Explanation:
Okay, so Bill is walking in different directions, and I need to figure out how far he is from his starting point in a straight line. Let me try to visualize this.

First, he walks 1/2 mile south. Then he walks 3/4 mile east, and finally another 1/2 mile south. Hmm, okay. So maybe I should draw a coordinate system to track his position. Let me set up coordinates where his starting point is at (0, 0).

If I consider north as positive y-direction and east as positive x-direction, then walking south would be in the negative y-direction, and east is positive x-direction. So let me convert each part of his journey into coordinates.

Starting at (0, 0). First, he goes 1/2 mile south. That would be moving along the negative y-axis. So his new position after the first leg is (0, -1/2).

Then he walks 3/4 mile east. East is the positive x-direction, so adding 3/4 to the x-coordinate. So his new position after the second leg is (3/4, -1/2).

Then he walks another 1/2 mile south. Again, that's negative y-direction. So subtract another 1/2 from the y-coordinate. His final position would be (3/4, -1/2 - 1/2) = (3/4, -1).

Now, I need to find the straight-line distance from his starting point (0, 0) to his final position (3/4, -1).

To find the straight-line distance between two points (x1, y1) and (x2, y2), we use the distance formula:

distance = √[(x2 - x1)² + (y2 - y1)²]

Plugging in the coordinates:

distance = √[(3/4 - 0)² + (-1 - 0)²] = √[(3/4)² + (-1)²]

Calculating each term:

(3/4)² is 9/16, and (-1)² is 1.

So the distance is √(9/16 + 1). Let me convert 1 to sixteenths to add them together. 1 is 16/16, so:

√(9/16 + 16/16) = √(25/16) = 5/4.

Wait, 5/4 is 1.25. So the distance is 1.25 miles? But the problem says to express the answer as a decimal to the nearest hundredth. 1.25 is already to the nearest hundredth. So is the answer 1.25?

Wait, let me double-check my steps. Maybe I made a mistake in coordinates? Let me go through again.

Starting point (0, 0). First, south 1/2 mile: so y-coordinate is -1/2. Then east 3/4 mile: x-coordinate is +3/4. Then south another 1/2 mile: y-coordinate is -1/2 -1/2 = -1. So final position (3/4, -1).

Distance from (0,0) is sqrt( (3/4)^2 + (-1)^2 ). Which is sqrt(9/16 + 1) = sqrt(25/16) = 5/4 = 1.25.

Wait, so that's 1.25 miles. So the answer is 1.25? But maybe I need to check if there's a different interpretation of the directions?

Wait, maybe I assigned the directions incorrectly? Let me make sure. If south is negative y, east is positive x. So the movements are:

First, moving south: y decreases. Then east: x increases. Then south again: y decreases. So the coordinates are correct.

Alternatively, could the problem be interpreted differently? For example, maybe the first movement is south, then east, then south again, but is there a possibility that the path is different? For instance, maybe he turns east after moving south, then turns south again? But regardless, the coordinate system should track his final position.

Alternatively, maybe I'm supposed to think of it as a triangle? Let me think. So he moves south twice, so total south distance is 1/2 + 1/2 = 1 mile south, and then east 3/4 mile. So if you imagine a right triangle where one leg is 1 mile south and the other leg is 3/4 mile east. Then the straight-line distance would be the hypotenuse.

Yes, so hypotenuse squared is (1)^2 + (3/4)^2 = 1 + 9/16 = 25/16, so sqrt(25/16) is 5/4. Which is 1.25.

So that's the same answer. So maybe 1.25 is correct. But why does the problem say "Express your answer as a decimal to the nearest hundredth"? Maybe because 5/4 is exactly 1.25, which is already to the nearest hundredth. So the answer is 1.25.

But maybe I made a mistake in thinking that the total south movement is 1 mile? Let me check. The first movement is 1/2 mile south, then another 1/2 mile south. So yes, that's 1 mile total south. So the legs are 1 mile and 3/4 mile.

Alternatively, maybe the problem is in the order? Wait, the problem says he walks 1/2 mile south, then 3/4 mile east, then 1/2 mile south. So the first movement is south, then east, then south again.

So in terms of coordinate system, that would be as I said. So the final position is 3/4 east and 1 south from the starting point.

Alternatively, if he walked 1/2 south, 3/4 east, then 1/2 south. So if you imagine plotting the points:

Start at (0, 0). After first move: (0, -0.5). Then east to (0.75, -0.5). Then south to (0.75, -1). So distance from (0,0) is sqrt( (0.75)^2 + (-1)^2 ) = sqrt(0.5625 + 1) = sqrt(1.5625) = 1.25.

Yes, that's 1.25. So the answer is 1.25.

Wait, but maybe the question is expecting a different answer? Let me check my calculation again.

0.75 squared is 0.5625. 1 squared is 1. Sum is 1.5625. Square root of that is 1.25. So that's correct. So 1.25 is the exact value.

But to make sure that there isn't a different interpretation. For example, maybe he walks 1/2 mile south, then 3/4 mile east, then 1/2 mile south. Maybe the problem is not in a straight line? But no, the question is asking for the direct line from starting point, so it's the straight line distance.

Alternatively, maybe the problem is in the direction? For example, if after moving east, he turns south again, but is that different? No, because moving east and then south would still result in the same coordinates.

Alternatively, maybe I need to consider that moving south twice is the same as moving south once? No, because he moves 1/2 mile south, then 3/4 east, then another 1/2 mile south. The total south movement is 1 mile.

Alternatively, perhaps there is an error in coordinate system? Let me try assigning different coordinates. Let me try with north as +y and east as +x.

Starting at (0, 0). Moving south 1/2 mile: that would be (0, -1/2). Then east 3/4 mile: (3/4, -1/2). Then south 1/2 mile: (3/4, -1). Same as before.

So the distance is sqrt( (3/4)^2 + (-1)^2 ) = same as before.

Alternatively, maybe the problem is a trick question, but I don't think so.

Alternatively, maybe the answer is supposed to be expressed as a decimal, and maybe 5/4 is 1.25, but let me check if there is a miscalculation here.

Wait, 3/4 is 0.75. So 0.75 squared is 0.5625. 1 squared is 1. Adding those gives 1.5625. The square root of 1.5625. Let me calculate that.

What is sqrt(1.5625)? Well, 1.25 squared is 1.5625. Because 1.2 squared is 1.44, 1.3 squared is 1.69. 1.25 squared: 1.25*1.25. Let me compute that:

1.25 * 1.25: 1 * 1 = 1, 1 * 0.25 = 0.25, 0.25 * 1 = 0.25, 0.25 * 0.25 = 0.0625. Adding them up: 1 + 0.25 + 0.25 + 0.0625 = 1.5625. Yes, so sqrt(1.5625) is indeed 1.25.

So the answer is 1.25. Therefore, the answer is 1.25, which is already to the nearest hundredth.

But just to make sure, maybe the problem had different numbers? Let me check again the original problem:

"Bill walks 1/2 mile south, then 3/4 mile east, and finally 1/2 mile south. How many miles is he, in a direct line, from his starting point? Express your answer as a decimal to the nearest hundredth."

Yes, that's what was given. So the answer is 1.25.

But maybe the problem expects me to convert the fractions to decimals first? Let me try that approach.

First movement: 1/2 mile south is 0.5 miles south.

Second movement: 3/4 mile east is 0.75 miles east.

Third movement: 0.5 miles south.

Total south movement: 0.5 + 0.5 = 1 mile south.

Total east movement: 0.75 miles east.

So displacement is 1 mile south and 0.75 miles east. So distance is sqrt(1^2 + 0.75^2) = sqrt(1 + 0.5625) = sqrt(1.5625) = 1.25.

Same result. So that's correct.

Therefore, I think the answer is 1.25. But just to check, maybe the problem is in some other way? For example, if he walked in some other path? But no, the problem says "in a direct line," so straight line.

Alternatively, maybe I need to consider that the total distance walked is something else? But no, the question is about the straight-line distance from start to finish.

Alternatively, maybe I misread the problem. Let me check again:

"Bill walks 1/2 mile south, then 3/4 mile east, and finally 1/2 mile south. How many miles is he, in a direct line, from his starting point?"

No, that's correct.

Hmm. Wait, maybe I should check if the answer is different if you consider the path as vectors. So, representing each movement as vectors:

First vector: 0i - 0.5j (south is negative y)

Second vector: +0.75i + 0j (east is positive x)

Third vector: 0i - 0.5j (south again)

Adding them together:

Total x-component: 0 + 0.75 + 0 = 0.75

Total y-component: -0.5 + 0 -0.5 = -1.0

So the displacement vector is (0.75, -1.0), so magnitude is sqrt(0.75² + (-1.0)²) = same as before.

So this also gives 1.25.

Therefore, I think the answer is definitely 1.25. Maybe the problem expects decimal, but since 1.25 is already exact, it's just 1.25.

Alternatively, perhaps the problem was expecting me to use some other method? But all methods lead to the same answer.

Alternatively, maybe the problem is from a different source where the answer is different? But I don't have that context.

Alternatively, maybe there's a miscalculation in my part? Let me check once more:

sqrt( (0.75)^2 + (1)^2 ) = sqrt(0.5625 + 1) = sqrt(1.5625).

What's sqrt(1.5625)? Let me try dividing 1.5625 by 1.25. 1.25 * 1.25 = 1.5625. So sqrt(1.5625) = 1.25.

Yes. Therefore, answer is 1.25.

Therefore, I think the answer is 1.25, so \boxed{1.25}.

**Final Answer**
\boxed{1.25}

To determine how far Bill is from his starting point in a direct line, we can model his movement using a coordinate system. Let’s assume the starting point is at the origin $(0, 0)$, with the positive y-axis pointing north and the positive x-axis pointing east.

---

### Step 1: Analyze the Movement

Bill’s journey consists of three movements:

1. **$ \frac{1}{2} $ mile south** – This means he moves 0.5 miles in the **negative y-direction**.
   - New position: $ (0, -0.5) $

2. **$ \frac{3}{4} = 0.75 $ mile east** – This means he moves 0.75 miles in the **positive x-direction**.
   - New position: $ (0.75, -0.5) $

3. **Another $ \frac{1}{2} $ mile south** – This again moves 0.5 miles in the **negative y-direction**.
   - Final position: $ (0.75, -1) $

---

### Step 2: Use the Distance Formula

To find the straight-line (i.e., direct) distance from the starting point $(0, 0)$ to his final position $(0.75, -1)$, we apply the **distance formula**:

$$
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
$$

Substitute the values:

$$
\text{Distance} = \sqrt{(0.75 - 0)^2 + (-1 - 0)^2} = \sqrt{0.75^2 + 1^2}
$$

Now compute:

$$
0.75^2 = 0.5625,\quad 1^2 = 1
$$

$$
\text{Distance} = \sqrt{0.5625 + 1} = \sqrt{1.5625}
$$

$$
\sqrt{1.5625} = 1.25
$$

---

### Final Answer

Thus, the straight-line distance from Bill's starting point to his final position is:

$$
\boxed{1.25}
$$

[Assistant]
Explanation:
Okay, so Bill is walking in different directions, and I need to figure out how far he is from his starting point in a straight line. Let me try to visualize this.

