Math Trail 2008

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Students: Spring, 2008 3rd, 4th and 5th Grade Gifted and Talented Education (GaTE) Math Students

Staff: Dr. J. Beyersdorfer, NBCT

Speeding Along with Math Problems in Our Community
Grade Five Students

Mike and Jacob

TRAINS

A train is going on a trip 600 miles long, from Minnesota to Illinois. During the ride, the train goes two speeds. Part of the time 95mph, another part of the time 110mph. It took the train 6 hours to go 600 miles. How long will the train go at each speed?

Will & Alex

MARS

Astronauts have taken George W. Bush and Dick Cheney captive. They are being taken to Mars for a present to reward them for being the President and Vice President for the past eight years. Bush and Cheney will be launched at Arlington Heights, Illinois because Illinois is flat land. They are busily preparing for the trip next week. To fill the rocket with fuel, they will need to pay $1000 total. They will need more fuel every 15 days. It will take approximately three 30-day months to get there and it is 38 million miles away. What is the speed they will be going? How many weeks will the trip take to get to the destination? How much money will they have to pay during the trip for fuel?

Charlie, Ryan, and Brendan

THE BIGGEST TRIP EVER

There is a plane and a jet. There is also a train. The train is going 70mph. The plane is going 500mph. The jet is going 700mph. The train is carrying 100 people. The plane is carrying 240 people. The jet is carrying 70 people. The plane and the jet are going 1568 miles from Key West to Juneau. Next they are going 2869 miles from Juneau to Tokyo. Next they are going 3423 miles from Tokyo to Sydney. Next they are going 4741 miles from Sydney to Paris. Next they are going 5985 miles from Paris to the Democratic Republic of Congo. Next they are going 6875 miles from the Democratic Republic of Congo to the South Sandwich Islands. Next they are going 7586 miles from the South Sandwich Islands to the British Islands. Oh no! The jets’ fuel tank broke and stopped at the British Islands for thirty hours. Then took off to Chicago, IL while the plane kept going on the 1,953 mile trip and they both landed at O’Hare airport the third largest airport in the U.S.A. Who got to the airport first and what was the difference in time between the plane and jet?

Rahul, Andrew, and Cameron

THE ROAD RACE

A car and a truck start at the same point in a race at O’Hare, the biggest airport in the Chicago land area. They also began to race at the same time. The truck moved 27 mph for a total of 42 minutes, and the car went 38 mph for a total of 37 minutes. Once each of the vehicles’ time was up they had to stop driving and the vehicle that was farther at that point wins. Halfway through the truck’s time the truck’s backdoor opened and onions came spilling out. Since the vehicles are traveling downhill the onions fall forward instead of backwards. As the onions fell they landed in chemicals spilled from a chemical truck, which cause them to grow motors and catch up with the truck. The onions then got caught in the truck’s wheels causing the truck to lose 5 minutes from his total time. The truck also gained 5 mph because truck’s load lightened. By the end of their time who had traveled farther and how far did they each travel?

Emily and Kate

THE CHASE

A bank robber is driving away from a cop who was born in Chicago, IL where Oprah Winfrey currently lives. The bank robber is driving at a speed of 45 mph, and is currently 13 miles away from the cop, Police Officer Jeff. If Jeff travels at a speed of 50 mph, can he catch the bank robber before he crosses the Mexican Border, which is 42 miles away from Jeff? How long will it take them to reach the border?

Marissa, Margarita and Brittany

THE ABOMINABLE CHAIRLIFT

Problem 1:
The Abominable Snowman was originally from Chicago, IL. He lived there during 1871, and needed to move because of the Chicago Fire. Legend has it that Mrs. O’Leary’s cow tipped over a lantern that started the fire. The raging fire then burned down his cabin. Then he moved to Antarctica to live on a mountain. One day in Antarctica the Abominable snowman ran down to the bottom of his mountain. At the bottom, he realized that he left the water running in his sink. The fastest way to get back to his cabin was the chairlift. If the chairlift normally goes 42 mph, but it only took him 8 minutes to get up to his cabin, how far up the mountain is his cabin?

