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### Horsin' Around * - posted February 14, 2005

Zachary travels on a journey of 50 miles. He spends half of his time riding on his horse and half of the time walking. When he rides his horse, he goes 9 miles every hour. When he walks, he goes 3 1/2 miles every hour. How long does it take him to complete the journey?

Be sure to explain your strategy and how you know your answer is correct.

Extra: How many miles did he travel by horseback? How many miles did he walk?

Learn About Our Scoring System

Meet the mentors of this puzzle:
Swarthmore College Winter 2005
Western Oregon University Winter 2005A

This problem was inspired by a problem in Seaver & Walton's Mental Arithmetic (1884). It involves logical reasoning as well as addition and multiplication of whole numbers and simple fractions (halves). It may be solved through a variety of approaches. The key concept is that Zachary rode his horse the same amount of time as he walked.

One successful strategy can be systematic Guess and Check. To score Practitioner in Strategy using this method the solver needs to make use of the information she/he gets from incorrect attempts in order to make a better next guess. Solvers who get lucky on a first guess need to explain why that guess made sense as a starting point – achieving success through skill and understanding, not purely luck. See Francis.

Repeated addition or doubling of 12.5 mi (9 mi + 3.5 mi, the combined distance covered by one hour each of riding and walking) was one of the most common methods students used. The most successful ones kept track of their work in a table, which made answering the Extra easier. Riley's solution makes use of multiples of 9 and 7, knowing that multiplication is a more efficient way to do repeated addition. Frederick adds 9 and 3.5 alternately until he reaches 50. Blake keeps track of distances in 1/2 hr increments. See also Douglas, Rebecca, and Anna.

Some students made their jobs easier by dividing 50 miles by 12.5. Still another variation involved finding the average speed and dividing 50 by 6.25 mph. This demonstrates excellent number sense. Beware: Averaging two speeds in problems where the times are not equal can lead to trouble! See Anderson and Angela.

A number of solvers used algebra to solve the problem very effectively. See Hannah for a good example of how to use text to explain the math.

We were happy to see so many solvers check their own work by using a second method to calculate the answer. See Anderson, Frederick, Angela, Hannah.

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4. Write a comment about what you learned from this problem.
5. Revise your solution and tell what you changed.
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### Highlighted solutions:

 From: Riley Ambrose, age 9 School: Swarthmore Rutledge School, Swarthmore, PA
```Zachary completed his journey in 8 hours. On horseback he rode 36
miles and walking he walked 14 miles.
```

```First I started with my nine times tables. I started off with 27
because that's almost half of 50. But then I could only get 48 if I
kept adding 3 1/2 since I need to use 3 1/2 to make it sort of even.
So then I made the 27 36. I know that 3 1/2 plus 3 1/2 is 7 so I can
count by sevens. It is much easier than adding fractions. Taking the
36 I can add 7  to get close to fifty or fifty to find how many hours
it took Zachary to get there. Then 36 plus  7 is 43 and that'd close
to fifty so then adding another seven would get me exactly to fifty.
So, every hour on horseback it's nine miles. I used nine four times
because of nine times four so that's four hours. Then for the walking
every time I add 3 1/2 that's an hour. Then I used seven twice but
really I used 3 1/2 four times because 3 1/2 two times is seven. Next,
since I used 3 1/2 four times that's four hours. So, four hours plus
four hours is eight hours. That is almost half and half. So, then 14
is the miles of walking and 36 is the miles of horseback. That is how
I found my answer.
```