First, he walks 1/2 mile south. Then he walks 3/4 mile east, and finally another 1/2 mile south. Hmm, okay. So maybe I should draw a coordinate system to track his position. Let me set up coordinates where his starting point is at (0, 0).

If I consider north as positive y-direction and east as positive x-direction, then walking south would be in the negative y-direction, and east is positive x-direction. So let me convert each part of his journey into coordinates.

Starting at (0, 0). First, he goes 1/2 mile south. That would be moving along the negative y-axis. So his new position after the first leg is (0, -1/2).

Then he walks 3/4 mile east. East is the positive x-direction, so adding 3/4 to the x-coordinate. So his new position after the second leg is (3/4, -1/2).

Then he walks another 1/2 mile south. Again, that's negative y-direction. So subtract another 1/2 from the y-coordinate. His final position would be (3/4, -1/2 - 1/2) = (3/4, -1).

Now, I need to find the straight-line distance from his starting point (0, 0) to his final position (3/4, -1).

To find the straight-line distance between two points (x1, y1) and (x2, y2), we use the distance formula:

distance = √[(x2 - x1)² + (y2 - y1)²]

Plugging in the coordinates:

distance = √[(3/4 - 0)² + (-1 - 0)²] = √[(3/4)² + (-1)²]

Calculating each term:

(3/4)² is 9/16, and (-1)² is 1.

So the distance is √(9/16 + 1). Let me convert 1 to sixteenths to add them together. 1 is 16/16, so:

√(9/16 + 16/16) = √(25/16) = 5/4.

Wait, 5/4 is 1.25. So the distance is 1.25 miles? But the problem says to express the answer as a decimal to the nearest hundredth. 1.25 is already to the nearest hundredth. So is the answer 1.25?

Wait, let me double-check my steps. Maybe I made a mistake in coordinates? Let me go through again.

Starting point (0, 0). First, south 1/2 mile: so y-coordinate is -1/2. Then east 3/4 mile: x-coordinate is +3/4. Then south another 1/2 mile: y-coordinate is -1/2 -1/2 = -1. So final position (3/4, -1).

Distance from (0,0) is sqrt( (3/4)^2 + (-1)^2 ). Which is sqrt(9/16 + 1) = sqrt(25/16) = 5/4 = 1.25.

Wait, so that's 1.25 miles. So the answer is 1.25? But maybe I need to check if there's a different interpretation of the directions?

Wait, maybe I assigned the directions incorrectly? Let me make sure. If south is negative y, east is positive x. So the movements are:

First, moving south: y decreases. Then east: x increases. Then south again: y decreases. So the coordinates are correct.

Alternatively, could the problem be interpreted differently? For example, maybe the first movement is south, then east, then south again, but is there a possibility that the path is different? For instance, maybe he turns east after moving south, then turns south again? But regardless, the coordinate system should track his final position.

Alternatively, maybe I'm supposed to think of it as a triangle? Let me think. So he moves south twice, so total south distance is 1/2 + 1/2 = 1 mile south, and then east 3/4 mile. So if you imagine a right triangle where one leg is 1 mile south and the other leg is 3/4 mile east. Then the straight-line distance would be the hypotenuse.

Yes, so hypotenuse squared is (1)^2 + (3/4)^2 = 1 + 9/16 = 25/16, so sqrt(25/16) is 5/4. Which is 1.25.

So that's the same answer. So maybe 1.25 is correct. But why does the problem say "Express your answer as a decimal to the nearest hundredth"? Maybe because 5/4 is exactly 1.25, which is already to the nearest hundredth. So the answer is 1.25.

But maybe I made a mistake in thinking that the total south movement is 1 mile? Let me check. The first movement is 1/2 mile south, then another 1/2 mile south. So yes, that's 1 mile total south. So the legs are 1 mile and 3/4 mile.

Alternatively, maybe the problem is in the order? Wait, the problem says he walks 1/2 mile south, then 3/4 mile east, then 1/2 mile south. So the first movement is south, then east, then south again.

So in terms of coordinate system, that would be as I said. So the final position is 3/4 east and 1 south from the starting point.

Alternatively, if he walked 1/2 south, 3/4 east, then 1/2 south. So if you imagine plotting the points:

Start at (0, 0). After first move: (0, -0.5). Then east to (0.75, -0.5). Then south to (0.75, -1). So distance from (0,0) is sqrt( (0.75)^2 + (-1)^2 ) = sqrt(0.5625 + 1) = sqrt(1.5625) = 1.25.

Yes, that's 1.25. So the answer is 1.25.

Wait, but maybe the question is expecting a different answer? Let me check my calculation again.

0.75 squared is 0.5625. 1 squared is 1. Sum is 1.5625. Square root of that is 1.25. So that's correct. So 1.25 is the exact value.

But to make sure that there isn't a different interpretation. For example, maybe he walks 1/2 mile south, then 3/4 mile east, then 1/2 mile south. Maybe the problem is not in a straight line? But no, the question is asking for the direct line from starting point, so it's the straight line distance.

Alternatively, maybe the problem is in the direction? For example, if after moving east, he turns south again, but is that different? No, because moving east and then south would still result in the same coordinates.

Alternatively, maybe I need to consider that moving south twice is the same as moving south once? No, because he moves 1/2 mile south, then 3/4 east, then another 1/2 mile south. The total south movement is 1 mile.

Alternatively, perhaps there is an error in coordinate system? Let me try assigning different coordinates. Let me try with north as +y and east as +x.

Starting at (0, 0). Moving south 1/2 mile: that would be (0, -1/2). Then east 3/4 mile: (3/4, -1/2). Then south 1/2 mile: (3/4, -1). Same as before.

So the distance is sqrt( (3/4)^2 + (-1)^2 ) = same as before.

Alternatively, maybe the problem is a trick question, but I don't think so.

Alternatively, maybe the answer is supposed to be expressed as a decimal, and maybe 5/4 is 1.25, but let me check if there is a miscalculation here.

Wait, 3/4 is 0.75. So 0.75 squared is 0.5625. 1 squared is 1. Adding those gives 1.5625. The square root of 1.5625. Let me calculate that.

What is sqrt(1.5625)? Well, 1.25 squared is 1.5625. Because 1.2 squared is 1.44, 1.3 squared is 1.69. 1.25 squared: 1.25*1.25. Let me compute that:

1.25 * 1.25: 1 * 1 = 1, 1 * 0.25 = 0.25, 0.25 * 1 = 0.25, 0.25 * 0.25 = 0.0625. Adding them up: 1 + 0.25 + 0.25 + 0.0625 = 1.5625. Yes, so sqrt(1.5625) is indeed 1.25.

So the answer is 1.25. Therefore, the answer is 1.25, which is already to the nearest hundredth.

But just to make sure, maybe the problem had different numbers? Let me check again the original problem:

"Bill walks 1/2 mile south, then 3/4 mile east, and finally 1/2 mile south. How many miles is he, in a direct line, from his starting point? Express your answer as a decimal to the nearest hundredth."

Yes, that's what was given. So the answer is 1.25.

But maybe the problem expects me to convert the fractions to decimals first? Let me try that approach.

First movement: 1/2 mile south is 0.5 miles south.

Second movement: 3/4 mile east is 0.75 miles east.

Third movement: 0.5 miles south.

Total south movement: 0.5 + 0.5 = 1 mile south.

Total east movement: 0.75 miles east.

So displacement is 1 mile south and 0.75 miles east. So distance is sqrt(1^2 + 0.75^2) = sqrt(1 + 0.5625) = sqrt(1.5625) = 1.25.

Same result. So that's correct.

Therefore, I think the answer is 1.25. But just to check, maybe the problem is in some other way? For example, if he walked in some other path? But no, the problem says "in a direct line," so straight line.

Alternatively, maybe I need to consider that the total distance walked is something else? But no, the question is about the straight-line distance from start to finish.

Alternatively, maybe I misread the problem. Let me check again:

"Bill walks 1/2 mile south, then 3/4 mile east, and finally 1/2 mile south. How many miles is he, in a direct line, from his starting point?"

No, that's correct.

Hmm. Wait, maybe I should check if the answer is different if you consider the path as vectors. So, representing each movement as vectors:

First vector: 0i - 0.5j (south is negative y)

Second vector: +0.75i + 0j (east is positive x)

Third vector: 0i - 0.5j (south again)

Adding them together:

Total x-component: 0 + 0.75 + 0 = 0.75

Total y-component: -0.5 + 0 -0.5 = -1.0

So the displacement vector is (0.75, -1.0), so magnitude is sqrt(0.75² + (-1.0)²) = same as before.

So this also gives 1.25.

Therefore, I think the answer is definitely 1.25. Maybe the problem expects decimal, but since 1.25 is already exact, it's just 1.25.

Alternatively, perhaps the problem was expecting me to use some other method? But all methods lead to the same answer.

Alternatively, maybe the problem is from a different source where the answer is different? But I don't have that context.

Alternatively, maybe there's a miscalculation in my part? Let me check once more:

sqrt( (0.75)^2 + (1)^2 ) = sqrt(0.5625 + 1) = sqrt(1.5625).

What's sqrt(1.5625)? Let me try dividing 1.5625 by 1.25. 1.25 * 1.25 = 1.5625. So sqrt(1.5625) = 1.25.

Yes. Therefore, answer is 1.25.

Therefore, I think the answer is 1.25, so \boxed{1.25}.

**Final Answer**
\boxed{1.25}

To determine how far Bill is from his starting point in a direct line, we can model his movement using a coordinate system. Let’s assume the starting point is at the origin $(0, 0)$, with the positive y-axis pointing north and the positive x-axis pointing east.

