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Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards.
When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coffee shop.
When she walks $s+2$ kilometers per hour, the walk takes her 2 hours and 24 minutes, including $t$ minutes spent in the coffee shop.
Suppose Aya walks at $s+rac{1}{2}$ kilometers per hour.
Find the number of minutes the walk takes her, including the $t$ minutes spent in the coffee shop.

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<｜User｜>
Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards.
When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coffee shop.
When she walks $s+2$ kilometers per hour, the walk takes her 2 hours and 24 minutes, including $t$ minutes spent in the coffee shop.
Suppose Aya walks at $s+rac{1}{2}$ kilometers per hour.
Find the number of minutes the walk takes her, including the $t$ minutes spent in the coffee shop.<｜Assistant｜><think>
Okay, so I have this problem where Aya walks a certain distance and stops at a coffee shop. The problem gives me two scenarios with different walking speeds and the corresponding time it takes her, including the time she spends in the coffee shop. I need to find out how long it would take her if she walks at a slightly slower speed.

Let me try to break this down step by step.

First, let's parse the information given:

1. When Aya walks at a speed of \( s \) kilometers per hour, the total time taken for her walk, including the time spent in the coffee shop, is 4 hours. The walk itself is 9 kilometers long.

2. When her walking speed increases to \( s + 2 \) kilometers per hour, the total time taken becomes 2 hours and 24 minutes, which includes the same \( t \) minutes spent in the coffee shop.

The question is asking: If Aya walks at \( s + \frac{1}{2} \) kilometers per hour, how many minutes will her total walk time be, including the \( t \) minutes in the coffee shop?

Alright, so I need to find the total time when her speed is \( s + \frac{1}{2} \) km/h.

Let me denote the total time as \( T \) hours. Since the problem mentions minutes, I might need to convert \( T \) into minutes at the end.

First, let's analyze the first scenario:

- Walking speed: \( s \) km/h
- Total time: 4 hours
- Distance: 9 km

I can use the formula:

\[
\text{Time} = \frac{\text{Distance}}{\text{Speed}}
\]

So, the time she spends walking is:

\[
\frac{9}{s} \text{ hours}
\]

But the total time is 4 hours, which includes the time in the coffee shop. So, the time spent in the coffee shop \( t \) is:

\[
t = 4 - \frac{9}{s} \text{ hours}
\]

Now, moving on to the second scenario:

- Walking speed: \( s + 2 \) km/h
- Total time: 2 hours and 24 minutes

First, I need to convert 2 hours and 24 minutes into hours. Since 24 minutes is \( \frac{24}{60} = 0.4 \) hours, the total time is:

\[
2.4 \text{ hours}
\]

Again, the time spent walking is:

\[
\frac{9}{s + 2} \text{ hours}
\]

And the time spent in the coffee shop is:

\[
t = 2.4 - \frac{9}{s + 2} \text{ hours}
\]

Now, I have two expressions for \( t \):

1. \( t = 4 - \frac{9}{s} \)
2. \( t = 2.4 - \frac{9}{s + 2} \)

Since both equal \( t \), I can set them equal to each other:

\[
4 - \frac{9}{s} = 2.4 - \frac{9}{s + 2}
\]

Now, I need to solve this equation for \( s \).

Let me rearrange the equation:

\[
4 - 2.4 = \frac{9}{s} - \frac{9}{s + 2}
\]

Simplify the left side:

\[
1.6 = \frac{9}{s} - \frac{9}{s + 2}
\]

Factor out the 9:

\[
1.6 = 9 \left( \frac{1}{s} - \frac{1}{s + 2} \right )
\]

Combine the fractions on the right side:

\[
\frac{1}{s} - \frac{1}{s + 2} = \frac{(s + 2) - s}{s(s + 2)} = \frac{2}{s(s + 2)}
\]

So, substituting back:

\[
1.6 = 9 \times \frac{2}{s(s + 2)}
\]

Simplify:

\[
1.6 = \frac{18}{s(s + 2)}
\]

Multiply both sides by \( s(s + 2) \):

\[
1.6 s(s + 2) = 18
\]

Divide both sides by 1.6 to simplify:

\[
s(s + 2) = \frac{18}{1.6} = 11.25
\]

So, we have:

\[
s^2 + 2s - 11.25 = 0
\]

This is a quadratic equation. Let me solve for \( s \) using the quadratic formula:

\[
s = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \( a = 1 \), \( b = 2 \), and \( c = -11.25 \).

