:
Okay, so I need to rewrite the quadratic equation $x^2 + 2.6x + 3.6$ in the form $(x + b)^2 + c$, where $b$ and $c$ are constants. Then, I have to find $b + c$ as a decimal. Hmm, let me think about how to approach this.

First, I remember that completing the square is the method used to convert a quadratic equation from standard form $ax^2 + bx + c$ to vertex form $a(x + b)^2 + c$. Wait, actually, in this case, the coefficient of $x^2$ is 1, so it's simpler. Let me recall the steps for completing the square.

The general process is: take the coefficient of $x$, divide it by 2, square it, and add and subtract that value to complete the square. Let me try applying that here.

Given the quadratic $x^2 + 2.6x + 3.6$. Let me write it down:

$x^2 + 2.6x + 3.6$

First, the coefficient of $x$ is 2.6. So, half of that is 1.3, right? Then, squaring that gives me $1.3^2$. Let me calculate that. 1.3 times 1.3... 1 times 1 is 1, 1 times 0.3 is 0.3, 0.3 times 1 is 0.3, and 0.3 times 0.3 is 0.09. Adding those up: 1 + 0.3 + 0.3 + 0.09 = 1.69. So, $1.3^2 = 1.69$.

So, to complete the square, I need to add and subtract 1.69. Let me do that:

$x^2 + 2.6x + 1.69 - 1.69 + 3.6$

Wait, but the original equation is $x^2 + 2.6x + 3.6$. So, if I add and subtract 1.69, the equation becomes:

$(x^2 + 2.6x + 1.69) - 1.69 + 3.6$

Simplifying the terms inside the parentheses first. Since $x^2 + 2.6x + 1.69$ is a perfect square trinomial, it can be written as $(x + 1.3)^2$. Let me check that:

$(x + 1.3)^2 = x^2 + 2*1.3*x + (1.3)^2 = x^2 + 2.6x + 1.69$. Yes, that's correct.

So, substituting back into the equation:

$(x + 1.3)^2 - 1.69 + 3.6$

Now, let me compute the constants: -1.69 + 3.6. Let me do that subtraction. 3.6 minus 1.69. Hmm, 3.6 is the same as 3.60. So, 3.60 - 1.69. Subtracting 1.69 from 3.60:

Starting from the right, 0 - 9. Can't do that, so borrow. 10 - 9 = 1. Then, the next digit: 5 (since we borrowed 1) - 6. Again, can't do that, so borrow. 15 - 6 = 9. Then, the next digit: 2 (since we borrowed 1) - 1 = 1. So, the result is 1.91? Wait, let me check again.

Wait, 3.60 minus 1.69:

Let me write it vertically:

3.60
-1.69
=?

Starting from the hundredths place: 0 - 9. Can't do, borrow from tenths place. Tenths place is 6, so becomes 5, and hundredths becomes 10. 10 - 9 = 1.

Tenths place: 5 - 6. Can't do, borrow from units place. Units place is 3, becomes 2, tenths becomes 15. 15 - 6 = 9.

Units place: 2 - 1 = 1.

So, the result is 1.91. Yes, that's correct. So, -1.69 + 3.6 = 1.91.

Therefore, the equation becomes:

$(x + 1.3)^2 + 1.91$

So, comparing this with the form $(x + b)^2 + c$, we can see that $b = 1.3$ and $c = 1.91$. Therefore, $b + c = 1.3 + 1.91$.

Calculating that: 1.3 + 1.91. Let me add them. 1 + 1 = 2, 0.3 + 0.91 = 1.21. So total is 2 + 1.21 = 3.21. Wait, no. Wait, 1.3 is 1.30, so adding 1.30 + 1.91:

1.30
+1.91
=?

Adding the hundredths: 0 + 1 = 1.
Tenths: 3 + 9 = 12. So, carry over 1.
Units: 1 + 1 + 1 (carry) = 3.
So, total is 3.21. Yes, that's correct. So, $b + c = 3.21$.

But let me double-check my steps to make sure I didn't make any mistakes. Let me verify by expanding $(x + 1.3)^2 + 1.91$ and see if it equals the original quadratic.

Expanding $(x + 1.3)^2$:

$x^2 + 2*1.3*x + (1.3)^2 = x^2 + 2.6x + 1.69$. Then adding 1.91 gives $x^2 + 2.6x + 1.69 + 1.91 = x^2 + 2.6x + (1.69 + 1.91)$. Let me compute 1.69 + 1.91. 1.69 + 1.91: 1 + 1 = 2, 0.69 + 0.91 = 1.60. So total is 2 + 1.60 = 3.60. Which matches the original constant term, 3.6. Perfect! So that checks out.

Therefore, my values for $b$ and $c$ are correct. So $b = 1.3$ and $c = 1.91$, so $b + c = 1.3 + 1.91 = 3.21$.

Wait, but the question says "What is $b + c$ (as a decimal)?" So, 3.21 is already a decimal. So, is that the answer? Let me check once more.