---

### Step 1: Analyze the Movement

Bill’s journey consists of three movements:

1. **$ \frac{1}{2} $ mile south** – This means he moves 0.5 miles in the **negative y-direction**.
   - New position: $ (0, -0.5) $

2. **$ \frac{3}{4} = 0.75 $ mile east** – This means he moves 0.75 miles in the **positive x-direction**.
   - New position: $ (0.75, -0.5) $

3. **Another $ \frac{1}{2} $ mile south** – This again moves 0.5 miles in the **negative y-direction**.
   - Final position: $ (0.75, -1) $

---

### Step 2: Use the Distance Formula

To find the straight-line (i.e., direct) distance from the starting point $(0, 0)$ to his final position $(0.75, -1)$, we apply the **distance formula**:

$$
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
$$

Substitute the values:

$$
\text{Distance} = \sqrt{(0.75 - 0)^2 + (-1 - 0)^2} = \sqrt{0.75^2 + 1^2}
$$

Now compute:

$$
0.75^2 = 0.5625,\quad 1^2 = 1
$$

$$
\text{Distance} = \sqrt{0.5625 + 1} = \sqrt{1.5625}
$$

$$
\sqrt{1.5625} = 1.25
$$

---

### Final Answer

Thus, the straight-line distance from Bill's starting point to his final position is:

$$
\boxed{1.25}
$$

Solution:
The diagram on the left shows the path of Bill's walk. As the diagram on the right illustrates, he could also have walked from $A$ to $B$ by first walking 1 mile south then $\frac{3}{4}$ mile east. [asy]
pair a=(0,1), b=(.75, 0), c=(0,.5), d=(.75,.5), o=(0,0);
draw(a--b--d--c--cycle);
label("$A$", a, NW);
label("$B$", b, SE);
label("$\frac{1}{2}$", (0,0.75), W);
label("$\frac{3}{4}$", (.7, 0.66),W);
label("$\frac{1}{2}$", (.75, .25), E);

picture pic;
draw(pic, a--b--o--cycle);
label(pic, "$A$", a, NW);
label(pic, "$B$", b, SE);
label(pic, "$\frac{3}{4}$", (.375,0), S);
label(pic, "1", (0, .5), W);
add(shift(1.5,0)*pic);
[/asy] By the Pythagorean Theorem, \[(AB)^2=1^2+\left(\frac{3}{4}\right)^2=1+\frac{9}{16}=\frac{25}{16},\]so $AB=\frac{5}{4}=1\frac{1}{4}$, or $\boxed{1.25}$.


Explanation:
Okay, so I need to find the vertical asymptotes and the horizontal asymptote of the function f(x) = (2x)/(x² - 5x - 14), and then add up their values a + b + c. Let me start by recalling what vertical and horizontal asymptotes are.

Vertical asymptotes occur where the denominator of a rational function is zero, provided that the numerator isn't also zero at those points. So, first step: factor the denominator and find its roots.

The denominator is x² - 5x - 14. Let me try to factor this quadratic. I need two numbers that multiply to -14 and add to -5. Hmm, let's see. Let me think: factors of -14 could be -7 and 2 because -7 * 2 = -14, and -7 + 2 = -5. Yes, that works! So the denominator factors as (x - 7)(x + 2). Let me check that: (x - 7)(x + 2) = x² + 2x - 7x -14 = x² -5x -14. Perfect! So the denominator factors into (x - 7)(x + 2).

So the vertical asymptotes are at the values of x that make the denominator zero, so x = 7 and x = -2. Therefore, a = -2 and b = 7? Wait, but the question says vertical asymptotes are x = a and x = b. They might not specify order, so a and b could be -2 and 7. But when they say a + b, it would be (-2) + 7 = 5. But I need to confirm that these are indeed vertical asymptotes. Let me check that the numerator isn't zero at these points. The numerator is 2x. So when x = -2, numerator is 2*(-2) = -4, which isn't zero. Similarly, at x = 7, numerator is 2*7 = 14, also not zero. So yes, both are vertical asymptotes. So a and b are -2 and 7, so a + b is 5.

Now, moving on to the horizontal asymptote. For horizontal asymptotes of rational functions, the rules depend on the degrees of the numerator and denominator. Let me recall: if the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If the degrees are equal, it's the ratio of the leading coefficients. If the degree of the numerator is higher, there's no horizontal asymptote (but there might be an oblique asymptote).

So here, the numerator is 2x, which is degree 1, and the denominator is x² -5x -14, which is degree 2. Since degree of numerator (1) is less than degree of denominator (2), the horizontal asymptote is y = 0. Therefore, c = 0.

So then a + b + c would be (-2) + 7 + 0 = 5. Wait, but let me confirm that there isn't a horizontal asymptote at some other value. Wait, maybe I need to double-check the horizontal asymptote calculation.

Alternatively, maybe I can do the limit as x approaches infinity of f(x). Let me compute the limit as x approaches infinity of (2x)/(x² -5x -14). To evaluate this, divide numerator and denominator by x², which is the highest power in the denominator.

So, numerator becomes (2x)/x² = 2/x. Denominator becomes (x² -5x -14)/x² = 1 - 5/x -14/x². Then as x approaches infinity, 2/x approaches 0, 5/x approaches 0, and 14/x² approaches 0. Therefore, the limit is 0 / (1 - 0 - 0) = 0. So yes, horizontal asymptote is y = 0. Therefore, c = 0.

Therefore, a is -2, b is 7, c is 0, so a + b + c = -2 + 7 + 0 = 5.

But wait, let me make sure that they didn't consider the horizontal asymptote in another way. For example, sometimes people might think about the leading terms. The leading term of the numerator is 2x and the denominator is x², so dividing gives 2x/x² = 2/x, which tends to 0 as x approaches infinity. So that also confirms that horizontal asymptote is y = 0.

So, I think that's correct. Therefore, the answer is 5. Wait, but maybe I need to check if the problem says "horizontal asymptote y = c" and maybe there's another horizontal asymptote? For example, sometimes if the degree of the numerator is exactly one more than the denominator, you get an oblique asymptote. But here, the degree of numerator is one less than denominator, so horizontal asymptote is y = 0. So I think that's correct.

Wait, another thing to check: sometimes when people say horizontal asymptote, they might mean the horizontal line that the function approaches, and in this case, since it's y = 0, which is the x-axis. So that is the horizontal asymptote. So I think that's solid.

Therefore, the sum a + b + c is (-2) + 7 + 0 = 5. So, 5 is the answer.

But let me check once again. Maybe there's a mistake in factoring the denominator? Let me check again. The denominator is x² -5x -14. Factoring, looking for two numbers m and n such that m * n = -14 and m + n = -5. Let me think. The factors of -14 could be ( -7, 2 ) since -7*2 = -14 and -7 + 2 = -5. Yes, that's correct. So the denominator factors into (x - 7)(x + 2). Therefore, vertical asymptotes at x = 7 and x = -2. So that is correct.

So, I think all steps are correct. Therefore, the answer is 5.

Alternatively, maybe a and b are supposed to be ordered? For example, if they are written as x=a and x=b, maybe a is the smaller one and b is the larger one. So, a is -2 and b is 7. Then a + b is 5, and c is 0. So, 5 + 0 = 5.

Yes, that seems right.

Therefore, I think the answer is 5.

**Final Answer**
The value of $a + b + c$ is \boxed{5}.

To solve the problem, we are given the rational function:

$$
f(x) = \frac{2x}{x^2 - 5x - 14}
$$

We are asked to find:

- The vertical asymptotes $ x = a $ and $ x = b $
- The horizontal asymptote $ y = c $
- Then compute $ a + b + c $

---

### Step 1: Find the vertical asymptotes

Vertical asymptotes occur where the **denominator is zero**, provided the numerator is **not also zero** at that point.

First, factor the denominator:

$$
x^2 - 5x - 14 = (x - 7)(x + 2)
$$

So, the denominator is zero when:

$$
x = 7 \quad \text{or} \quad x = -2
$$

We now check whether the numerator is zero at these values:

- At $ x = -2 $, numerator is $ 2(-2) = -4 $ (not zero)
- At $ x = 7 $, numerator is $ 2(7) = 14 $ (also not zero)

Thus, vertical asymptotes are:

$$
x = -2 \quad \text{and} \quad x = 7
$$

So, $ a = -2 $, $ b = 7 $

---

### Step 2: Find the horizontal asymptote

To find the horizontal asymptote, compare the **degrees** of the numerator and denominator.

- Numerator is of degree 1 (linear: $ 2x $)
- Denominator is of degree 2 (quadratic: $ x^2 $)

Since the degree of the numerator is **less than** the degree of the denominator, the horizontal asymptote is:

$$
y = 0
$$

So, $ c = 0 $

---

### Step 3: Compute $ a + b + c $

$$
a + b + c = (-2) + 7 + 0 = 5
$$

---

### Final Answer:

$$
\boxed{5}
$$

[Assistant]
Explanation:
Okay, so I need to find the vertical asymptotes and the horizontal asymptote of the function f(x) = (2x)/(x² - 5x - 14), and then add up their values a + b + c. Let me start by recalling what vertical and horizontal asymptotes are.

Vertical asymptotes occur where the denominator of a rational function is zero, provided that the numerator isn't also zero at those points. So, first step: factor the denominator and find its roots.

The denominator is x² - 5x - 14. Let me try to factor this quadratic. I need two numbers that multiply to -14 and add to -5. Hmm, let's see. Let me think: factors of -14 could be -7 and 2 because -7 * 2 = -14, and -7 + 2 = -5. Yes, that works! So the denominator factors as (x - 7)(x + 2). Let me check that: (x - 7)(x + 2) = x² + 2x - 7x -14 = x² -5x -14. Perfect! So the denominator factors into (x - 7)(x + 2).

So the vertical asymptotes are at the values of x that make the denominator zero, so x = 7 and x = -2. Therefore, a = -2 and b = 7? Wait, but the question says vertical asymptotes are x = a and x = b. They might not specify order, so a and b could be -2 and 7. But when they say a + b, it would be (-2) + 7 = 5. But I need to confirm that these are indeed vertical asymptotes. Let me check that the numerator isn't zero at these points. The numerator is 2x. So when x = -2, numerator is 2*(-2) = -4, which isn't zero. Similarly, at x = 7, numerator is 2*7 = 14, also not zero. So yes, both are vertical asymptotes. So a and b are -2 and 7, so a + b is 5.

Now, moving on to the horizontal asymptote. For horizontal asymptotes of rational functions, the rules depend on the degrees of the numerator and denominator. Let me recall: if the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If the degrees are equal, it's the ratio of the leading coefficients. If the degree of the numerator is higher, there's no horizontal asymptote (but there might be an oblique asymptote).

So here, the numerator is 2x, which is degree 1, and the denominator is x² -5x -14, which is degree 2. Since degree of numerator (1) is less than degree of denominator (2), the horizontal asymptote is y = 0. Therefore, c = 0.

So then a + b + c would be (-2) + 7 + 0 = 5. Wait, but let me confirm that there isn't a horizontal asymptote at some other value. Wait, maybe I need to double-check the horizontal asymptote calculation.

Alternatively, maybe I can do the limit as x approaches infinity of f(x). Let me compute the limit as x approaches infinity of (2x)/(x² -5x -14). To evaluate this, divide numerator and denominator by x², which is the highest power in the denominator.

So, numerator becomes (2x)/x² = 2/x. Denominator becomes (x² -5x -14)/x² = 1 - 5/x -14/x². Then as x approaches infinity, 2/x approaches 0, 5/x approaches 0, and 14/x² approaches 0. Therefore, the limit is 0 / (1 - 0 - 0) = 0. So yes, horizontal asymptote is y = 0. Therefore, c = 0.

