#### A Math Forum Project

 ESCOT Problem of the Week: Archive of Problems, Submissions, & Commentary

Student Version

### Marabyn - posted May 20, 2001

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 Marabyn

Marabyn usually takes a bus home from school. The weather today is beautiful and the bus is traveling so slowly that she decides to get off and just walk the rest of the way home. She'd like to walk at least 10 minutes but she doesn't want to walk more than 2 kilometers to her home which is 9 kilometers from school.

Using the tools in the Java applet, select a distance for Marabyn to have traveled in the bus and to use as the start of her walk by moving the slider, and choose a walking time from the pull-down menu.

### Questions

1. How far (distance) can Marabyn travel on the bus if she wants to walk at least 10 minutes but not more than 2 kilometers to get home? How long (time) will she walk?

2. List two more bus-travel distances and walking times that also get Marabyn home.

3. Write an equation that uses Marabyn's walking speed of 0.1 km/min and the variables "t" for walking time and "d" for bus-travel distance to express the distance of Marabyn's journey home.

Teacher Support Page

This week I was impressed by how the students continued to think about the problem and send in their revised answers. In the end, 10 out of 27 students were given credit for their answers. Everyone who revised but did not receive credit for all three questions, showed improvement with each try.

Some of the problems students had included giving the same answer for question 1 and 2. Students showed that they understood that if Marabyn rode the bus to the 8 kilometer mark it would mean that she would walk 1 kilometer. They also showed that they knew that she would walk for 10 minutes. I'm not sure why, but students gave this answer for question 1 and then gave it again for question 2.

To have correct responses for question 1 and 2 you needed to provide three specific answers that met the conditions given in the problem. You needed to understand that the total distance was 9 km. and combine that with the thought that to walk "no more than 2 km." meant that Marabyn would get off the bus "not before" the 7 km. mark.

Next if you took the time into account you understood that Marabyn would get off the bus "not after" the 8 km. mark. After establishing the range of distances, it should have been clear that the answers had to be 7.0 km, 7.5 km. and 8.0 km. with the accompanying times of 20 minutes, 15 minutes, and 10 minutes.

Students who originally gave answers including 6.5 km. or 8.5 km. quickly revised their thinking once the "at least 10 minutes" and "not more than 2 km." parts of the problem were pointed out.

To figure out the answer to question 3 it helped if you first thought of this familiar equation:

d = r x t

where d = distance, r = rate, and t = time.

Using the information in the problem would give: d = 0.1 x t. Then if you remembered that Marabyn was traveling a total distance of 9 km. and she was riding the bus for a part of that distance, you should arrive at the equation:

9 - d = 0.1 x t

where t = time she walked and d = distance she rode on the bus.

We've got four nice examples of good solutions to this probem below.

### Highlighted solutions:

 From: Katie L., age 13 School: Taipei American School, Taipei, Taiwan

```1. How far (distance) can Marabyn travel on the bus if she wants to
walk at least 10 minutes but not more than 2 kilometers to get home?
How long (time) will she walk?

Marabyn can travel on the bus for 8 kilometers, and then she'd walk
the last kilometers in 10 minutes.

2. List two more bus-travel distances and walking times that also get
Marabyn home.

1) Marabyn can get off the bus after 7 kilometers, and walk the last
2 kilometers in 20 minutes.
2) Marabyn can get off the bus after 7.5 kilometers, and walk the
last 1.5 kilometers in 15 minutes.

3. Write an equation that uses Marabyn's walking speed of 0.1 km/min
and the variables "t" for walking time and "d" for bus-travel
distance to express the distance of Marabyn's journey home.

The formula for distance is...
distance = time * rate
So we know that r = 0.1, is Marabyn's walking rate, and so following
the formula for distance, we find that the distance of the part that
Marabyn walks is
d = 0.1t
Then there's also the bus travel distance, and the distance for bus
travel is simply represented as d. So if those two were added
together, then they equal the distance of Marabyn's journey home.
But we already know that the distance of Marabyn's journey home is 9
kilometers, so we can use that and substitute it into the equation.
So the equation that expresses the distance of Marabyn's journey home
is...
9 = 0.1t + d
```

 From: Mindy H., age 13 School: Taipei American School, Taipei, Taiwan

```1. How far (distance) can Marabyn travel on the bus if she wants to
walk at least 10 minutes but not more than 2 kilometers to get home?
How long (time) will she walk?

Marabyn van travel 8.0 km with the school bus to walk at least 10
minutes but not more than 2 kilometers away, and he will walk for 10
minutes from 8.0 km to hime.

2. List two more bus-travel distances and walking times that also get
Marabyn home.

Marabyn can travel 7.5 km with the bus, and he will walk for 15
minutes. He also can travel to 7.0 km, and walk for 20 minutes.

3. Write an equation that uses Marabyn's walking speed of 0.1 km/min
and the variables "t" for walking time and "d" for bus-travel
distance to express the distance of Marabyn's journey home.

- We have to find the formula of distance:
Distance = speed * time

- We can substitute Marabyn's walking speed into the equation abive
to get the distance Marabyn has to walk.
D = 0.1 * t

- Then we add the bus-travel distance and the distance Marabyn has to
walk to arrive home.

d + 0.1t = 9

Check:
?
8.0 + 0.1(10) = 9
8 + 1 = 9

?
7.5 + 0.1(15) = 9
7.5 + 1.5 = 9

?
7.0 + 0.1(20) = 9
7 + 2 = 9
```