Alternatively, maybe I made a mistake in calculating $ -1.69 + 3.6 $. Let me verify that again. 3.6 minus 1.69. Let me think of 3.6 as 3.60. So, 3.60 - 1.69. If I subtract 1.69 from 3.60, that's the same as 3.60 - 1.69. Let me do this another way. 3.60 - 1.69 = (3.60 - 1.00) - 0.69 = 2.60 - 0.69 = 1.91. Yes, same result. So that's correct.

Alternatively, maybe I can use fractions to check? Let me try converting the decimals to fractions to see if that gives the same result.

Original quadratic: $x^2 + 2.6x + 3.6$. Let me convert 2.6 and 3.6 to fractions.

2.6 is equal to 26/10, which simplifies to 13/5. Similarly, 3.6 is 36/10, which simplifies to 18/5.

So, the quadratic is $x^2 + \frac{13}{5}x + \frac{18}{5}$.

Now, completing the square. The coefficient of x is 13/5. Half of that is 13/10, which is 1.3, same as before. Squaring that gives (13/10)^2 = 169/100 = 1.69. So, adding and subtracting 169/100:

$x^2 + \frac{13}{5}x + \frac{169}{100} - \frac{169}{100} + \frac{18}{5}$

Which becomes:

$\left(x + \frac{13}{10}\right)^2 + \left(-\frac{169}{100} + \frac{18}{5}\right)$

Now, compute the constants:

Convert $\frac{18}{5}$ to 100 denominator: $\frac{18}{5} = \frac{360}{100}$. Then, $-\frac{169}{100} + \frac{360}{100} = \frac{191}{100} = 1.91$. So, same result as before. Therefore, $c = 191/100 = 1.91$, and $b = 13/10 = 1.3$. Therefore, $b + c = 13/10 + 191/100 = (130 + 191)/100 = 321/100 = 3.21$. So, same answer. Therefore, this confirms that my answer is correct.

Alternatively, maybe I can check by plugging in a value for x into both the original equation and the completed square form to see if they match. Let me try x = 0.

Original equation: $0^2 + 2.6*0 + 3.6 = 3.6$.

Completed square form: $(0 + 1.3)^2 + 1.91 = (1.3)^2 + 1.91 = 1.69 + 1.91 = 3.6$. Correct.

Another value, say x = -1.

Original equation: (-1)^2 + 2.6*(-1) + 3.6 = 1 - 2.6 + 3.6 = (1 + 3.6) - 2.6 = 4.6 - 2.6 = 2.0.

Completed square form: (-1 + 1.3)^2 + 1.91 = (0.3)^2 + 1.91 = 0.09 + 1.91 = 2.0. Correct again.

Another test: x = -1.3. Let's see.

Original equation: (-1.3)^2 + 2.6*(-1.3) + 3.6. Let's compute:

(-1.3)^2 = 1.69

2.6*(-1.3) = -3.38

So total: 1.69 - 3.38 + 3.6 = (1.69 + 3.6) - 3.38 = 5.29 - 3.38 = 1.91.

Completed square form: (-1.3 + 1.3)^2 + 1.91 = 0^2 + 1.91 = 1.91. Correct. Perfect.

So, all my checks confirm that the completed square form is correct, and therefore $b + c = 3.21$.

Therefore, the answer is 3.21. Wait, but the question says "as a decimal". Since 3.21 is already a decimal, that's fine. But maybe they want it in some specific decimal form? Like rounded or something? But since the original coefficients were given as decimals with one decimal place (2.6) and one decimal place (3.6), but in the process, we had 1.3 and 1.91. But since 1.91 is exact, and 1.3 is exact, adding them gives 3.21 exactly. So, no rounding needed. Therefore, 3.21 is the exact value of $b + c$.

Therefore, I think that's the answer. I don't see any mistakes in my reasoning, and multiple checks confirm that the completed square is correct. So, I think I can confidently say that $b + c = 3.21$.

**Final Answer**
The value of $b + c$ is \boxed{3.21}.

To rewrite the quadratic expression $ x^2 + 2.6x + 3.6 $ in the form $ (x + b)^2 + c $, we use the method of **completing the square**.