Therefore, a is -2, b is 7, c is 0, so a + b + c = -2 + 7 + 0 = 5.

But wait, let me make sure that they didn't consider the horizontal asymptote in another way. For example, sometimes people might think about the leading terms. The leading term of the numerator is 2x and the denominator is x², so dividing gives 2x/x² = 2/x, which tends to 0 as x approaches infinity. So that also confirms that horizontal asymptote is y = 0.

So, I think that's correct. Therefore, the answer is 5. Wait, but maybe I need to check if the problem says "horizontal asymptote y = c" and maybe there's another horizontal asymptote? For example, sometimes if the degree of the numerator is exactly one more than the denominator, you get an oblique asymptote. But here, the degree of numerator is one less than denominator, so horizontal asymptote is y = 0. So I think that's correct.

Wait, another thing to check: sometimes when people say horizontal asymptote, they might mean the horizontal line that the function approaches, and in this case, since it's y = 0, which is the x-axis. So that is the horizontal asymptote. So I think that's solid.

Therefore, the sum a + b + c is (-2) + 7 + 0 = 5. So, 5 is the answer.

But let me check once again. Maybe there's a mistake in factoring the denominator? Let me check again. The denominator is x² -5x -14. Factoring, looking for two numbers m and n such that m * n = -14 and m + n = -5. Let me think. The factors of -14 could be ( -7, 2 ) since -7*2 = -14 and -7 + 2 = -5. Yes, that's correct. So the denominator factors into (x - 7)(x + 2). Therefore, vertical asymptotes at x = 7 and x = -2. So that is correct.

So, I think all steps are correct. Therefore, the answer is 5.

Alternatively, maybe a and b are supposed to be ordered? For example, if they are written as x=a and x=b, maybe a is the smaller one and b is the larger one. So, a is -2 and b is 7. Then a + b is 5, and c is 0. So, 5 + 0 = 5.

Yes, that seems right.

Therefore, I think the answer is 5.

**Final Answer**
The value of $a + b + c$ is \boxed{5}.

To solve the problem, we are given the rational function:

$$
f(x) = \frac{2x}{x^2 - 5x - 14}
$$

We are asked to find:

- The vertical asymptotes $ x = a $ and $ x = b $
- The horizontal asymptote $ y = c $
- Then compute $ a + b + c $

---

### Step 1: Find the vertical asymptotes

Vertical asymptotes occur where the **denominator is zero**, provided the numerator is **not also zero** at that point.

First, factor the denominator:

$$
x^2 - 5x - 14 = (x - 7)(x + 2)
$$

So, the denominator is zero when:

$$
x = 7 \quad \text{or} \quad x = -2
$$

We now check whether the numerator is zero at these values:

- At $ x = -2 $, numerator is $ 2(-2) = -4 $ (not zero)
- At $ x = 7 $, numerator is $ 2(7) = 14 $ (also not zero)

Thus, vertical asymptotes are:

$$
x = -2 \quad \text{and} \quad x = 7
$$

So, $ a = -2 $, $ b = 7 $

---

### Step 2: Find the horizontal asymptote

To find the horizontal asymptote, compare the **degrees** of the numerator and denominator.

- Numerator is of degree 1 (linear: $ 2x $)
- Denominator is of degree 2 (quadratic: $ x^2 $)

Since the degree of the numerator is **less than** the degree of the denominator, the horizontal asymptote is:

$$
y = 0
$$

So, $ c = 0 $

---

### Step 3: Compute $ a + b + c $

$$
a + b + c = (-2) + 7 + 0 = 5
$$

---

### Final Answer:

$$
\boxed{5}
$$

Solution:
Vertical asymptotes occur at values of $x$ where the denominator is 0.  We can factor the denominator into $(x-7)(x+2)$, so the denominator equals 0 when $x=7$ or $x=-2$. Those $x$-values are where our vertical asymptotes are located.

For horizontal asymptotes, we look at the degree of $x$ in the numerator and the denominator. The degree of the numerator is 1, and the degree of the denominator is 2, so the denominator grows faster than the numerator for large values of $x$, and the function approaches the horizontal asymptote $y=0$. We can also see that when we divide $x$ out of the numerator and denominator, we get \[\frac{2x}{x^2 - 5x - 14} = \frac{\frac{2x}{x}}{\frac{x^2-5x-14}{x}}=\frac{2}{x-5-\frac{14}{x}}.\]As $x$ approaches infinity or negative infinity, the expression approaches 0.

So, our answer is $7 + (-2) + 0 = \boxed{5}$.


Explanation:
Okay, let me try to figure out this problem. So, the problem says I have seven bags of gold coins, each with the same number of coins. Then I find an additional bag of 53 coins, making it eight bags total. I redistribute all the coins so that all eight bags have the same number of coins. After redistribution, I have more than 200 coins. The question is asking for the smallest number of coins I could have had before finding the bag of 53 coins.

Hmm, let me break this down step by step. Let me denote the original number of coins in each of the seven bags as 'x'. So initially, the total number of coins is 7x. Then I find a bag of 53 coins, so the total number of coins becomes 7x + 53. Then I redistribute these coins into eight bags so that each has the same number. Let me call the number of coins in each of the eight bags as 'y'. Therefore, the total number of coins after redistribution is 8y.

Since all the coins are redistributed, we can set up the equation:

7x + 53 = 8y.

Also, the problem says that after redistribution, the total number of coins is more than 200. Therefore:

8y > 200.

Which simplifies to y > 25. Since y has to be an integer (you can't have a fraction of a coin), y must be at least 26.

But also, we need to find the smallest original number of coins before finding the 53-coin bag. That would be 7x, and we need to find the minimal 7x such that 7x + 53 is divisible by 8 (so that y is integer), and 8y > 200.

So, first, let's rephrase the equation:

7x + 53 ≡ 0 mod 8.

So, solving for x here. Let me rearrange the equation:

7x ≡ -53 mod 8.

But -53 mod 8 can be calculated as follows. Since 53 divided by 8 is 6*8=48, with a remainder of 5. Therefore, 53 ≡ 5 mod 8, so -53 ≡ -5 mod 8. Which is equivalent to 3 mod 8 (since -5 + 8 = 3). Therefore:

7x ≡ 3 mod 8.

So, we have 7x ≡ 3 mod 8. Now, to solve for x, we can find the multiplicative inverse of 7 modulo 8. Since 7 and 8 are coprime, the inverse exists. Let me find an integer 'a' such that 7a ≡ 1 mod 8. Testing a=1: 7*1=7≡7 mod8. Not 1. a=3: 7*3=21≡5 mod8. a=5: 7*5=35≡3 mod8. a=7: 7*7=49≡1 mod8. Ah, so inverse of 7 mod8 is 7.

Therefore, multiplying both sides by 7:

x ≡ 3*7 mod8 ⇒ x ≡21 mod8 ⇒ 21 divided by 8 is 2*8=16, remainder 5. So x ≡5 mod8.

Therefore, x=8k +5 for some integer k ≥0.

So the original number of coins was 7x =7*(8k +5)=56k +35.

Now, we also have that after adding 53 coins, total coins are 7x +53 =56k +35 +53=56k +88. This must equal 8y. Therefore:

56k +88=8y ⇒ divide both sides by 8: 7k + 11 = y.

So y=7k +11. And since we know that y>25, then:

7k +11 >25 ⇒ 7k >14 ⇒k>2. Since k must be integer, so k≥3.

But since we need the minimal original number of coins, which is 7x=56k +35, we need the minimal k≥3. Therefore, the smallest k is 3. Let me check that.

If k=3, then original coins 56*3 +35= 168 +35=203. Wait, but the problem says that after redistribution, the total coins are more than 200. Let's check: total coins after adding 53 is 203 +53=256? Wait, no. Wait, 7x was 203? Wait, original coins were 7x. Wait no, original coins is 7x. Then adding 53 coins makes total coins 7x +53. Which equals 8y. So for k=3:

Original coins: 56k +35=56*3 +35= 168+35=203. Then adding 53 coins: 203 +53=256. Which is 8y, so y=256 /8=32. Which is greater than 25. So that's okay.

But the problem says "you have more than 200 coins". Wait, does that refer to after redistribution? Let me check the problem statement again.

"You successfully manage to redistribute all the coins, and you also note that you have more than 200 coins." So "you have more than 200 coins" after redistribution. So total coins after redistribution is 8y>200. Which for y=32, 8*32=256>200. So that's okay.

But the question is asking for the smallest number of coins you could have had before finding the bag of 53 coins. Which is 7x. So if k=3 gives 203, but maybe there is a smaller k? Wait, but earlier we saw that k must be ≥3 because y=7k +11>25 ⇒k>2. So k must be at least 3. Therefore, the minimal original number is 203? But wait, let me check if k=2 gives something?

For k=2, original coins would be 56*2 +35=112 +35=147. Then total coins after adding 53 is 147 +53=200. But since we need total coins after redistribution to be more than 200, 200 is not allowed. Therefore, k=2 is invalid. Therefore, the minimal k is 3, leading to original coins 203.

But wait, hold on. Let me check again. Maybe there's a mistake here. Because 7x +53 must be divisible by 8. So maybe there is a smaller x which satisfies 7x ≡ -53 mod8, which we found x≡5 mod8. So x can be 5,13,21,29,...

Then original number of coins is 7x, which would be 35, 91, 147, 203,...

Then adding 53: 35+53=88. 88 divided by 8 is 11. So total coins after redistribution is 88, which is 8*11. But 88 is less than 200, which is not allowed. Then next x is 13. 7*13=91. Adding 53 is 144. 144/8=18. 18*8=144. Still less than 200. Next x=21: 7*21=147. Adding 53=200. Again, total coins is 200, which is not more than 200. Therefore, invalid. Next x=29: 7*29=203. Adding 53=256. Which is 256>200, okay.

Therefore, the minimal original number is 203? But the problem says "you have more than 200 coins" after redistribution. So 256 is okay. But maybe there's a smaller x?

Wait, but according to our previous equations, x must be of the form 8k +5, so k=0: x=5, k=1: x=13, k=2: x=21, k=3: x=29. Then original coins 7x would be 35,91,147,203,... So yes, 203 is the first one where after adding 53, total coins is more than 200.

But let me check if there's another way. Maybe I made a mistake in solving the congruence?

Original equation: 7x +53 = 8y. So 7x = 8y -53. Therefore, 7x must be equal to 8y -53. Therefore, 8y -53 must be divisible by 7. So 8y ≡53 mod7. Let me check that again.

Alternatively, maybe I can approach it differently. Let's see:

We have 7x +53 must be divisible by 8. So 7x ≡ -53 mod8. As before, -53 mod8 is same as (-53 + 64)=11 mod8. Wait, 53 divided by 8 is 6*8=48, 53-48=5. Therefore, -53 ≡ -5 mod8. Which is 3 mod8. So 7x ≡3 mod8. Which then gives x≡5 mod8. So that's correct.