Problem 2:
The abominable snowman and his brother, Yeti, were having a race down the Yeti’s mountain. If his mountain is 84 miles down hill and the snowman went down 3 miles per hour, while his brother went down 4 miles per hour, how long did it take to get to the bottom and who won?

Solutions

TRAINS:
Mike and Jacob

You take 110mph and 95mph and play around with the numbers until you get the total of 600 miles. We started with six 110 mph, it was too high so we decreased the number of 110 mph until there was two 110 mph and four 95 mph.

All units for all the numbers in the charts are mph

A	B	C	D	E	F	G
110	110	110	110	95	110	110
+110	+110	+110	+110	+95	+95	+110
+110	+110	+110	+110	+95	+95	+95
+110	+110	+110	+95	+95	+95	+95
+110	+110	+95	+95	+95	+95	+95
+110	+95	+95	+95	+95	+95	+95
660	645	630	615	570	585	600

These are all the number models that you can try, but will not be successful, except for letter G.

MARS:
Will & Alex

First you will have to divide 7 days into 90 days (three 30 day months which equals 90 days) to get to Mars. It will take 12.86 weeks (about 12 weeks and 6 days) for that answer. Then you divide 90 into 38,000,000 miles to get the answer of how many miles a day they are traveling which you will get 42,222.22 miles a day. To find out how much they will have to pay for fuel, you have to divide 15 into 90 and then multiply the answer by 1,000 and you will get $6,000.

Answers:
1) 12.86 weeks
2) 42,222.22 miles per day
3) $6,000

THE BIGGEST TRIP EVER:
Charlie, Ryan, and Brendan

Because the trip is a total of 35,000 miles and the plane is going 500 mph and the jet is going 700 mph you divide 35,000 by 500 and 700 and get 50 hours for the jet and 70 hours for the plane to land at O’Hare airport, but because the jet stopped for 30 hours you add that to the 50 hours and you get 80 hours long for the jet to land there so the plane got there 1st and by 10 hours because 80-70=10 hours.

Answer: The plane by 10 hours

THE ROAD RACE:
Rahul, Andrew and Cameron

Step 1: You have to find out the truck’s halftime so you know the moment when chaos strikes. You take the full time of the truck and divide it by 2 to find the halftime of the truck. Example: 42 min. divided by 2 is 21 minutes, which is the half time of the truck.

Step 2: You then have to find out how far the truck moved in 21 min. because you add that to the other half of the race to get the truck’s total time. You multiply the truck’s mph by the 21 minutes divided by the 60 min. you have in a hour to get the miles traveled by the truck in that half. Example: the problem should look like this:27 mph multiplied by [21 min. divided by 60 min.]=9.45 miles traveled in the first half.

Step 3: Now you get into the fun part. After the disaster, you have to subtract 5 min. from the 21 min. to get 16 min. and you also do 27 mph + 5 mph to get 32 mph. The truck’s new statistics are 32 mph for 16 min.

Step 4: You now have to do the same equation as step 2. You multiply the truck’s new mph by the new time, 16 min. multiplied by 60 min, which is how many minutes are in an hour. After that you should get 8.5333333333 etc. miles. Example: 32 mph multiplied by [16 min. divided by 60 min.]=8.53333333333 etc. miles

Step 5: Now you have to add up those 2 fractions to find out how far the truck traveled in all. Example: 9.45 miles + 8.53 miles [rounded]=17 and 59/60. To do that though you have to find common denominators, which are 9 and 27/60 and 8 and 32/60.

Step 6: Now to know who won the race we have to know the cars time. To do that we have to multiply the fractions 38 over 1 [The cars mph] and 37 over 60 [ The cars min. per hour ] to get 23 and 26/60 miles.

ANSWER
The answer is the car won the race. The truck drove for 17 and 59/60 miles and the car drove for 23 and 26/60 miles.

THE CHASE:
Emily and Kate

First you need to figure out how far the bank robber is form the Mexican border. Since you know the border is 42 miles away from Jeff, and the robber is 13 miles away from Jeff. You have to subtract 13 miles from 42 miles to figure out how far the robber is from the border.

42 miles - 13 miles = 29 miles
The robber is 29 miles away from the border.