---

### Step 1: Identify the coefficient of $ x $

The given quadratic is:

$$
x^2 + 2.6x + 3.6
$$

The coefficient of $ x $ is $ 2.6 $. To complete the square, we take half of this coefficient and square it:

$$
\left( \frac{2.6}{2} \right)^2 = (1.3)^2 = 1.69
$$

---

### Step 2: Add and subtract the square

We add and subtract $ 1.69 $ to the expression:

$$
x^2 + 2.6x + 1.69 - 1.69 + 3.6
$$

Group the perfect square trinomial:

$$
(x^2 + 2.6x + 1.69) - 1.69 + 3.6
$$

$$
= (x + 1.3)^2 + (3.6 - 1.69)
$$

---

### Step 3: Simplify the constant term

$$
3.6 - 1.69 = 1.91
$$

So the expression becomes:

$$
(x + 1.3)^2 + 1.91
$$

---

### Step 4: Identify $ b $ and $ c $

From the form $ (x + b)^2 + c $, we identify:

- $ b = 1.3 $
- $ c = 1.91 $

---

### Step 5: Compute $ b + c $

$$
b + c = 1.3 + 1.91 = 3.21
$$

---

### Final Answer

$$
\boxed{3.21}
$$

[Assistant]
Explanation:
Okay, so I need to rewrite the quadratic equation $x^2 + 2.6x + 3.6$ in the form $(x + b)^2 + c$, where $b$ and $c$ are constants. Then, I have to find $b + c$ as a decimal. Hmm, let me think about how to approach this.

First, I remember that completing the square is the method used to convert a quadratic equation from standard form $ax^2 + bx + c$ to vertex form $a(x + b)^2 + c$. Wait, actually, in this case, the coefficient of $x^2$ is 1, so it's simpler. Let me recall the steps for completing the square.

The general process is: take the coefficient of $x$, divide it by 2, square it, and add and subtract that value to complete the square. Let me try applying that here.

Given the quadratic $x^2 + 2.6x + 3.6$. Let me write it down:

$x^2 + 2.6x + 3.6$

First, the coefficient of $x$ is 2.6. So, half of that is 1.3, right? Then, squaring that gives me $1.3^2$. Let me calculate that. 1.3 times 1.3... 1 times 1 is 1, 1 times 0.3 is 0.3, 0.3 times 1 is 0.3, and 0.3 times 0.3 is 0.09. Adding those up: 1 + 0.3 + 0.3 + 0.09 = 1.69. So, $1.3^2 = 1.69$.

So, to complete the square, I need to add and subtract 1.69. Let me do that:

$x^2 + 2.6x + 1.69 - 1.69 + 3.6$

Wait, but the original equation is $x^2 + 2.6x + 3.6$. So, if I add and subtract 1.69, the equation becomes:

$(x^2 + 2.6x + 1.69) - 1.69 + 3.6$

Simplifying the terms inside the parentheses first. Since $x^2 + 2.6x + 1.69$ is a perfect square trinomial, it can be written as $(x + 1.3)^2$. Let me check that:

$(x + 1.3)^2 = x^2 + 2*1.3*x + (1.3)^2 = x^2 + 2.6x + 1.69$. Yes, that's correct.

So, substituting back into the equation:

$(x + 1.3)^2 - 1.69 + 3.6$

Now, let me compute the constants: -1.69 + 3.6. Let me do that subtraction. 3.6 minus 1.69. Hmm, 3.6 is the same as 3.60. So, 3.60 - 1.69. Subtracting 1.69 from 3.60:

Starting from the right, 0 - 9. Can't do that, so borrow. 10 - 9 = 1. Then, the next digit: 5 (since we borrowed 1) - 6. Again, can't do that, so borrow. 15 - 6 = 9. Then, the next digit: 2 (since we borrowed 1) - 1 = 1. So, the result is 1.91? Wait, let me check again.

Wait, 3.60 minus 1.69:

Let me write it vertically:

3.60
-1.69
=?

Starting from the hundredths place: 0 - 9. Can't do, borrow from tenths place. Tenths place is 6, so becomes 5, and hundredths becomes 10. 10 - 9 = 1.

Tenths place: 5 - 6. Can't do, borrow from units place. Units place is 3, becomes 2, tenths becomes 15. 15 - 6 = 9.

Units place: 2 - 1 = 1.

So, the result is 1.91. Yes, that's correct. So, -1.69 + 3.6 = 1.91.

Therefore, the equation becomes:

$(x + 1.3)^2 + 1.91$

So, comparing this with the form $(x + b)^2 + c$, we can see that $b = 1.3$ and $c = 1.91$. Therefore, $b + c = 1.3 + 1.91$.

Calculating that: 1.3 + 1.91. Let me add them. 1 + 1 = 2, 0.3 + 0.91 = 1.21. So total is 2 + 1.21 = 3.21. Wait, no. Wait, 1.3 is 1.30, so adding 1.30 + 1.91:

1.30
+1.91
=?

Adding the hundredths: 0 + 1 = 1.
Tenths: 3 + 9 = 12. So, carry over 1.
Units: 1 + 1 + 1 (carry) = 3.
So, total is 3.21. Yes, that's correct. So, $b + c = 3.21$.

But let me double-check my steps to make sure I didn't make any mistakes. Let me verify by expanding $(x + 1.3)^2 + 1.91$ and see if it equals the original quadratic.