 From: Douglas Lyman, age 9 School: West Bradford Elementary School, Downingtown, PA
```It took Zach 8 hrs. to complete the journey (4 hrs. on both horseback
and foot).
```

```The problem said that Zach went half the time on foot and half the
time on his horse so to make that time half and half I did this:

(1hr.)3 1/2(miles)+(1hr.)9(miles)=12 1/2 not fifty

so I multiplied 3 1/2(times)2=7
then I multiplied 9(times)2=18
then I added the product of my two multiplication problems above

(2hrs.)7+(2hrs.)18=25 not fifty

so this time I added 7+3 1/2=10 1/2
then I added 18+9=27
then I added the sum of the two addition problems I did above

(3hrs.)10 1/2+(3hrs.)27=37 1/2 not fifty

so I added again 10 1/2+3 1/2=14
then I added 27+9=36
then I added the sum of the two addition problems I did above

(4hrs.)14+(4hrs.)36=50 THAT'S IT

then I looked at the hrs. (the numbers in parentheses) and added them
and got my answer.

At first I thought that when the problem said "half the time on foot
and half the time on horse" I thought that "half the time"=half the
distance, but it ment that he traveled an equal amount of hrs. on
both foot and horseback.

There is another way of doing this problem with algebra that my dad
did but I forget how.
```

 From: Frederick Seymour, age 9 School: Holy Trinity School, El Dorado Hills, CA
```It will take Zachary 8 hours to complete his journey.
```

```He will ride the first hour on his horse and travel 9 miles.  Then I
added 3 1/2 + 9 = 12 1/2 miles after 2 hours.  Then I added 9 + 12
1/2 = 21 1/2 miles after 3 hours.  Then I added 21 1/2 + 3 1/2 = 25
miles after 4 hours.  I realized 25 miles was half of 50 miles,
which is the total length of his journey.  So, I multiplied 4 hours
X 2 = 8 hours to complete the journey.

Extra:  He will travel 36 miles by horse and 14 miles by walking.  I
added up 9 + 9 = 18 (from the explanation above).  Then I multiplied
18 X 2 = 36 miles he traveled by riding horseback.  Thenk I added 3
1/2 + 3 1/2 = 7 and multiplied 7 X 2 = 14 miles by walking.  I also
checked this math by subtracting 50 - 36 = 14.  I checked that
answer by adding 14 + 36 = 50.
```

 From: Blake S., age 9 School: Abraham Lincoln Elementary School, Medford, OR
```It will take Zach 8 hrs. To get to his destination
```

```I started on this problem by re-reading the question to look for
anything that might have slipped by me.when I re-read it I noticed
it said half the time not half the miles. When I was in math class
we discussed the problem and tried using half an hour periods.
Math
Time---miles on horse---miles walking ---total miles
1 hr.     4.50                    1.75                     6.25
2 hr.     9.00                    3.50                     12.50
3 hr.     13.50                  5.25                     18.75
4 hr.     18.00                  7.00                     25.00
When I got to 4 hr. I knew that if I just doubled everything I would
come up with the answer.
math:
4x2=8 18.00x2=36.00  7.00x2=14.00  25.00x2=50.00

My answer is: It will take Zach 8 hrs. To get to his destination.

My extra answer is: He will travel 36 mi. by horse and 14 mi by foot.

I checked my work by dividing each sum by the number of hours.
Math
36÷8=4.50     14÷8=1.75     50÷8=6.25
```

 From: Rebecca Lindenmaier, age 8 School: Moses Brown School, Providence, RI
```It would take 8 hours to complete the journey.On horseback he
traveled 36 miles, and he walked 14 miles.
```

```Explanation:  First I made a chart with “horseback” on one side
and “walking” on the other side.  Then under “horseback” I wrote the
distance he would travel on horseback in an hour, which was 9 miles.
Next to this, I did the same for “walking,” which was 3.5 miles.

I did that again, putting another 9 miles under horseback and another
3.5 miles under walking.

Then I added both sides up, and got 18 miles for horseback and 7
miles for walking.  This was the distance traveled in 4 hours, that
is, 2 hours on horseback and 2  hours walking.

Then I added 7 and 18 together and got 25.  I knew that 25 was half
of 50, which is how many miles the trip was.  So all I had to do was
repeat the 2 hours on horseback and the 2 hours walking to get 50
miles.

Therefore I knew it took 8 hours to make the trip.

For extra, from what I had already figured out I had also figured out
how many miles he traveled on horseback and how many miles he walked.

So he traveled 36 miles on horseback and he walked 14 miles.
```