Alternatively, maybe trying to find the minimal 7x such that 7x +53 is divisible by 8 and 7x +53 > 200. Wait, but 7x +53 >200 ⇒7x >147 ⇒x>21. So since x must be integer, x≥22. But x must be ≡5 mod8. So the minimal x≥22 with x≡5 mod8. Let's see:

Numbers ≡5 mod8 are 5,13,21,29,37,...

So x must be ≥22. The first one is 29. Therefore, x=29. Hence original coins is 7*29=203. Which matches the previous result. So that seems to be the answer.

But let me check for x=29. Original coins: 7*29=203. Adding 53: 203 +53=256. Divided by 8 is 32. So 32*8=256. So that works. And 256>200. So that's valid.

Is there a smaller x? Let's check. For x=21, which is 7*21=147. Adding 53: 147+53=200. Which is equal to 200, but problem says "more than 200", so not allowed. x=21 is invalid. Next is x=29. So yes, 29 is the minimal x which satisfies x≡5 mod8 and x≥22. Therefore, original number of coins is 203.

But wait, let me check another way. Suppose I didn't go through congruence, but tried to find the minimal total coins after adding 53, which is more than 200. So total coins after adding is T = 7x +53 >200 ⇒7x > 147 ⇒x>21. So x≥22. And T must be divisible by 8. So T=8y, with y>25. So T must be a multiple of 8 greater than 200. The minimal such T is 208 (since 200/8=25, so next multiple is 25*8=200, then 26*8=208). So T must be at least 208. Therefore, 7x +53 ≥208 ⇒7x≥155 ⇒x≥155/7≈22.14. Since x must be integer, x≥23.

But x must also satisfy that 7x +53 is divisible by 8. Let me check for x=23: 7*23=161. 161+53=214. Is 214 divisible by 8? 214 /8=26.75. No. Not divisible.

x=24: 7*24=168. 168+53=221. 221/8=27.625. Not integer.

x=25: 7*25=175. 175+53=228. 228/8=28.5. No.

x=26: 7*26=182. 182+53=235. 235/8=29.375. No.

x=27: 7*27=189. 189+53=242. 242/8=30.25. No.

x=28: 7*28=196. 196+53=249. 249/8=31.125. No.

x=29: 7*29=203. 203+53=256. 256/8=32. Yes, that's integer.

So x=29 is the first x≥23 which makes T=256 divisible by 8. Therefore, original coins is 7*29=203.

So that seems to confirm the answer is 203. Therefore, the smallest number of coins you could have had before finding the bag of 53 coins is 203.

But let me check again if there is a smaller T between 208 and 256 that is divisible by 8 and satisfies 7x +53=T.

For example, T=208: 7x=208 -53=155. 155/7≈22.14. Not integer.

T=216: 7x=216-53=163. 163/7≈23.28. Not integer.

T=224: 7x=224-53=171. 171/7≈24.428. No.

T=232: 232-53=179. 179/7≈25.57. No.

T=240: 240-53=187. 187/7≈26.71. No.

T=248: 248-53=195. 195/7≈27.85. No.

T=256: 256-53=203. 203/7=29. Yes. So that's the first one.

Therefore, the answer is 203.

But wait, the problem says "you have seven bags of gold coins. Each bag has the same number of gold coins." So before adding the 53-coin bag, the total coins were 7x. So 7x must be a multiple of 7, which it is. But also, the original number of coins in each bag must be an integer. Which it is. So x=29, so each original bag had 29 coins.

But let me check if there is any other constraint. For example, when you redistribute, you have eight bags. So the number of coins per bag after redistribution is y=32, as we saw. So each bag has 32 coins.

But is there any restriction on the number of coins per bag? The problem doesn't mention any, so as long as it's an integer, which it is.

Therefore, unless I made a mistake in the congruence or in the calculations, 203 should be the answer.

But let me check once more with equations.

We had 7x +53 = 8y. And 8y > 200 ⇒ y ≥ 26.

Therefore, 7x = 8y -53. Since y must be ≥26, let's compute 8*26 -53 = 208 -53 = 155. So 7x=155 ⇒x≈22.14. Not integer.

Next y=27: 8*27=216. 216-53=163. 163/7≈23.28. Not integer.

y=28: 224-53=171. 171/7≈24.428. No.

y=29: 232-53=179. 179/7≈25.57. No.

y=30: 240-53=187. 187/7≈26.71. No.

y=31: 248-53=195. 195/7≈27.85. No.

y=32: 256-53=203. 203/7=29. Yes.

Therefore, the minimal y is 32, leading to x=29, original total coins 7*29=203.

Therefore, the answer is 203.

But wait, just to be thorough, is there another way where maybe the total coins after adding the 53 coins is not the minimal multiple of 8 above 200? For example, maybe there's some other combination where the total coins is higher, but the original coins is smaller? But no, because original coins is 7x, and we found that for y=32, the original coins is 203, and for lower y's, the original coins would have been lower but would not satisfy the divisibility.

Moreover, we checked all the y from 26 up to 32 and only y=32 gives an integer x. Therefore, 203 is indeed the minimal original number.

So I think that's the answer.

**Final Answer**
The smallest number of coins you could have had before finding the bag of 53 coins is \boxed{203}.

To determine the smallest number of coins you could have had **before** finding the additional bag of 53 coins, we need to analyze the situation carefully.

---

### Step 1: Define the Variables

Let:
- $ x $ be the number of coins originally in **each of the 7 bags**
- Then the **initial total number of coins** is $ 7x $

You discover an additional bag of **53 coins**, so the **new total number of coins** becomes:

$$
7x + 53
$$

You redistribute all these coins into **8 bags**, making the number of coins in each of the new 8 bags equal to $ y $, giving the equation:

$$
7x + 53 = 8y
$$

Furthermore, we're told the **final total number of coins** is more than 200:

$$
8y > 200 \Rightarrow y > 25
$$

So $ y $ must be an **integer greater than 25**.

---

### Step 2: Reformulate the Equation

Rewriting the equation:

$$
7x + 53 = 8y \Rightarrow 7x = 8y - 53
$$

This tells us that $ 8y - 53 $ must be divisible by 7. That is:

$$
8y - 53 \equiv 0 \pmod{7} \Rightarrow 8y \equiv 53 \pmod{7}
$$

Since $ 53 \equiv 53 - 7 \times 7 = 53 - 49 = 4 \pmod{7} $, and $ 8 \equiv 1 \pmod{7} $, we get:

$$
8y \equiv y \pmod{7} \Rightarrow y \equiv 4 \pmod{7}
$$

Therefore, $ y = 7k + 4 $ for some integer $ k \geq 3 $ (since $ y > 25 $).

Substituting back:

$$
7x = 8y - 53 = 8(7k + 4) - 53 = 56k + 32 - 53 = 56k - 21
$$
$$
x = \frac{56k - 21}{7} = 8k - 3
$$

So the number of coins in each original bag is $ x = 8k - 3 $, and the original total number of coins was $ 7x = 7(8k - 3) = 56k - 21 $

We now check for the **smallest valid $ k $** that ensures $ y > 25 $ and $ x $ is a positive integer:

Try $ k = 3 $:

- $ x = 8(3) - 3 = 24 - 3 = 21 $
- $ 7x = 7 \times 21 = 147 $
- $ 7x + 53 = 147 + 53 = 200 $

But we are told the **total after redistribution is more than 200**, so we must raise $ k $.

Try $ k = 4 $:

- $ x = 8(4) - 3 = 32 - 3 = 29 $
- $ 7x = 7 \times 29 = 203 $
- $ 7x + 53 = 203 + 53 = 256 $

Now check if 256 is divisible by 8:

$$
256 \div 8 = 32 \Rightarrow \text{Yes}
$$

And $ 256 > 200 $, satisfying all conditions.

Thus, the **smallest number of coins you could have had originally** is:

$$
\boxed{203}
$$

[Assistant]
Explanation:
Okay, let me try to figure out this problem. So, the problem says I have seven bags of gold coins, each with the same number of coins. Then I find an additional bag of 53 coins, making it eight bags total. I redistribute all the coins so that all eight bags have the same number of coins. After redistribution, I have more than 200 coins. The question is asking for the smallest number of coins I could have had before finding the bag of 53 coins.

Hmm, let me break this down step by step. Let me denote the original number of coins in each of the seven bags as 'x'. So initially, the total number of coins is 7x. Then I find a bag of 53 coins, so the total number of coins becomes 7x + 53. Then I redistribute these coins into eight bags so that each has the same number. Let me call the number of coins in each of the eight bags as 'y'. Therefore, the total number of coins after redistribution is 8y.

Since all the coins are redistributed, we can set up the equation:

7x + 53 = 8y.

Also, the problem says that after redistribution, the total number of coins is more than 200. Therefore:

8y > 200.

Which simplifies to y > 25. Since y has to be an integer (you can't have a fraction of a coin), y must be at least 26.

But also, we need to find the smallest original number of coins before finding the 53-coin bag. That would be 7x, and we need to find the minimal 7x such that 7x + 53 is divisible by 8 (so that y is integer), and 8y > 200.

So, first, let's rephrase the equation:

7x + 53 ≡ 0 mod 8.

So, solving for x here. Let me rearrange the equation:

7x ≡ -53 mod 8.

But -53 mod 8 can be calculated as follows. Since 53 divided by 8 is 6*8=48, with a remainder of 5. Therefore, 53 ≡ 5 mod 8, so -53 ≡ -5 mod 8. Which is equivalent to 3 mod 8 (since -5 + 8 = 3). Therefore:

7x ≡ 3 mod 8.

So, we have 7x ≡ 3 mod 8. Now, to solve for x, we can find the multiplicative inverse of 7 modulo 8. Since 7 and 8 are coprime, the inverse exists. Let me find an integer 'a' such that 7a ≡ 1 mod 8. Testing a=1: 7*1=7≡7 mod8. Not 1. a=3: 7*3=21≡5 mod8. a=5: 7*5=35≡3 mod8. a=7: 7*7=49≡1 mod8. Ah, so inverse of 7 mod8 is 7.

Therefore, multiplying both sides by 7:

x ≡ 3*7 mod8 ⇒ x ≡21 mod8 ⇒ 21 divided by 8 is 2*8=16, remainder 5. So x ≡5 mod8.

Therefore, x=8k +5 for some integer k ≥0.

So the original number of coins was 7x =7*(8k +5)=56k +35.

Now, we also have that after adding 53 coins, total coins are 7x +53 =56k +35 +53=56k +88. This must equal 8y. Therefore:

56k +88=8y ⇒ divide both sides by 8: 7k + 11 = y.

So y=7k +11. And since we know that y>25, then:

7k +11 >25 ⇒ 7k >14 ⇒k>2. Since k must be integer, so k≥3.