Expanding $(x + 1.3)^2$:

$x^2 + 2*1.3*x + (1.3)^2 = x^2 + 2.6x + 1.69$. Then adding 1.91 gives $x^2 + 2.6x + 1.69 + 1.91 = x^2 + 2.6x + (1.69 + 1.91)$. Let me compute 1.69 + 1.91. 1.69 + 1.91: 1 + 1 = 2, 0.69 + 0.91 = 1.60. So total is 2 + 1.60 = 3.60. Which matches the original constant term, 3.6. Perfect! So that checks out.

Therefore, my values for $b$ and $c$ are correct. So $b = 1.3$ and $c = 1.91$, so $b + c = 1.3 + 1.91 = 3.21$.

Wait, but the question says "What is $b + c$ (as a decimal)?" So, 3.21 is already a decimal. So, is that the answer? Let me check once more.

Alternatively, maybe I made a mistake in calculating $ -1.69 + 3.6 $. Let me verify that again. 3.6 minus 1.69. Let me think of 3.6 as 3.60. So, 3.60 - 1.69. If I subtract 1.69 from 3.60, that's the same as 3.60 - 1.69. Let me do this another way. 3.60 - 1.69 = (3.60 - 1.00) - 0.69 = 2.60 - 0.69 = 1.91. Yes, same result. So that's correct.

Alternatively, maybe I can use fractions to check? Let me try converting the decimals to fractions to see if that gives the same result.

Original quadratic: $x^2 + 2.6x + 3.6$. Let me convert 2.6 and 3.6 to fractions.

2.6 is equal to 26/10, which simplifies to 13/5. Similarly, 3.6 is 36/10, which simplifies to 18/5.

So, the quadratic is $x^2 + \frac{13}{5}x + \frac{18}{5}$.

Now, completing the square. The coefficient of x is 13/5. Half of that is 13/10, which is 1.3, same as before. Squaring that gives (13/10)^2 = 169/100 = 1.69. So, adding and subtracting 169/100:

$x^2 + \frac{13}{5}x + \frac{169}{100} - \frac{169}{100} + \frac{18}{5}$

Which becomes:

$\left(x + \frac{13}{10}\right)^2 + \left(-\frac{169}{100} + \frac{18}{5}\right)$

Now, compute the constants:

Convert $\frac{18}{5}$ to 100 denominator: $\frac{18}{5} = \frac{360}{100}$. Then, $-\frac{169}{100} + \frac{360}{100} = \frac{191}{100} = 1.91$. So, same result as before. Therefore, $c = 191/100 = 1.91$, and $b = 13/10 = 1.3$. Therefore, $b + c = 13/10 + 191/100 = (130 + 191)/100 = 321/100 = 3.21$. So, same answer. Therefore, this confirms that my answer is correct.

Alternatively, maybe I can check by plugging in a value for x into both the original equation and the completed square form to see if they match. Let me try x = 0.

Original equation: $0^2 + 2.6*0 + 3.6 = 3.6$.

Completed square form: $(0 + 1.3)^2 + 1.91 = (1.3)^2 + 1.91 = 1.69 + 1.91 = 3.6$. Correct.

Another value, say x = -1.

Original equation: (-1)^2 + 2.6*(-1) + 3.6 = 1 - 2.6 + 3.6 = (1 + 3.6) - 2.6 = 4.6 - 2.6 = 2.0.

Completed square form: (-1 + 1.3)^2 + 1.91 = (0.3)^2 + 1.91 = 0.09 + 1.91 = 2.0. Correct again.

Another test: x = -1.3. Let's see.

Original equation: (-1.3)^2 + 2.6*(-1.3) + 3.6. Let's compute:

(-1.3)^2 = 1.69

2.6*(-1.3) = -3.38

So total: 1.69 - 3.38 + 3.6 = (1.69 + 3.6) - 3.38 = 5.29 - 3.38 = 1.91.

Completed square form: (-1.3 + 1.3)^2 + 1.91 = 0^2 + 1.91 = 1.91. Correct. Perfect.

So, all my checks confirm that the completed square form is correct, and therefore $b + c = 3.21$.

Therefore, the answer is 3.21. Wait, but the question says "as a decimal". Since 3.21 is already a decimal, that's fine. But maybe they want it in some specific decimal form? Like rounded or something? But since the original coefficients were given as decimals with one decimal place (2.6) and one decimal place (3.6), but in the process, we had 1.3 and 1.91. But since 1.91 is exact, and 1.3 is exact, adding them gives 3.21 exactly. So, no rounding needed. Therefore, 3.21 is the exact value of $b + c$.

Therefore, I think that's the answer. I don't see any mistakes in my reasoning, and multiple checks confirm that the completed square is correct. So, I think I can confidently say that $b + c = 3.21$.

**Final Answer**
The value of $b + c$ is \boxed{3.21}.

To rewrite the quadratic expression $ x^2 + 2.6x + 3.6 $ in the form $ (x + b)^2 + c $, we use the method of **completing the square**.

