Stefanie's train will leave the station in 24 minutes and she is y miles from the station. To catch the train, she walks at a speed of 4 miles per hour and later runs at a speed of 8 mph. Write an equation in terms of y for the distance she has to walk, w, to reach the station in 24 minutes. My question is...why is the answer w=16-5y/5 or w=3.2-y. I don't know how to figure it out

Since the total distance is y, if she walks a distance w, then she runs the rest of the way (y-w).

since time = distance/speed, and her total time is 2/5 hours,

w/4 + (y-w)/8 = 2/5
2w + y-w = 16/5
w+y = 16/5
w = 16/5 - y
w = (16-5y)/5

Wait... where did yo get the 2/5

To answer your question, let's break down the situation step by step:

1. First, we need to determine the time Stefanie will spend walking and running.
- The total time Stefanie has to catch the train is 24 minutes.
- Let's assume she walks for t minutes and runs for the remaining time, which is (24 - t) minutes.

2. Now, let's calculate the distance she covered while walking.
- Stefanie walks at a speed of 4 miles per hour, which means she covers 4 * (t/60) miles in t minutes.
(We divide t by 60 to convert minutes to hours.)

3. Next, let's calculate the distance she covered while running.
- Stefanie runs at a speed of 8 miles per hour, which means she covers 8 * ((24 - t)/60) miles in (24 - t) minutes.
(We divide (24 - t) by 60 to convert minutes to hours.)

4. The total distance Stefanie needs to cover is y miles.
- The distance she walks plus the distance she runs should add up to y.
- Setting up the equation: distance walked + distance run = y

Therefore, we have:
4 * (t/60) + 8 * ((24 - t)/60) = y

Now, let's simplify this equation step by step:

1. Multiply both sides of the equation by 60 to clear the fractions:
4t + 8(24 - t) = 60y

2. Distribute the 8 to simplify:
4t + 192 - 8t = 60y

3. Combine like terms:
-4t + 192 = 60y

4. Rearrange the equation to isolate t:
-4t = 60y - 192
divide both sides by -4
t = (192 - 60y)/4

5. Simplify the right side of the equation:
t = 48 - 15y/4

And that is the equation for the time Stefanie spends walking, in terms of y.

Now, let's calculate the distance she has to walk, w, using the formula: distance = speed * time.

1. Substituting t = (192 - 60y)/4 into the formula for distance walked:
distance = speed * time
distance = 4 * ((192 - 60y)/4)/60

2. Simplify the expression:
distance = (192 - 60y)/60
distance = (192/60) - (60y/60)
distance = 16/5 - y

Thus, we get the equation: w = 16/5 - y, which is equivalent to w = 3.2 - y.

So, the answer to your question is w = 16/5 - y or w = 3.2 - y, depending on how it is presented.

To solve this problem, we need to understand that Stefanie is walking for some time and then running for the remaining time to catch the train.

Let's break down the problem step-by-step:

1. We know that Stefanie's train will leave the station in 24 minutes. This is the total time she has to reach the station.

2. Stefanie walks at a speed of 4 miles per hour. To find the distance she walks, we need to calculate how far she can travel in the time it takes to walk before she starts running.

To do this, we need to convert the 24 minutes into hours. Since there are 60 minutes in an hour, we divide 24 by 60 to get the time in hours: 24/60 = 0.4 hours.

Now we can find the distance Stefanie walks at a speed of 4 miles per hour by using the formula: distance = speed × time.
So, the distance she walks is 4 × 0.4 = 1.6 miles.

Therefore, the distance Stefanie walks is 1.6 miles.

3. Now, let's find the remaining distance she needs to cover by running.

We know that Stefanie is y miles away from the station initially. Since she only walks 1.6 miles, the remaining distance is y - 1.6 miles.

4. Stefanie runs at a speed of 8 miles per hour. To find the time it takes for her to cover the remaining distance (y - 1.6 miles) at this speed, we use the formula: time = distance / speed.

The time it takes for Stefanie to run is (y - 1.6) / 8 hours.

5. We already know that Stefanie has a total of 24 minutes (or 0.4 hours) to reach the station, so the time taken for walking and running should add up to 0.4 hours.

Setting up this equation: 0.4 = 0.4 hours = 0.4 = (y - 1.6) / 8.

Now, let's solve for y:

Multiply both sides of the equation by 8:

0.4 * 8 = y - 1.6

3.2 = y - 1.6

Add 1.6 to both sides of the equation:

3.2 + 1.6 = y

4.8 = y

Therefore, the distance Stefanie has to walk, w, to reach the station in 24 minutes is y - 1.6 miles, which can be simplified as w = 4.8 - 1.6 = 3.2 miles.

So, the equation in terms of y for the distance she has to walk is w = 3.2 - y.

Please note that w = 16 - 5y/5 is incorrect. It seems to be a mistake in the calculation or understanding of the problem. The correct equation is w = 3.2 - y.