But since we need the minimal original number of coins, which is 7x=56k +35, we need the minimal k≥3. Therefore, the smallest k is 3. Let me check that.

If k=3, then original coins 56*3 +35= 168 +35=203. Wait, but the problem says that after redistribution, the total coins are more than 200. Let's check: total coins after adding 53 is 203 +53=256? Wait, no. Wait, 7x was 203? Wait, original coins were 7x. Wait no, original coins is 7x. Then adding 53 coins makes total coins 7x +53. Which equals 8y. So for k=3:

Original coins: 56k +35=56*3 +35= 168+35=203. Then adding 53 coins: 203 +53=256. Which is 8y, so y=256 /8=32. Which is greater than 25. So that's okay.

But the problem says "you have more than 200 coins". Wait, does that refer to after redistribution? Let me check the problem statement again.

"You successfully manage to redistribute all the coins, and you also note that you have more than 200 coins." So "you have more than 200 coins" after redistribution. So total coins after redistribution is 8y>200. Which for y=32, 8*32=256>200. So that's okay.

But the question is asking for the smallest number of coins you could have had before finding the bag of 53 coins. Which is 7x. So if k=3 gives 203, but maybe there is a smaller k? Wait, but earlier we saw that k must be ≥3 because y=7k +11>25 ⇒k>2. So k must be at least 3. Therefore, the minimal original number is 203? But wait, let me check if k=2 gives something?

For k=2, original coins would be 56*2 +35=112 +35=147. Then total coins after adding 53 is 147 +53=200. But since we need total coins after redistribution to be more than 200, 200 is not allowed. Therefore, k=2 is invalid. Therefore, the minimal k is 3, leading to original coins 203.

But wait, hold on. Let me check again. Maybe there's a mistake here. Because 7x +53 must be divisible by 8. So maybe there is a smaller x which satisfies 7x ≡ -53 mod8, which we found x≡5 mod8. So x can be 5,13,21,29,...

Then original number of coins is 7x, which would be 35, 91, 147, 203,...

Then adding 53: 35+53=88. 88 divided by 8 is 11. So total coins after redistribution is 88, which is 8*11. But 88 is less than 200, which is not allowed. Then next x is 13. 7*13=91. Adding 53 is 144. 144/8=18. 18*8=144. Still less than 200. Next x=21: 7*21=147. Adding 53=200. Again, total coins is 200, which is not more than 200. Therefore, invalid. Next x=29: 7*29=203. Adding 53=256. Which is 256>200, okay.

Therefore, the minimal original number is 203? But the problem says "you have more than 200 coins" after redistribution. So 256 is okay. But maybe there's a smaller x?

Wait, but according to our previous equations, x must be of the form 8k +5, so k=0: x=5, k=1: x=13, k=2: x=21, k=3: x=29. Then original coins 7x would be 35,91,147,203,... So yes, 203 is the first one where after adding 53, total coins is more than 200.

But let me check if there's another way. Maybe I made a mistake in solving the congruence?

Original equation: 7x +53 = 8y. So 7x = 8y -53. Therefore, 7x must be equal to 8y -53. Therefore, 8y -53 must be divisible by 7. So 8y ≡53 mod7. Let me check that again.

Alternatively, maybe I can approach it differently. Let's see:

We have 7x +53 must be divisible by 8. So 7x ≡ -53 mod8. As before, -53 mod8 is same as (-53 + 64)=11 mod8. Wait, 53 divided by 8 is 6*8=48, 53-48=5. Therefore, -53 ≡ -5 mod8. Which is 3 mod8. So 7x ≡3 mod8. Which then gives x≡5 mod8. So that's correct.

Alternatively, maybe trying to find the minimal 7x such that 7x +53 is divisible by 8 and 7x +53 > 200. Wait, but 7x +53 >200 ⇒7x >147 ⇒x>21. So since x must be integer, x≥22. But x must be ≡5 mod8. So the minimal x≥22 with x≡5 mod8. Let's see:

Numbers ≡5 mod8 are 5,13,21,29,37,...

So x must be ≥22. The first one is 29. Therefore, x=29. Hence original coins is 7*29=203. Which matches the previous result. So that seems to be the answer.

But let me check for x=29. Original coins: 7*29=203. Adding 53: 203 +53=256. Divided by 8 is 32. So 32*8=256. So that works. And 256>200. So that's valid.

Is there a smaller x? Let's check. For x=21, which is 7*21=147. Adding 53: 147+53=200. Which is equal to 200, but problem says "more than 200", so not allowed. x=21 is invalid. Next is x=29. So yes, 29 is the minimal x which satisfies x≡5 mod8 and x≥22. Therefore, original number of coins is 203.

But wait, let me check another way. Suppose I didn't go through congruence, but tried to find the minimal total coins after adding 53, which is more than 200. So total coins after adding is T = 7x +53 >200 ⇒7x > 147 ⇒x>21. So x≥22. And T must be divisible by 8. So T=8y, with y>25. So T must be a multiple of 8 greater than 200. The minimal such T is 208 (since 200/8=25, so next multiple is 25*8=200, then 26*8=208). So T must be at least 208. Therefore, 7x +53 ≥208 ⇒7x≥155 ⇒x≥155/7≈22.14. Since x must be integer, x≥23.

But x must also satisfy that 7x +53 is divisible by 8. Let me check for x=23: 7*23=161. 161+53=214. Is 214 divisible by 8? 214 /8=26.75. No. Not divisible.

x=24: 7*24=168. 168+53=221. 221/8=27.625. Not integer.

x=25: 7*25=175. 175+53=228. 228/8=28.5. No.

x=26: 7*26=182. 182+53=235. 235/8=29.375. No.

x=27: 7*27=189. 189+53=242. 242/8=30.25. No.

x=28: 7*28=196. 196+53=249. 249/8=31.125. No.

x=29: 7*29=203. 203+53=256. 256/8=32. Yes, that's integer.

So x=29 is the first x≥23 which makes T=256 divisible by 8. Therefore, original coins is 7*29=203.

So that seems to confirm the answer is 203. Therefore, the smallest number of coins you could have had before finding the bag of 53 coins is 203.

But let me check again if there is a smaller T between 208 and 256 that is divisible by 8 and satisfies 7x +53=T.

For example, T=208: 7x=208 -53=155. 155/7≈22.14. Not integer.

T=216: 7x=216-53=163. 163/7≈23.28. Not integer.

T=224: 7x=224-53=171. 171/7≈24.428. No.

T=232: 232-53=179. 179/7≈25.57. No.

T=240: 240-53=187. 187/7≈26.71. No.

T=248: 248-53=195. 195/7≈27.85. No.

T=256: 256-53=203. 203/7=29. Yes. So that's the first one.

Therefore, the answer is 203.

But wait, the problem says "you have seven bags of gold coins. Each bag has the same number of gold coins." So before adding the 53-coin bag, the total coins were 7x. So 7x must be a multiple of 7, which it is. But also, the original number of coins in each bag must be an integer. Which it is. So x=29, so each original bag had 29 coins.

But let me check if there is any other constraint. For example, when you redistribute, you have eight bags. So the number of coins per bag after redistribution is y=32, as we saw. So each bag has 32 coins.

But is there any restriction on the number of coins per bag? The problem doesn't mention any, so as long as it's an integer, which it is.

Therefore, unless I made a mistake in the congruence or in the calculations, 203 should be the answer.

But let me check once more with equations.

We had 7x +53 = 8y. And 8y > 200 ⇒ y ≥ 26.

Therefore, 7x = 8y -53. Since y must be ≥26, let's compute 8*26 -53 = 208 -53 = 155. So 7x=155 ⇒x≈22.14. Not integer.

Next y=27: 8*27=216. 216-53=163. 163/7≈23.28. Not integer.

y=28: 224-53=171. 171/7≈24.428. No.

y=29: 232-53=179. 179/7≈25.57. No.

y=30: 240-53=187. 187/7≈26.71. No.

y=31: 248-53=195. 195/7≈27.85. No.

y=32: 256-53=203. 203/7=29. Yes.

Therefore, the minimal y is 32, leading to x=29, original total coins 7*29=203.

Therefore, the answer is 203.

But wait, just to be thorough, is there another way where maybe the total coins after adding the 53 coins is not the minimal multiple of 8 above 200? For example, maybe there's some other combination where the total coins is higher, but the original coins is smaller? But no, because original coins is 7x, and we found that for y=32, the original coins is 203, and for lower y's, the original coins would have been lower but would not satisfy the divisibility.

Moreover, we checked all the y from 26 up to 32 and only y=32 gives an integer x. Therefore, 203 is indeed the minimal original number.

So I think that's the answer.

**Final Answer**
The smallest number of coins you could have had before finding the bag of 53 coins is \boxed{203}.

To determine the smallest number of coins you could have had **before** finding the additional bag of 53 coins, we need to analyze the situation carefully.

---

### Step 1: Define the Variables

Let:
- $ x $ be the number of coins originally in **each of the 7 bags**
- Then the **initial total number of coins** is $ 7x $

You discover an additional bag of **53 coins**, so the **new total number of coins** becomes:

$$
7x + 53
$$

You redistribute all these coins into **8 bags**, making the number of coins in each of the new 8 bags equal to $ y $, giving the equation:

$$
7x + 53 = 8y
$$

Furthermore, we're told the **final total number of coins** is more than 200:

$$
8y > 200 \Rightarrow y > 25
$$

So $ y $ must be an **integer greater than 25**.

---

### Step 2: Reformulate the Equation

Rewriting the equation:

$$
7x + 53 = 8y \Rightarrow 7x = 8y - 53
$$

This tells us that $ 8y - 53 $ must be divisible by 7. That is:

$$
8y - 53 \equiv 0 \pmod{7} \Rightarrow 8y \equiv 53 \pmod{7}
$$

Since $ 53 \equiv 53 - 7 \times 7 = 53 - 49 = 4 \pmod{7} $, and $ 8 \equiv 1 \pmod{7} $, we get:

$$
8y \equiv y \pmod{7} \Rightarrow y \equiv 4 \pmod{7}
$$

Therefore, $ y = 7k + 4 $ for some integer $ k \geq 3 $ (since $ y > 25 $).

Substituting back:

$$
7x = 8y - 53 = 8(7k + 4) - 53 = 56k + 32 - 53 = 56k - 21
$$
$$
x = \frac{56k - 21}{7} = 8k - 3
$$

So the number of coins in each original bag is $ x = 8k - 3 $, and the original total number of coins was $ 7x = 7(8k - 3) = 56k - 21 $

We now check for the **smallest valid $ k $** that ensures $ y > 25 $ and $ x $ is a positive integer:

Try $ k = 3 $:

- $ x = 8(3) - 3 = 24 - 3 = 21 $
- $ 7x = 7 \times 21 = 147 $
- $ 7x + 53 = 147 + 53 = 200 $

But we are told the **total after redistribution is more than 200**, so we must raise $ k $.