Plugging in the values:

\[
s = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-11.25)}}{2(1)} = \frac{-2 \pm \sqrt{4 + 45}}{2} = \frac{-2 \pm \sqrt{49}}{2}
\]

Since \( \sqrt{49} = 7 \):

\[
s = \frac{-2 \pm 7}{2}
\]

We have two solutions:

1. \( s = \frac{-2 + 7}{2} = \frac{5}{2} = 2.5 \) km/h
2. \( s = \frac{-2 - 7}{2} = \frac{-9}{2} = -4.5 \) km/h

Since speed cannot be negative, we discard \( s = -4.5 \) km/h.

Thus, \( s = 2.5 \) km/h.

Now, we need to find the total time when Aya walks at \( s + \frac{1}{2} = 2.5 + 0.5 = 3 \) km/h.

First, let's find the time spent walking at 3 km/h:

\[
\text{Time walking} = \frac{9}{3} = 3 \text{ hours}
\]

The total time is the sum of walking time and coffee shop time. We already have the coffee shop time from the first scenario.

From the first scenario, when \( s = 2.5 \) km/h, the total time was 4 hours, including 24 minutes in the coffee shop.

Wait, let me verify that.

When \( s = 2.5 \) km/h:

- Walking time: \( \frac{9}{2.5} = 3.6 \) hours
- Coffee shop time: \( 4 - 3.6 = 0.4 \) hours, which is 24 minutes.

Yes, that's correct.

So, when walking at 3 km/h:

- Walking time: 3 hours
- Coffee shop time: 0.4 hours (24 minutes)

Total time: \( 3 + 0.4 = 3.4 \) hours

Convert 0.4 hours to minutes: \( 0.4 \times 60 = 24 \) minutes

So, total time is 3 hours and 24 minutes.

But the question asks for the number of minutes, including the \( t \) minutes spent in the coffee shop.

Wait, but in the first scenario, the total time was 4 hours, which includes 24 minutes in the coffee shop. So, when she walks at 3 km/h, the total time is 3 hours and 24 minutes, which is 204 minutes.

But let me double-check.

Alternatively, perhaps I should compute the total time as:

Total time = Walking time + Coffee shop time

Walking time at 3 km/h: 9 / 3 = 3 hours

Coffee shop time: 24 minutes

Total time: 3 hours + 24 minutes = 3 hours and 24 minutes = 204 minutes.

Yes, that seems correct.

So, the answer is 204 minutes.
</think>

The total time Aya spends walking, including the time in the coffee shop, when she walks at \( s + \frac{1}{2} \) km/h is 204 minutes.

**Step-by-Step Explanation:**

1. **Define Variables:**
   - Let \( s \) be Aya's walking speed in km/h.
   - Let \( t \) be the time spent in the coffee shop in hours.

2. **First Scenario:**
   - Walking speed: \( s \) km/h
   - Total time: 4 hours
   - Walking time: \( \frac{9}{s} \) hours
   - Coffee shop time: \( t = 4 - \frac{9}{s} \) hours

3. **Second Scenario:**
   - Walking speed: \( s + 2 \) km/h
   - Total time: 2 hours 24 minutes = 2.4 hours
   - Walking time: \( \frac{9}{s + 2} \) hours
   - Coffee shop time: \( t = 2.4 - \frac{9}{s + 2} \) hours

4. **Set Equations Equal:**
   \[
   4 - \frac{9}{s} = 2.4 - \frac{9}{s + 2}
   \]

5. **Solve for \( s \):**
   \[
   1.6 = \frac{18}{s(s + 2)} \Rightarrow s^2 + 2s - 11.25 = 0
   \]
   Using the quadratic formula:
   \[
   s = 2.5 \text{ km/h}
   \]

6. **Calculate Total Time at \( s + \frac{1}{2} = 3 \) km/h:**
   - Walking time: \( \frac{9}{3} = 3 \) hours
   - Coffee shop time: 24 minutes
   - Total time: 3 hours 24 minutes = 204 minutes

**Answer:** 204 minutes