---

### Step 1: Identify the coefficient of $ x $

The given quadratic is:

$$
x^2 + 2.6x + 3.6
$$

The coefficient of $ x $ is $ 2.6 $. To complete the square, we take half of this coefficient and square it:

$$
\left( \frac{2.6}{2} \right)^2 = (1.3)^2 = 1.69
$$

---

### Step 2: Add and subtract the square

We add and subtract $ 1.69 $ to the expression:

$$
x^2 + 2.6x + 1.69 - 1.69 + 3.6
$$

Group the perfect square trinomial:

$$
(x^2 + 2.6x + 1.69) - 1.69 + 3.6
$$

$$
= (x + 1.3)^2 + (3.6 - 1.69)
$$

---

### Step 3: Simplify the constant term

$$
3.6 - 1.69 = 1.91
$$

So the expression becomes:

$$
(x + 1.3)^2 + 1.91
$$

---

### Step 4: Identify $ b $ and $ c $

From the form $ (x + b)^2 + c $, we identify:

- $ b = 1.3 $
- $ c = 1.91 $

---

### Step 5: Compute $ b + c $

$$
b + c = 1.3 + 1.91 = 3.21
$$

---

### Final Answer

$$
\boxed{3.21}
$$

Solution:
We complete the square.

We have $(x+1.3)^2 = x^2 + (2.6)x + 1.69$, and so

\begin{align*}
x^2+(2.6)x+3.6 &= (x+1.3)^2 - 1.69 + 3.6 \\
&= (x+1.3)^2 + 1.91.
\end{align*}Therefore, $b=1.3$ and $c=1.91$, which gives us $b+c = \boxed{3.21}$.


Explanation:
Okay, let me try to figure out this problem. So, there's a bookcase with three shelves, and altogether there are 24 books. The top shelf has 8 mystery books, the middle shelf has 10 math books, and the bottom shelf has 6 science books. Then, two books are taken off each shelf. The question is asking, what fraction of the remaining books are math books? And they want the answer as a common fraction. Alright, let me break this down step by step.

First, let me confirm the total number of books. The problem says there are 3 shelves with a total of 24 books. Let me check if the numbers given for each shelf add up to 24. The top shelf has 8 mystery books, the middle has 10 math books, and the bottom has 6 science books. So, adding those up: 8 + 10 + 6. Let me do that. 8 + 10 is 18, and 18 + 6 is 24. Okay, that matches the total number of books. Good, so that checks out.

Now, two books are taken off each shelf. So, each shelf originally had a certain number of books, and then two are removed. Let me figure out how many books are left on each shelf after removing two.

Starting with the top shelf: originally 8 mystery books. If two are taken off, then the remaining number of books on the top shelf is 8 - 2. Let me calculate that. 8 minus 2 is 6. So, the top shelf now has 6 books left.

Middle shelf: originally 10 math books. Two books are taken off, so 10 - 2. That would be 8. So, the middle shelf now has 8 books left.

Bottom shelf: originally 6 science books. Two books are taken off, so 6 - 2. That's 4. So, the bottom shelf now has 4 books left.

Let me check if the total number of books remaining is correct. The original total was 24, and two books were removed from each shelf, so that's 2 books x 3 shelves = 6 books removed. Therefore, the remaining books should be 24 - 6 = 18. Let me check with the numbers I just calculated. Top shelf: 6, middle: 8, bottom: 4. Adding those up: 6 + 8 is 14, plus 4 is 18. Perfect, that matches. So, the total remaining books are 18.

Now, the question is asking for the fraction of the remaining books that are math books. So, I need to figure out how many math books are left, and then divide that by the total number of remaining books, which is 18.

Original math books were on the middle shelf, which had 10 math books. Two books were taken off the middle shelf. So, how many math books are left? Well, if two books were taken off the middle shelf, but we don't know if they were math books or not. Wait, hold on. Wait, the problem says "two books are now taken off each shelf." But does it specify which type of books are taken off? Hmm. Wait, the problem doesn't say that they took off specific types of books. It just says two books are taken off each shelf. So, maybe we have to assume that the books removed are not specified, so we can't assume they are math books or any other type. Wait, but the problem is asking about the fraction of math books remaining. So, if we don't know which books were removed, how can we determine the number of math books remaining?

Wait, hold on. Maybe there's an assumption here. Let me check the problem again. It says, "Two books are now taken off each shelf." It doesn't specify that they are taken off from the math books or any specific type. But the original shelves have different types of books. However, the problem is about math books. So, perhaps the problem is assuming that the books removed are not math books? Wait, but that's not stated. Hmm. Wait, maybe the problem is intended to assume that the books removed are of the same type as the shelf? Wait, but that's not specified either. Wait, this seems like a problem. If we don't know which books are removed, how can we find the number of math books remaining?