Try $ k = 4 $:

- $ x = 8(4) - 3 = 32 - 3 = 29 $
- $ 7x = 7 \times 29 = 203 $
- $ 7x + 53 = 203 + 53 = 256 $

Now check if 256 is divisible by 8:

$$
256 \div 8 = 32 \Rightarrow \text{Yes}
$$

And $ 256 > 200 $, satisfying all conditions.

Thus, the **smallest number of coins you could have had originally** is:

$$
\boxed{203}
$$

Solution: If there are $b$ gold coins in each of the original 7 bags, then $7b+53$ is divisible by 8. In other words, $7b + 53 \equiv 0 \pmod{8}$. Since $53 \equiv 5 \pmod{8}$ and $7 \equiv -1 \pmod{8}$, we have that $-b \equiv -5 \pmod{8}$. Multiplying both sides by $-1$, we get that $b \equiv 5 \pmod{8}$. Now, we want $7b + 53 > 200$, so as a result, $b > \frac{200-53}{7} \implies b > 21$. Therefore, we want an integer greater than 21 which leaves a remainder of 5 when divided by 8. The least such integer is 29, so you had $29 \cdot 7 = \boxed{203}$ coins before finding the bag of 53 coins.


Explanation:
Okay, so I need to simplify this expression: the square root of (2.5 squared minus 0.7 squared) divided by (2.7 minus 2.5). Let me write that down first to make sure I have it right.

So, it's √(2.5² - 0.7²) divided by (2.7 - 2.5). Let me check if I copied that correctly. Yep, that's the expression. Alright, let me think about how to approach this.

First, maybe I should compute the numerator and the denominator separately. Let me start with the numerator: √(2.5² - 0.7²). Hmm, that looks like a difference of squares. Remember, a² - b² factors into (a - b)(a + b). Maybe that can help simplify it?

Let me try that. If I factor the numerator inside the square root, it becomes √[(2.5 - 0.7)(2.5 + 0.7)]. Let me compute those values inside the parentheses.

Calculating 2.5 minus 0.7: 2.5 - 0.7. Let's see, 2.5 minus 0.5 is 2.0, so minus another 0.2 is 1.8. So, 2.5 - 0.7 is 1.8.

Then, 2.5 plus 0.7: 2.5 + 0.7. That would be 3.2. So, the numerator simplifies to √(1.8 * 3.2). Let me compute 1.8 multiplied by 3.2.

Hmm, 1.8 times 3.2. Let me do that step by step. 1 times 3.2 is 3.2, 0.8 times 3.2. Let me compute 0.8 * 3.2. Well, 0.8*3 = 2.4 and 0.8*0.2 = 0.16, so total is 2.4 + 0.16 = 2.56. So, adding that to the 3.2 gives 3.2 + 2.56 = 5.76. Therefore, the numerator is √5.76.

Wait, √5.76... Hmm, what's the square root of 5.76? Let me think. I know that 2.4 squared is 5.76 because 2.4*2.4 = 5.76. Let me check that. 2*2 is 4, 2*0.4 is 0.8, 0.4*2 is 0.8, and 0.4*0.4 is 0.16. So adding up, (2 + 0.4)^2 = 2^2 + 2*2*0.4 + 0.4^2 = 4 + 1.6 + 0.16 = 5.76. Yep, that's right. So the numerator simplifies to 2.4.

Okay, so now the numerator is 2.4. Let me check the denominator: 2.7 - 2.5. That's straightforward. 2.7 minus 2.5 is 0.2. So the entire expression simplifies to 2.4 divided by 0.2. Let me compute that.

Dividing 2.4 by 0.2. Well, dividing by 0.2 is the same as multiplying by 5, since 1/0.2 = 5. So 2.4 * 5. Let me calculate that. 2 * 5 is 10, 0.4 * 5 is 2, so total is 10 + 2 = 12. So the result is 12. Wait, so the entire expression simplifies to 12? Let me verify that again to make sure I didn't make any mistakes.

Starting again: numerator is sqrt(2.5² - 0.7²). Let me compute 2.5 squared. 2.5 squared is 6.25. Then 0.7 squared is 0.49. Subtracting, 6.25 - 0.49 = 5.76. Then square root of 5.76 is indeed 2.4. Denominator: 2.7 - 2.5 is 0.2. So 2.4 divided by 0.2 is indeed 12. So, that seems right.

But let me think again if there's another way to approach this or if there's a possibility of error. For instance, maybe I made a mistake when factoring the numerator? Let me check.

Original numerator inside the square root: 2.5² - 0.7². Using the difference of squares formula: (a² - b²) = (a - b)(a + b). So substituting a = 2.5 and b = 0.7, we have (2.5 - 0.7)(2.5 + 0.7) = 1.8 * 3.2 = 5.76. Which is correct. Then sqrt(5.76) is 2.4. All steps check out.

Alternatively, maybe I could have calculated the numerator directly without factoring? Let me try that as a check. 2.5 squared is 6.25, 0.7 squared is 0.49, subtract them to get 6.25 - 0.49 = 5.76. Square root of 5.76 is 2.4. Yep, same result. So that's consistent.

Denominator is straightforward: 2.7 - 2.5 = 0.2. Then 2.4 divided by 0.2. Let me do this division step by step. 2.4 divided by 0.2. Since both numbers are decimals, I can multiply numerator and denominator by 10 to eliminate the decimal. So 2.4 * 10 = 24, and 0.2 * 10 = 2. So now it's 24 divided by 2, which is 12. Yep, same answer. So that seems consistent.

Alternatively, thinking in fractions. Maybe converting the decimals to fractions might be another approach? Let me try that.

First, let's convert all decimals to fractions. 2.5 is equal to 5/2, 0.7 is 7/10, 2.7 is 27/10. Let me write that down.

So numerator: sqrt( (5/2)^2 - (7/10)^2 ). Let me compute each term:

(5/2)^2 = 25/4

(7/10)^2 = 49/100

Subtracting them: 25/4 - 49/100. To subtract these fractions, need a common denominator. The least common denominator of 4 and 100 is 100. So convert 25/4 to 625/100 (since 25*25=625 and 4*25=100). Wait, hold on. Let me check that again. 25 divided by 4 is equal to (25 * 25)/ (4 * 25)? Wait, no. Let me do it properly. To convert 25/4 to a denominator of 100, multiply numerator and denominator by 25, since 4*25 = 100. So numerator becomes 25*25 = 625. Therefore, 25/4 is 625/100. Then subtract 49/100: 625/100 - 49/100 = (625 - 49)/100 = 576/100. So the expression inside the square root is 576/100. Taking the square root gives sqrt(576/100) = sqrt(576)/sqrt(100) = 24/10 = 12/5 = 2.4. Which matches our previous result.

Then the denominator is 27/10 - 5/2. Let me compute that. Convert to common denominator. 27/10 - 5/2. 5/2 is equal to 25/10. So 27/10 - 25/10 = 2/10 = 1/5 = 0.2. So denominator is 1/5. Therefore, the entire expression is numerator divided by denominator, which is (12/5) divided by (1/5) = (12/5) * (5/1) = 12. Exactly. So this method also gives 12. So that confirms the answer again.

So, whether I do decimal calculations or convert to fractions, I get 12. So I think that's solid. Therefore, the simplified value is 12.

But just to make sure I didn't make any calculation errors, let me recheck the subtraction in the fractions part again. So 25/4 minus 49/100. Converting 25/4 to 625/100. 25*25 is 625? Wait, 25*25 is 625? Wait, 25*20 is 500, 25*5 is 125, so 500 + 125 = 625. Yes. Then subtract 49 gives 625 - 49. Let me compute that. 625 - 40 = 585, then -9 more is 576. Yes, so 576/100. Square root of 576 is 24, square root of 100 is 10. 24/10 = 2.4. Correct.

Denominator: 27/10 - 5/2. 5/2 is 2.5, which is 25/10. So 27/10 - 25/10 = 2/10 = 1/5. Correct. So division of 2.4 / 0.2 is 12. All steps are consistent.

Alternatively, if I didn't factor the numerator, but just computed the square root directly. Let me try that again. Compute 2.5 squared: 6.25. 0.7 squared: 0.49. Subtract: 6.25 - 0.49. Let me compute this again. 6.25 minus 0.49. 6.25 minus 0.4 is 5.85, minus another 0.09 is 5.76. Correct. Square root of 5.76 is 2.4. Divided by 0.2. So 2.4 divided by 0.2. Another way to think is 0.2 times 10 is 2, so 0.2 times 12 is 2.4. So 0.2 times 12 = 2.4. Hence, 2.4 divided by 0.2 is 12. Yep.

Therefore, regardless of the method, the answer seems to be 12. Therefore, I think that's the correct simplified result.

But just to make sure there isn't some hidden complexity or a trick in the question that I'm missing. The original problem is √(2.5² - 0.7²)/(2.7 - 2.5). All numbers are positive, so the square root is defined, and the denominator is positive because 2.7 - 2.5 = 0.2. So no issues with division by zero or negative numbers under the square root.

Alternatively, maybe the problem expects some other form? Like a fraction? But 12 is an integer, so it's already simplified. Therefore, I think the answer is 12.

Wait, but let me think again. Maybe there is another way to simplify? For instance, maybe recognizing some relation between the numbers? Let me check.

Looking at the numbers: 2.5, 0.7, 2.7. Hmm. 2.5 is 5/2, 0.7 is 7/10, 2.7 is 27/10. Wait, 2.5 is 5/2, 2.7 is 27/10, which is 2.7. Is there a relationship between these? Not sure. Maybe not. Alternatively, maybe if I think of 2.5 and 2.7 as numbers close to each other? But I think the previous approaches have already confirmed the answer.

Alternatively, if I use decimal approximations for square roots, but since we already have exact values, that might not be necessary. For example, sqrt(5.76) is exactly 2.4 because 2.4 squared is 5.76. So that's precise. Therefore, the entire calculation is exact, not an approximation.

Therefore, I can be confident that the simplified value is 12. Therefore, the answer is 12.

**Final Answer**
\boxed{12}

To simplify the expression:

$$
\frac{\sqrt{2.5^2 - 0.7^2}}{2.7 - 2.5}
$$

---

### Step 1: Evaluate the Denominator

$$
2.7 - 2.5 = 0.2
$$

---

### Step 2: Simplify the Numerator

We are given:

$$
\sqrt{2.5^2 - 0.7^2}
$$

This expression matches the **difference of squares** identity:

$$
a^2 - b^2 = (a - b)(a + b)
$$

Apply this:

$$
2.5^2 - 0.7^2 = (2.5 - 0.7)(2.5 + 0.7) = (1.8)(3.2)
$$

Now multiply:

$$
1.8 \times 3.2 = 5.76
$$

So the numerator becomes:

$$
\sqrt{5.76} = 2.4
$$

---

### Step 3: Compute the Full Expression

Now we substitute the simplified parts:

$$
\frac{\sqrt{2.5^2 - 0.7^2}}{2.7 - 2.5} = \frac{2.4}{0.2}
$$

This can be simplified as:

$$
\frac{2.4}{0.2} = 12
$$

---

### Final Answer:

$$
\boxed{12}
$$

[Assistant]
Explanation:
Okay, so I need to simplify this expression: the square root of (2.5 squared minus 0.7 squared) divided by (2.7 minus 2.5). Let me write that down first to make sure I have it right.