 From: Esther W., age 14 School: Taipei American School, Taipei, Taiwan

```1. How far (distance) can Marabyn travel on the bus if she wants to
walk at least 10 minutes but not more than 2 kilometers to get home?
How long (time) will she walk?

Marabyn can travel 7 km on the bus, and walk for 2 km. It would take her 20
minutes.

I solved this problem by finding the maxium and minimum "bus travel distance".
Because Marabyn's walking speed is 0.1km/min, it would also equal 10min = 1km.

In that case, the maximum bus ride she gets is 8 km, because she would like to
walk for at least 10 minutes which equals to 1km. 9km total - 8km bus ride =
1km walking

Because she doesn't want to walk more than 2km, her bus ride would be:
9km total - 2km walk = 7km bus ride

As long as she rides the bus for no longer than 8km no less than 7km, walks no
farther than 2 km no less than 1 km, walks no longer than 20 minutes no less
than 10 minutes, she would be fine.

2. List two more bus-travel distances and walking times that also get
Marabyn home.

Marabyn can also travel:

- 7.5 km on the bus, walk 1.5 km for 15 minutes
- 8 km on the bus, walk 1 km for 10 minutes

3. Write an equation that uses Marabyn's walking speed of 0.1 km/min
and the variables "t" for walking time and "d" for bus-travel distance
to express the distance of Marabyn's journey home.

0.1t + d = 9
(the "9"  is the "distance of Marabyn's journey home)
```

 From: Andy L., age 14 School: Taipei American School, Taipei, Taiwan

```1. How far (distance) can Marabyn travel on the bus if she wants to
walk at least 10 minutes but not more than 2 kilometers to get home?
How long (time) will she walk?

She can travel 8 kilometers on the bus if she wants to walk at least
10 minutes but no more than 2 kilometers to get home.  She will walk
for 10 minutes.
I got this answer by using the java applet and calculating.  Since we
want to know how far she can travel on the bus if she wants to walk
for at least 10 minutes, we want her to walk the least possible, so i
set the applet to 10 minutes for her walking time.  Then we need to
know how far she travels in 10 minutes.  Using the applet, it shows
that the rate of her walking is 1 kilometer per 10 minutes. So then
we can set the traveled distance of the bus to 8 kilometers, and the
walking time of Marabyn to 10 minutes, and once we predict, it shows
that it is correct!

2. List two more bus-travel distances and walking times that also get
Marabyn home.

Marabyn will also get home if the bus travels 7.5 kilometers, and she
walks for 15 minutes.  When the bus travels for 7 kilometers and she
walks for 20 minutes, she will also get home.
I got these answers because Marabyn walks 1 kilometer every 10
minutes.  This means that she walks 0.5 kilometers every 5 minutes.
Using this rate, we know that every 0.5 kilometer added or subtracted
from the bus-travel distances, we can just add or subtract a 5 minute
walking time, and she would still get home.

3. Write an equation that uses Marabyn's walking speed of 0.1 km/min
and the variables "t" for walking time and "d" for bus-travel
distance
to express the distance of Marabyn's journey home.

I will name the distance of Marabyn's journey home x.
First, I started out with this, because she will only walk the
distance of the difference of the total distance of her school to her
house and the bus-traveled distance.  Then we multiply it by 10,
since the rate of her walk speed is 0.1 km/min.
t = (x - d) * 10
Then we use the distributive property and multiply both x and d by 10.
t = 10x - 10d
Then we take the variable and switch it to the left side.  We also
take t and put it to the right side, since the variable must be alone.
-10x = -10d - t
Since x, the variable, cannot be negative, we multiply both sides of
the equation by -1 to turn it positive.
10x = 10d + t
And then we divide both sides by 10, because we want the variable to
be standing by itself.
x = d + (t/10)
And that is the equation we end up with.
```

### 10 students received credit this week.

Cansu A., age 13 - Uskudar SEV Elementary School, Istanbul, Turkey
Emily C., age 13 - Taipei American School, Taipei, Taiwan
Mindy H., age 13 - Taipei American School, Taipei, Taiwan
Andy L., age 14 - Taipei American School, Taipei, Taiwan
Katie L., age 13 - Taipei American School, Taipei, Taiwan
Michael L., age 13 - Taipei American School, Taipei, Taiwan
Max S., age 14 - Taipei American School, Taipei, Taiwan
Albert W., age 13 - Taipei American School, Taipei, Taiwan
Denise W., age 13 - Taipei American School, Taipei, Taiwan
Esther W., age 14 - Taipei American School, Taipei, Taiwan