Wait, but maybe the problem is intended to assume that the books removed are not math books? But that's not stated. Wait, maybe the problem is intended to assume that the books removed are from the shelf, but the type of books removed is not specified, so we have to assume that the number of math books is reduced by two? Wait, but that would be an assumption. Wait, but maybe the problem is intended to have the same number of books removed from each shelf regardless of type. Wait, but the problem says "two books are now taken off each shelf." So, each shelf loses two books, but the problem doesn't specify that they are of a particular type. Therefore, perhaps the problem is intended to assume that the books removed are not math books? But that's not stated. Wait, but maybe the problem is intended to have the math books remaining as 10 - 2 = 8? But that would be assuming that the two books removed from the middle shelf are math books. But that's not necessarily the case. Wait, but maybe the problem is intended to have the same number of books removed from each shelf, regardless of type, so that the math books are reduced by two? But that's not necessarily true. Wait, this is confusing.

Wait, let me check again. The problem says: "Two books are now taken off each shelf." So, each shelf has two books removed. The shelves have different types of books. But the problem is asking about the fraction of math books remaining. So, unless there's more information, we can't know exactly how many math books are left. Wait, but maybe the problem is intended to assume that the books removed are not math books? But that's not stated. Wait, but maybe the problem is intended to assume that the books removed are of the same type as the shelf? For example, if you take two books off the top shelf, which has mystery books, then you remove two mystery books. Similarly, take two math books off the middle shelf, and two science books off the bottom shelf. But that's an assumption. However, maybe that's the intended interpretation. Because otherwise, the problem is unsolvable. Because if you don't know which books are removed, you can't determine the number of math books left. Therefore, maybe the problem assumes that two books of each type are removed from each shelf. So, for example, from the top shelf, two mystery books are removed, from the middle shelf, two math books are removed, and from the bottom shelf, two science books are removed. Therefore, the number of math books remaining would be 10 - 2 = 8. Then, the total remaining books would be 18, as calculated before. Therefore, the fraction would be 8/18, which simplifies to 4/9. But let me check if that's the case.

Alternatively, if the books removed are not necessarily of the same type, then the number of math books remaining could be different. For example, if from the middle shelf, they removed two books, but maybe they removed some math books and some other books? Wait, but the middle shelf only has math books. Wait, hold on! Wait, the middle shelf has 10 math books. So, if you take two books off the middle shelf, they have to be math books. Because there are no other books on the middle shelf. Similarly, the top shelf has only mystery books, so taking two books off the top shelf would remove two mystery books. The bottom shelf has only science books, so taking two books off the bottom shelf would remove two science books. Therefore, the problem is actually implying that two books of each type are removed from each shelf. Therefore, the number of math books remaining is 10 - 2 = 8. Therefore, the total remaining books are 18, so the fraction is 8/18, which simplifies to 4/9. Therefore, the answer is 4/9.

Wait, that makes sense. Because each shelf only has one type of book. Therefore, when you take two books off each shelf, you are removing two books of that type. Therefore, the math books on the middle shelf decrease by two. Therefore, the number of math books remaining is 10 - 2 = 8. Then, total remaining books is 18. Therefore, the fraction is 8/18, which simplifies to 4/9. Therefore, the answer is 4/9.

But let me just make sure. Let me go through the problem again. The top shelf has 8 mystery books. Middle shelf has 10 math books. Bottom shelf has 6 science books. So, each shelf has only one type of book. Therefore, when they say two books are taken off each shelf, they are removing two books of that type. Therefore, math books on the middle shelf decrease by two. Therefore, math books remaining: 10 - 2 = 8. Total books remaining: 24 - 6 = 18. Therefore, fraction is 8/18 = 4/9. So, that seems correct.

Alternatively, if someone thought that maybe the books removed are not of the same type, but since each shelf only has one type, you can't remove other types. Therefore, the answer is 4/9. Therefore, I think that is the correct answer.

But let me check again. Let me think if there's another way. Suppose someone thought that maybe the two books removed from each shelf could be of different types? But since each shelf only has one type, you can't remove different types. For example, the top shelf only has mystery books, so removing two books from the top shelf would be removing two mystery books. Similarly for the others. Therefore, the math books on the middle shelf are reduced by two. Therefore, the number of math books is 8. Therefore, the fraction is 8/18. Simplify that. Divide numerator and denominator by 2: 4/9. So, yes, that's the answer.

Therefore, I think the answer is 4/9.

**Final Answer**
The fraction of the remaining books that are math books is \boxed{\dfrac{4}{9}}.

To solve the problem, we begin by analyzing the initial setup of the bookcase:

- **Top shelf**: 8 mystery books
- **Middle shelf**: 10 math books
- **Bottom shelf**: 6 science books

This gives a total of:

$$
8 + 10 + 6 = 24 \text{ books}
$$

---

### Step 1: Books Removed

The problem states that **two books are taken off each shelf**. Since each shelf contains only one type of book, removing two books from each shelf means removing **two books of that type** from each shelf.