So, it's √(2.5² - 0.7²) divided by (2.7 - 2.5). Let me check if I copied that correctly. Yep, that's the expression. Alright, let me think about how to approach this.

First, maybe I should compute the numerator and the denominator separately. Let me start with the numerator: √(2.5² - 0.7²). Hmm, that looks like a difference of squares. Remember, a² - b² factors into (a - b)(a + b). Maybe that can help simplify it?

Let me try that. If I factor the numerator inside the square root, it becomes √[(2.5 - 0.7)(2.5 + 0.7)]. Let me compute those values inside the parentheses.

Calculating 2.5 minus 0.7: 2.5 - 0.7. Let's see, 2.5 minus 0.5 is 2.0, so minus another 0.2 is 1.8. So, 2.5 - 0.7 is 1.8.

Then, 2.5 plus 0.7: 2.5 + 0.7. That would be 3.2. So, the numerator simplifies to √(1.8 * 3.2). Let me compute 1.8 multiplied by 3.2.

Hmm, 1.8 times 3.2. Let me do that step by step. 1 times 3.2 is 3.2, 0.8 times 3.2. Let me compute 0.8 * 3.2. Well, 0.8*3 = 2.4 and 0.8*0.2 = 0.16, so total is 2.4 + 0.16 = 2.56. So, adding that to the 3.2 gives 3.2 + 2.56 = 5.76. Therefore, the numerator is √5.76.

Wait, √5.76... Hmm, what's the square root of 5.76? Let me think. I know that 2.4 squared is 5.76 because 2.4*2.4 = 5.76. Let me check that. 2*2 is 4, 2*0.4 is 0.8, 0.4*2 is 0.8, and 0.4*0.4 is 0.16. So adding up, (2 + 0.4)^2 = 2^2 + 2*2*0.4 + 0.4^2 = 4 + 1.6 + 0.16 = 5.76. Yep, that's right. So the numerator simplifies to 2.4.

Okay, so now the numerator is 2.4. Let me check the denominator: 2.7 - 2.5. That's straightforward. 2.7 minus 2.5 is 0.2. So the entire expression simplifies to 2.4 divided by 0.2. Let me compute that.

Dividing 2.4 by 0.2. Well, dividing by 0.2 is the same as multiplying by 5, since 1/0.2 = 5. So 2.4 * 5. Let me calculate that. 2 * 5 is 10, 0.4 * 5 is 2, so total is 10 + 2 = 12. So the result is 12. Wait, so the entire expression simplifies to 12? Let me verify that again to make sure I didn't make any mistakes.

Starting again: numerator is sqrt(2.5² - 0.7²). Let me compute 2.5 squared. 2.5 squared is 6.25. Then 0.7 squared is 0.49. Subtracting, 6.25 - 0.49 = 5.76. Then square root of 5.76 is indeed 2.4. Denominator: 2.7 - 2.5 is 0.2. So 2.4 divided by 0.2 is indeed 12. So, that seems right.

But let me think again if there's another way to approach this or if there's a possibility of error. For instance, maybe I made a mistake when factoring the numerator? Let me check.

Original numerator inside the square root: 2.5² - 0.7². Using the difference of squares formula: (a² - b²) = (a - b)(a + b). So substituting a = 2.5 and b = 0.7, we have (2.5 - 0.7)(2.5 + 0.7) = 1.8 * 3.2 = 5.76. Which is correct. Then sqrt(5.76) is 2.4. All steps check out.

Alternatively, maybe I could have calculated the numerator directly without factoring? Let me try that as a check. 2.5 squared is 6.25, 0.7 squared is 0.49, subtract them to get 6.25 - 0.49 = 5.76. Square root of 5.76 is 2.4. Yep, same result. So that's consistent.

Denominator is straightforward: 2.7 - 2.5 = 0.2. Then 2.4 divided by 0.2. Let me do this division step by step. 2.4 divided by 0.2. Since both numbers are decimals, I can multiply numerator and denominator by 10 to eliminate the decimal. So 2.4 * 10 = 24, and 0.2 * 10 = 2. So now it's 24 divided by 2, which is 12. Yep, same answer. So that seems consistent.

Alternatively, thinking in fractions. Maybe converting the decimals to fractions might be another approach? Let me try that.

First, let's convert all decimals to fractions. 2.5 is equal to 5/2, 0.7 is 7/10, 2.7 is 27/10. Let me write that down.

So numerator: sqrt( (5/2)^2 - (7/10)^2 ). Let me compute each term:

(5/2)^2 = 25/4

(7/10)^2 = 49/100

Subtracting them: 25/4 - 49/100. To subtract these fractions, need a common denominator. The least common denominator of 4 and 100 is 100. So convert 25/4 to 625/100 (since 25*25=625 and 4*25=100). Wait, hold on. Let me check that again. 25 divided by 4 is equal to (25 * 25)/ (4 * 25)? Wait, no. Let me do it properly. To convert 25/4 to a denominator of 100, multiply numerator and denominator by 25, since 4*25 = 100. So numerator becomes 25*25 = 625. Therefore, 25/4 is 625/100. Then subtract 49/100: 625/100 - 49/100 = (625 - 49)/100 = 576/100. So the expression inside the square root is 576/100. Taking the square root gives sqrt(576/100) = sqrt(576)/sqrt(100) = 24/10 = 12/5 = 2.4. Which matches our previous result.

Then the denominator is 27/10 - 5/2. Let me compute that. Convert to common denominator. 27/10 - 5/2. 5/2 is equal to 25/10. So 27/10 - 25/10 = 2/10 = 1/5 = 0.2. So denominator is 1/5. Therefore, the entire expression is numerator divided by denominator, which is (12/5) divided by (1/5) = (12/5) * (5/1) = 12. Exactly. So this method also gives 12. So that confirms the answer again.

So, whether I do decimal calculations or convert to fractions, I get 12. So I think that's solid. Therefore, the simplified value is 12.

But just to make sure I didn't make any calculation errors, let me recheck the subtraction in the fractions part again. So 25/4 minus 49/100. Converting 25/4 to 625/100. 25*25 is 625? Wait, 25*25 is 625? Wait, 25*20 is 500, 25*5 is 125, so 500 + 125 = 625. Yes. Then subtract 49 gives 625 - 49. Let me compute that. 625 - 40 = 585, then -9 more is 576. Yes, so 576/100. Square root of 576 is 24, square root of 100 is 10. 24/10 = 2.4. Correct.

Denominator: 27/10 - 5/2. 5/2 is 2.5, which is 25/10. So 27/10 - 25/10 = 2/10 = 1/5. Correct. So division of 2.4 / 0.2 is 12. All steps are consistent.

Alternatively, if I didn't factor the numerator, but just computed the square root directly. Let me try that again. Compute 2.5 squared: 6.25. 0.7 squared: 0.49. Subtract: 6.25 - 0.49. Let me compute this again. 6.25 minus 0.49. 6.25 minus 0.4 is 5.85, minus another 0.09 is 5.76. Correct. Square root of 5.76 is 2.4. Divided by 0.2. So 2.4 divided by 0.2. Another way to think is 0.2 times 10 is 2, so 0.2 times 12 is 2.4. So 0.2 times 12 = 2.4. Hence, 2.4 divided by 0.2 is 12. Yep.

Therefore, regardless of the method, the answer seems to be 12. Therefore, I think that's the correct simplified result.

But just to make sure there isn't some hidden complexity or a trick in the question that I'm missing. The original problem is √(2.5² - 0.7²)/(2.7 - 2.5). All numbers are positive, so the square root is defined, and the denominator is positive because 2.7 - 2.5 = 0.2. So no issues with division by zero or negative numbers under the square root.

Alternatively, maybe the problem expects some other form? Like a fraction? But 12 is an integer, so it's already simplified. Therefore, I think the answer is 12.

Wait, but let me think again. Maybe there is another way to simplify? For instance, maybe recognizing some relation between the numbers? Let me check.

Looking at the numbers: 2.5, 0.7, 2.7. Hmm. 2.5 is 5/2, 0.7 is 7/10, 2.7 is 27/10. Wait, 2.5 is 5/2, 2.7 is 27/10, which is 2.7. Is there a relationship between these? Not sure. Maybe not. Alternatively, maybe if I think of 2.5 and 2.7 as numbers close to each other? But I think the previous approaches have already confirmed the answer.

Alternatively, if I use decimal approximations for square roots, but since we already have exact values, that might not be necessary. For example, sqrt(5.76) is exactly 2.4 because 2.4 squared is 5.76. So that's precise. Therefore, the entire calculation is exact, not an approximation.

Therefore, I can be confident that the simplified value is 12. Therefore, the answer is 12.

**Final Answer**
\boxed{12}

To simplify the expression:

$$
\frac{\sqrt{2.5^2 - 0.7^2}}{2.7 - 2.5}
$$

---

### Step 1: Evaluate the Denominator

$$
2.7 - 2.5 = 0.2
$$

---

### Step 2: Simplify the Numerator

We are given:

$$
\sqrt{2.5^2 - 0.7^2}
$$

This expression matches the **difference of squares** identity:

$$
a^2 - b^2 = (a - b)(a + b)
$$

Apply this:

$$
2.5^2 - 0.7^2 = (2.5 - 0.7)(2.5 + 0.7) = (1.8)(3.2)
$$

Now multiply:

$$
1.8 \times 3.2 = 5.76
$$

So the numerator becomes:

$$
\sqrt{5.76} = 2.4
$$

---

### Step 3: Compute the Full Expression

Now we substitute the simplified parts:

$$
\frac{\sqrt{2.5^2 - 0.7^2}}{2.7 - 2.5} = \frac{2.4}{0.2}
$$

This can be simplified as:

$$
\frac{2.4}{0.2} = 12
$$

---

### Final Answer:

$$
\boxed{12}
$$

Solution:
We have  \begin{align*}
\frac{\sqrt{2.5^2 - 0.7^2}}{2.7-2.5} &= \frac{\sqrt{6.25 - 0.49}}{2.7-2.5} = \frac{\sqrt{5.76}}{0.2} = \frac{\sqrt{576/100}}{0.2}\\
&= \frac{\sqrt{576}/\sqrt{100}}{0.2} = \frac{24/10}{0.2} = \frac{2.4}{0.2} = \boxed{12}.\end{align*}