 From: Anderson Wang, age 10 School: Lower Gwynedd Elementary, Ambler, PA
```It takes him 8 hours to complete the journey.Extra: He traveled 36
miles on horseback and 14 miles walking.
```

```First, I did (9+3.5)/2 (which is the average of 9 miles per hour and
3.5 miles per hour) and I got 6.25 miles per hour, so on average, he
traveled 6.25 miles per hour. Since he traveled 6.25 miles per hour
and he traveled for 50 miles, I divided 50 miles by 6.25 miles per
hour, and I got 8 hours so he took 8 hours to complete the journey.
For the extra, I knew that half of 8 hours is 4 hours so he did 4
hours on horseback and 4 hours walking. On horseback, he traveled 9
miles per hour for 4 hours so I multiplied 9 miles per hour by 4
hours and I got 36 miles so he traveled 36 miles on horse. When he
walked, he traveled 3.5 miles per hour for 4 hours so I multiplied
3.5 miles per hour by 4 hours and I got 14 miles so he traveled 14
miles walking. To check my question, I add 36 miles and 14 miles
(those are the distances Zachary traveled on horseback and walking)
and I get 50 miles and thats correct because the problem says that
his journey was 50 miles long.
```

 From: Francis Gonzalez, age 11 School: Northside Middle School, Norfolk, VA
```It will take eight hours for the journey.
EXTRA:Zachary rode 36 miles on horsebackand 14 miles walking.
```

```In order to solve this problem, I needed to make a trial. I have
already known that they(horseback and walking) took the same amount
of time and that the journey took the same amount of time and that
the journey took 50 miles. So what I did to find out how many hours
each way of transportation took was to estimate. I did the
following.....

9(horses mph)     3.5(walking mph)
x6(estimated hours)x 6(estimated hours)
--                 ---
54       +         21  =75 miles
- 9                -3.5
---                ----
45        +        17.5 =62.5 miles
- 9                -3.5
---                ----
36        +         14  =50 miles

So what I found out from my data was that it took 4 hours for each
mode of transportation. If each mode of transportation took half the
amount of time, I added 4+4 to get a total of 8 hours.

EXTRA:Sinceit took four hours for both modes of transportation, I
multiplied how fast the horse traveled and how fast Zachary walked by
4.

Reflection: After the completion of this problem, I believed that
this was a fair problem. This problem could be solved, with a certain
amount of interperation, by novice and expert problem
solvers.
```

 From: Anna B., age 8 School: Mary Scroggs Elementary School, Chapel Hill, NC
```My final answer is 8 hours. Here is why.
```

```First, I decided I needed to know how many miles it took them for
two hours -- one hour of horseback and one hour of walking. So I did
9 + 3 1/2 = 12 1/2.

Now, I thought to figure out the rest I will double the amount of
time and miles.  So in 4 hours I will have gone 25 miles.  I
realized that was my halfway point to 50.  To get the final answer I
did 4 hours + 4 hours = 8 hours.  I did 4 + 4 because I needed
another 25 miles to reach 50.

EXTRA:
In 4 hours of riding he will go 4 times 9 or 36 miles on horseback.
In 4 hours of walking he will go 4 times 3 1/2 miles.  4 times 3 is
12, and 1/2 times 4 is 2 so the total walking is 14 miles.  This
checks out because 36 + 14 = 50 the total miles gone.
```

 From: Hannah Huang, age 11 School: Transit Middle School, East Amherst, NY
```It takes him 8 hours to complete the journey.

Extra: He travels 36 miles by horseback and 14 miles by foot.
```

```First I assumed T was the total time for the journey.

I made the equation 1/2 T * 9 mph + 1/2 T * 3.5 mph = 50 miles to
represent the journey.

1/2 T * 9 mph stood for 1/2 of the total time when he rode the horse
at 9 miles per hour.  1/2 T * 3.5 mph stood for 1/2 of the total time
when he walked at 3.5 mph.

I multiplied this by 2 to simplify the equation to 9 T + 3.5 T = 100
miles,
or 12.5 T = 100 miles.

To find out T, I divided 100 miles by 12.5 which equaled 8 hours.
This meant T equaled 8 hours.

Extra: Since T = 8 hours, that meant he rode the horse for 4 hours and
he walked for 4 hours.  The distance he traveled by horseback equaled
4 hours * 9 mph = 36 miles.  The distance he traveled by foot equaled
4 hours * 3.5 mph = 14 miles.

To check my answers I added 14 and 36 together, which equaled 50 miles.
```