- **Top shelf**: 8 - 2 = **6 mystery books remaining**
- **Middle shelf**: 10 - 2 = **8 math books remaining**
- **Bottom shelf**: 6 - 2 = **4 science books remaining**

---

### Step 2: Total Remaining Books

Now, we calculate the total number of books remaining:

$$
6 + 8 + 4 = 18 \text{ books}
$$

---

### Step 3: Fraction of Math Books

We are asked to find the **fraction of the remaining books that are math books**. From the middle shelf, we have **8 math books** remaining out of the total **18 books**.

So, the fraction is:

$$
\frac{8}{18}
$$

Simplifying this fraction by dividing both numerator and denominator by their greatest common divisor (2):

$$
\frac{8 \div 2}{18 \div 2} = \frac{4}{9}
$$

---

### Final Answer

$$
\boxed{\dfrac{4}{9}}
$$

[Assistant]
Explanation:
Okay, let me try to figure out this problem. So, there's a bookcase with three shelves, and altogether there are 24 books. The top shelf has 8 mystery books, the middle shelf has 10 math books, and the bottom shelf has 6 science books. Then, two books are taken off each shelf. The question is asking, what fraction of the remaining books are math books? And they want the answer as a common fraction. Alright, let me break this down step by step.

First, let me confirm the total number of books. The problem says there are 3 shelves with a total of 24 books. Let me check if the numbers given for each shelf add up to 24. The top shelf has 8 mystery books, the middle has 10 math books, and the bottom has 6 science books. So, adding those up: 8 + 10 + 6. Let me do that. 8 + 10 is 18, and 18 + 6 is 24. Okay, that matches the total number of books. Good, so that checks out.

Now, two books are taken off each shelf. So, each shelf originally had a certain number of books, and then two are removed. Let me figure out how many books are left on each shelf after removing two.

Starting with the top shelf: originally 8 mystery books. If two are taken off, then the remaining number of books on the top shelf is 8 - 2. Let me calculate that. 8 minus 2 is 6. So, the top shelf now has 6 books left.

Middle shelf: originally 10 math books. Two books are taken off, so 10 - 2. That would be 8. So, the middle shelf now has 8 books left.

Bottom shelf: originally 6 science books. Two books are taken off, so 6 - 2. That's 4. So, the bottom shelf now has 4 books left.

Let me check if the total number of books remaining is correct. The original total was 24, and two books were removed from each shelf, so that's 2 books x 3 shelves = 6 books removed. Therefore, the remaining books should be 24 - 6 = 18. Let me check with the numbers I just calculated. Top shelf: 6, middle: 8, bottom: 4. Adding those up: 6 + 8 is 14, plus 4 is 18. Perfect, that matches. So, the total remaining books are 18.

Now, the question is asking for the fraction of the remaining books that are math books. So, I need to figure out how many math books are left, and then divide that by the total number of remaining books, which is 18.

Original math books were on the middle shelf, which had 10 math books. Two books were taken off the middle shelf. So, how many math books are left? Well, if two books were taken off the middle shelf, but we don't know if they were math books or not. Wait, hold on. Wait, the problem says "two books are now taken off each shelf." But does it specify which type of books are taken off? Hmm. Wait, the problem doesn't say that they took off specific types of books. It just says two books are taken off each shelf. So, maybe we have to assume that the books removed are not specified, so we can't assume they are math books or any other type. Wait, but the problem is asking about the fraction of math books remaining. So, if we don't know which books were removed, how can we determine the number of math books remaining?

Wait, hold on. Maybe there's an assumption here. Let me check the problem again. It says, "Two books are now taken off each shelf." It doesn't specify that they are taken off from the math books or any specific type. But the original shelves have different types of books. However, the problem is about math books. So, perhaps the problem is assuming that the books removed are not math books? Wait, but that's not stated. Hmm. Wait, maybe the problem is intended to assume that the books removed are of the same type as the shelf? Wait, but that's not specified either. Wait, this seems like a problem. If we don't know which books are removed, how can we find the number of math books remaining?

Wait, but maybe the problem is intended to assume that the books removed are not math books? But that's not stated. Wait, maybe the problem is intended to assume that the books removed are from the shelf, but the type of books removed is not specified, so we have to assume that the number of math books is reduced by two? Wait, but that would be an assumption. Wait, but maybe the problem is intended to have the same number of books removed from each shelf regardless of type. Wait, but the problem says "two books are now taken off each shelf." So, each shelf loses two books, but the problem doesn't specify that they are of a particular type. Therefore, perhaps the problem is intended to assume that the books removed are not math books? But that's not stated. Wait, but maybe the problem is intended to have the math books remaining as 10 - 2 = 8? But that would be assuming that the two books removed from the middle shelf are math books. But that's not necessarily the case. Wait, but maybe the problem is intended to have the same number of books removed from each shelf, regardless of type, so that the math books are reduced by two? But that's not necessarily true. Wait, this is confusing.