Next you have to figure out how long it will take the robber to cross the border. To do this, you must remember

Time = Speed / Distance. The speed in this problem is 45MPH and the distance is 29 miles. When working with this problem the line between the speed and distance means divide. So you must divide 45MPH by 29 miles.
45 MPH / 29 miles = 1.551724135 hours
If you rounded it to the hundredths place you would have 1.55 hours.

We know that the 1 in 1.55 hours is equal to 60 minutes, because 1 hour is 60 minutes. The .55 of an hour must also be converted to minutes. To do this, we used the formula to convert hours to minutes. The formula is multiply the hour(s) by 60 minutes.

.55 of an hour * 60 minutes = 33 minutes
Then you have to find out how many minutes it took the robber to reach the Mexican border. To do this, you must add 60 minutes and 33 minutes.
60 minutes + 33 minutes = 93 minutes = 1 hour and 33 minutes

Now we have to figure out how long it will take Jeff to reach the border. His speed is 50 MPH, and his distance is 42 miles.

50 MPH/42 miles = 1.2 hours.
We know that the 1 in 1.2 hours stands for 1 hour or 60 minutes. Now we have to do
.2 hours * 60 minutes = 12 minutes.
Finally you have to add
60 minutes + 12 minutes = 72 minutes, or 1 hour 12 minutes,
to find out the total minutes it took Jeff to reach the border.

ANSWER
Bank Robber= 1 hour 33 minutes (93 minutes)
Police Officer = 1 hour 12 minutes (72 minutes)
Police Officer Jeff will eventually catch the bank robber before he reaches the Mexican Border.

THE ABOMINABLE CHAIRLIFT:
Marissa, Margarita, Brittany

Problem 1:
To find the answer of how high the mountain is, you need to know that speed = distance/time. So, 42 mph=? /8 minutes. To find the answer you must figure out how many times, 8 goes into 42,so that you know how far up the mountain is (8 minutes to get up in the chairlift going 42 mph). Now you divide 42 by 8,which is: 42 /8=5.25. To make sure the answer is correct we checked by multiplying: 8*5.25=42. Our final answer is that he went up 5.25 miles.

Problem 2:
Yeti; 21 minutes
Snowman; 28 minutes
Yeti won

Explanation
You had to divide both miles per hour by 84 to see how long it took.

Snowman: 84/3=28 minutes

Yeti: 84/4= 21 minutes

Yeti won because he took less time to get down his hill.

in our Neighborhood Grade Four Students
Problem One: Brian		Problem One: Konrad
The Thrill Riders Roller coaster Company has orders for eight roller coasters. Each coaster requires the same number of rails. There was a problem at the manufacturing plant and 1,287 rails were not packaged with the orders! Questions: How many rails must each order receive? Are there any remaining rails? Using the remainder, how many rails must be added to the order so each order receives the same amount of rails? Next, the Thrill Riders received emails from seven of the eight companies asking for more rails to expand their roller coaster. Each company wants to expand their coaster by two hundred seventy-six feet of rail. Each rail is six feet long. Questions: 4. How many additional rails will each company buy from the Thrill Rider Company? 5. How many feet long is the coaster they are building?		There are seventy-three children waiting to go on a roller coaster with a minimum capacity of fifteen and a maximum of twenty riders. You want to fill the ride to the maximum as often as possible. Question: 1. How many trips will the roller coaster take if all the children ride on it one time? The three supervisors joined the children on the ride. The roller coaster operator decided to make more than four trips with as close to the minimum number (15) of people possible. Question: 2. How many people will go on each ride?
Solution to Problem One: Brian		Solution to Problem One: Konrad
Q 1. 160 First, you have to divide1,287 by eight to find the number of rails that each company did receive, which is one hundred sixty. Q 2. Yes 7 Q 3. One, because we had seven rails remaining and there are eight companies Q 4. Forty-six, since each rail is six feet, I divided six into two hundred seventy six to determine there were forty-six rails. Q 5. 1,236 feet. Multiply 160 rails by 6 (because that’s how many feet each rail is and each company got one hundred sixty feet of rails to get 960 feet. Then you add 276 because that’s how many extra feet they wanted. The total length was 1,236 feet.		Q1. First, divide the seventy-three children into groups of twenty (the maximum per ride) so that you know that you have three groups of twenty children with thirteen children remaining. Then, because the minimum is fifteen, you have to take away two from one of the groups with twenty riders and put them into the group with thirteen riders. Lastly, you have to count the number of groups and you have your answer. There will be four trips with two trips of twenty riders (40 riders) one trip of fifteen riders (15 riders) and one trip with eighteen riders (18 riders). Q2. First you have to add the three riders to the group of seventy-three children for a total of seventy-six people. Then you have to divide seventy-six (sum) by fifteen (minimum capacity) to determine the number of trips with the minimum number of riders. There are four trips of fifteen with a of remainder sixteen riders. So there are four trips of fifteen and one trip of sixteen.