 From: Angela Boullata, age 11 School: Upper Moreland MS - Schuck, Hatboro, PA
```It takes him 8 hours to complete his journey.

Extra: He traveled 36 miles by horseback and 14 miles walking.
```

```First, I added the two miles/hour together and got the sum of 12.5.
Next I divided 12.5 by two to find the average.  The average was
6.25.  Then I divided 50 miles by 6.25 to find how much time it took
him to travel.  That answer is 8 hours.  I double checked by
multiplying 4 hours (Because the answer 8 hours divided by two is 4.)
x 9 miles/hour, and 4 hours x 3.5 miles/hour.  I added those two
products up and the sum was 50.  That is how I found the answer 8
hours.

Extra: First I multiplied 9 miles/hour and 4 hours to find how many
miles he rode on his horse.  The product was 36 miles.  Next I needed
to find out how many miles he took walking. So I multiplied 3.5
miles/hour by 4 hours and the product was 14 miles.  To check my
answer, I added 36 miles and 14 miles and it was 50 miles, the total
amount that Zachary went.  That is how I found my answer.

```

The Gold List
(for more about our scoring rubric, visit our scoring information page)

E A, age 10 - Boyle Road Elementary School, Port Jefferson Station, NY
Riley Ambrose, age 9 - Swarthmore Rutledge School, Swarthmore, PA
Anna B., age 8 - Mary Scroggs Elementary School, Chapel Hill, NC
Angela Boullata, age 11 - Upper Moreland MS - Schuck, Hatboro, PA
Sherley Chabur, age 12 - Northside Middle School, Norfolk, VA
Martín Díaz, age 12 - Caobos, Bogotá, Colombia
Jack Flowers, age 12 - St. James Academy, Monkton, MD
Spencer Funk, age 10 - Abraham Lincoln Elementary School, Medford, OR
J G, age 10 - Boyle Road Elementary School, Port Jefferson Station, NY
Steven Glick, age 10 - University of Chicago Laboratory School, Chicago, IL
Francis Gonzalez, age 11 - Northside Middle School, Norfolk, VA
Lance Goodridge, age 9 - McDonogh School, Owings Mills, MD
Googoplex, average age 11 - Hebron Elementary School, Hebron, CT
Grasseaters, average age 11 - Hebron Elementary School, Hebron, CT
Payton Gross, age 8 - Kegonsa Elementary School, Stoughton, WI
Lena Gunzl, age 11 - Lake Garda Elementary School, Burlington, CT
Nishant Gupta, age 15 - Anandalaya School, Anand, India
Julia Herrle, age 7 - Wexford Elementary School, Wexford, PA
Hannah Huang, age 11 - Transit Middle School, East Amherst, NY
Tyler Johnson, age 11 - Downsville Elementary School, Downsville, LA
Michael Judge, age 25+ - University of Saint Francis, Joliet, IL
Eric Klingsberg, age 13 - St. Mary's Elementary School, Manhasset, NY
Ben Lee, age 10 - McDonogh School, Owings Mills, MD
Ryan Lee, age 10 - Mary Scroggs Elementary School, Chapel Hill, NC
Steven Lee, age 15 - Rosmini College, Glenfield North Shore City, New Zealand
Maren Lightheart, age 9 - Abraham Lincoln Elementary School, Medford, OR
Rebecca Lindenmaier, age 8 - Moses Brown School, Providence, RI
Douglas Lyman, age 9 - West Bradford Elementary School, Downingtown, PA
Biagio Marino, age 13 - St. Mary's Elementary School, Manhasset, NY
PS2Scooters, average age 18 - Hebron Elementary School, Hebron, CT
Sydney Raisbeck, age 9 - Kegonsa Elementary School, Stoughton, WI
Elizabeth Rizzo, age 12 - St. Mary's Elementary School, Manhasset, NY
YunSu Ryu, age 10 - Mary Scroggs Elementary School, Chapel Hill, NC
Conrad S, age 10 - Abraham Lincoln Elementary School, Medford, OR
Blake S., age 9 - Abraham Lincoln Elementary School, Medford, OR
Frederick Seymour, age 9 - Holy Trinity School, El Dorado Hills, CA
Hannah Sonsalla, age 9 - Kegonsa Elementary School, Stoughton, WI
Alicia Stith, age 12 - Northside Middle School, Norfolk, VA
Teeter-Totters, average age 11 - Hebron Elementary School, Hebron, CT
TidsClass, average age 9 - Lake Hills Elementary School, Ferrysburg, MI
Letizia Vaccaro, age 12 - St. Mary's Elementary School, Manhasset, NY
Ravi Vaidya, age 12 - Nyack Middle School, Nyack, NY
V W, age 12 - Clague School, Ann Arbor, MI
Katie W., age 8 - Mary Scroggs Elementary School, Chapel Hill, NC
Anderson Wang, age 10 - Lower Gwynedd Elementary, Ambler, PA
Ross Widom, age 11 - Seven Bridges Middle School, Chappaqua, NY
Nick Zuraw, age 18 - Seymour High School, Seymour, CT