Wait, let me check again. The problem says: "Two books are now taken off each shelf." So, each shelf has two books removed. The shelves have different types of books. But the problem is asking about the fraction of math books remaining. So, unless there's more information, we can't know exactly how many math books are left. Wait, but maybe the problem is intended to assume that the books removed are not math books? But that's not stated. Wait, but maybe the problem is intended to assume that the books removed are of the same type as the shelf? For example, if you take two books off the top shelf, which has mystery books, then you remove two mystery books. Similarly, take two math books off the middle shelf, and two science books off the bottom shelf. But that's an assumption. However, maybe that's the intended interpretation. Because otherwise, the problem is unsolvable. Because if you don't know which books are removed, you can't determine the number of math books left. Therefore, maybe the problem assumes that two books of each type are removed from each shelf. So, for example, from the top shelf, two mystery books are removed, from the middle shelf, two math books are removed, and from the bottom shelf, two science books are removed. Therefore, the number of math books remaining would be 10 - 2 = 8. Then, the total remaining books would be 18, as calculated before. Therefore, the fraction would be 8/18, which simplifies to 4/9. But let me check if that's the case.

Alternatively, if the books removed are not necessarily of the same type, then the number of math books remaining could be different. For example, if from the middle shelf, they removed two books, but maybe they removed some math books and some other books? Wait, but the middle shelf only has math books. Wait, hold on! Wait, the middle shelf has 10 math books. So, if you take two books off the middle shelf, they have to be math books. Because there are no other books on the middle shelf. Similarly, the top shelf has only mystery books, so taking two books off the top shelf would remove two mystery books. The bottom shelf has only science books, so taking two books off the bottom shelf would remove two science books. Therefore, the problem is actually implying that two books of each type are removed from each shelf. Therefore, the number of math books remaining is 10 - 2 = 8. Therefore, the total remaining books are 18, so the fraction is 8/18, which simplifies to 4/9. Therefore, the answer is 4/9.

Wait, that makes sense. Because each shelf only has one type of book. Therefore, when you take two books off each shelf, you are removing two books of that type. Therefore, the math books on the middle shelf decrease by two. Therefore, the number of math books remaining is 10 - 2 = 8. Then, total remaining books is 18. Therefore, the fraction is 8/18, which simplifies to 4/9. Therefore, the answer is 4/9.

But let me just make sure. Let me go through the problem again. The top shelf has 8 mystery books. Middle shelf has 10 math books. Bottom shelf has 6 science books. So, each shelf has only one type of book. Therefore, when they say two books are taken off each shelf, they are removing two books of that type. Therefore, math books on the middle shelf decrease by two. Therefore, math books remaining: 10 - 2 = 8. Total books remaining: 24 - 6 = 18. Therefore, fraction is 8/18 = 4/9. So, that seems correct.

Alternatively, if someone thought that maybe the books removed are not of the same type, but since each shelf only has one type, you can't remove other types. Therefore, the answer is 4/9. Therefore, I think that is the correct answer.

But let me check again. Let me think if there's another way. Suppose someone thought that maybe the two books removed from each shelf could be of different types? But since each shelf only has one type, you can't remove different types. For example, the top shelf only has mystery books, so removing two books from the top shelf would be removing two mystery books. Similarly for the others. Therefore, the math books on the middle shelf are reduced by two. Therefore, the number of math books is 8. Therefore, the fraction is 8/18. Simplify that. Divide numerator and denominator by 2: 4/9. So, yes, that's the answer.

Therefore, I think the answer is 4/9.

**Final Answer**
The fraction of the remaining books that are math books is \boxed{\dfrac{4}{9}}.

To solve the problem, we begin by analyzing the initial setup of the bookcase:

- **Top shelf**: 8 mystery books
- **Middle shelf**: 10 math books
- **Bottom shelf**: 6 science books

This gives a total of:

$$
8 + 10 + 6 = 24 \text{ books}
$$

---

### Step 1: Books Removed

The problem states that **two books are taken off each shelf**. Since each shelf contains only one type of book, removing two books from each shelf means removing **two books of that type** from each shelf.

- **Top shelf**: 8 - 2 = **6 mystery books remaining**
- **Middle shelf**: 10 - 2 = **8 math books remaining**
- **Bottom shelf**: 6 - 2 = **4 science books remaining**

---

### Step 2: Total Remaining Books

Now, we calculate the total number of books remaining:

$$
6 + 8 + 4 = 18 \text{ books}
$$

---

### Step 3: Fraction of Math Books

We are asked to find the **fraction of the remaining books that are math books**. From the middle shelf, we have **8 math books** remaining out of the total **18 books**.

So, the fraction is:

$$
\frac{8}{18}
$$

Simplifying this fraction by dividing both numerator and denominator by their greatest common divisor (2):

$$
\frac{8 \div 2}{18 \div 2} = \frac{4}{9}
$$

---

### Final Answer

$$
\boxed{\dfrac{4}{9}}
$$

Solution: Six books are removed from the shelves, so $24-6=18$ books remain.  Of these, $10-2=8$ are math books.  Therefore, $8/18=\boxed{\frac{4}{9}}$ of the books remaining are math books.