#### View most of the solutions submitted by the students above

The Silver List

Kristina B, age 9 - Abraham Lincoln Elementary School, Medford, OR
Maryssa B, age 10 - Abraham Lincoln Elementary School, Medford, OR
Haley C, age 10 - Abraham Lincoln Elementary School, Medford, OR
Michael Cervino, age 12 - St. James Academy, Monkton, MD
Christina Chen, age 10 - Underwood Elementary School, Newton, MA
Casey Clark, age 9 - McDonogh School, Owings Mills, MD
Teresa Cong, age 14 - Loreto Kirribilli, Sydney, Australia
Chaitri Desai, age 11 - Upper Moreland MS - Ziegler, Hatboro, PA
Mieke Dykhouse, age 13 - VanDellen Christian Middle School, Denver, CO
Erasers, average age 11 - Hebron Elementary School, Hebron, CT
Flying Pink Bunnies from Above, average age 10 - Tustin Memorial Academy, Santa Ana, CA
Lucy Helveso, age 11 - Cedar Grove Elementary School, San Jose, CA
Austin K, age 9 - Abraham Lincoln Elementary School, Medford, OR
Spencer K, age 13 - Hardyston Middle School, Hamburg, NJ
Matthew Linardi, age 10 - McDonogh School, Owings Mills, MD
Ray Manatad, age 25+ - Mapua Institute of Technology, Intramuros, Manila, Philippines
Kara McGee, age 10 - Swarthmore Rutledge School, Swarthmore, PA
Ian Morse, age 10 - Swarthmore Rutledge School, Swarthmore, PA
Daniel Mutunga, age 25+ - University of Central Florida, Orlando, FL
Devon Nech, age 12 - West Briar Middle School, Houston, TX
K P, age 22 - N/A, N/A, IL
Packers, average age 10 - Kegonsa Elementary School, Stoughton, WI
Jeffrey Perkins, age 8 - Mary Scroggs Elementary School, Chapel Hill, NC
Gavin Quinn, age 13 - East Alternative School of Toronto, Toronto, Ontario, Canada
Elizabeth Sheehan, age 12 - St. Mary's Elementary School, Manhasset, NY
Arthur Smith, age 25+ - Peter Cooper Elementary School, Superior, WI
Meredith Sparks, age 11 - Transit Middle School, East Amherst, NY
Anthony Villar, age 12 - Upper Moreland MS - Schuck, Hatboro, PA
Katie W, age 9 - Abraham Lincoln Elementary School, Medford, OR
Arnold Zhang, age 11 - Wallingford Elementary School, Wallingford, PA

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