Problem 2: Brian		Problem 2: Konrad
Jon wants to buy pizza. Jon looks at the sign for prices. Judy and Bob do the same. A large pizza has eight slices, a medium has six slices, and a small has four slices. Jon orders a large and a small pizza. Judy ordered a medium. But Bob ordered one large. Jon ate eleven slices. Bob ate seven slices of his large pizza. Judy ate four slices. Questions What fraction of his/her pizza did each person eat? How many pieces of each order did each person not eat? How many pieces did the three kids eat in all? How many pieces did the three kids have as leftovers? (Reduce the fractions into lowest terms)		The Happy Ville Water Park has three rides, Kiddie Falls, Exploding River, and Super Toilet. The park decided to take away three hundred twenty-eight gallons of water of the nine hundred eighty-four gallons they already use. They did the following to reach their goal. The park manager decided to take away all the water in Kiddie Falls and rename it Kiddie Canyon. That change saved them 24/32ths of the nine hundred eighty-four gallons. Then one of the crew members suggested to take away the water guns on the Exploding River, the team agreed. When they took away the guns it saved them 1/8th of the three hundred twenty-eight gallons of water they wanted to save. Then the life guard soggiest to fix the leaks on the Super Toilet. They called the Wisconsin Pluming Co. to fix those pipes. That saved them 2/16ths of their goal. Question: 1. What fraction of the total gallons saved was contributed by each ride? 2. How many gallons did the park save on each ride?
Solution		Solution
Q1. You have to know the number of each of pizza slices each person ordered and the number of slices each child ate. Judy had a medium pizza worth 6 slices, she ate four slices (4/6 = 2/3) Jon ordered a large (8 slices) and small pizza (4) which was 12 total; he ate eleven slices (11/12), and Bob ordered one large that had 8 slices and he ate seven slices (7/8). Q2. you have to add the fractions of the pizza that they did not eat. Then you have to reduce the fraction. Jon has 1/12 left, Judy has 1/3 left, and Bob has 1/8 left. Next, find the total number of slices the three kids in the kids’ pizzas. (12 + 8 + 6 = 26). Q3. you have to add all of the pieces eaten together, to get twenty-two slices in all (11 + 7 + 4 = 22). Q4. you subtract how many slices in all the pizzas by how many slices the kids ate (26 + 22 = 4 slices). There are four slices left over. The fraction is 4/26 = 2/13.		Q1. To find out how many gallons the park saved, you first have to convert all the fractions to the lowest possible common denominator. Then you can add the fractions (24/32 + 1/8 + 2/16 = 12/16 reduced this is 12/16 + 2/16 + 2/16 = 16/16 all the gallons saved). Q2. You should find out how many gallons the fraction of the total amount of water is. 12/16 or ¾ of 328 gal = 246 gallons 1/8 of 328 gal = 41gallons 2/16 or 1/8 of 328 gal =41 gallons This amounts to 246 gallons for Kiddie Canyon and 41 gallons each for Super Toilet and Exploding River.

Fraction Problems by Grade Three Students: Around Our Homes

Christopher

At my house I have 105 utensils. 15 are forks, 30 are spoons, 15 are knives, and 45 are other types. What fraction are the knives and spoons out of all utensils?

Benjamin

Thirty-one people share twenty-four pies equally. If there were four pieces in each pie, how many pieces are in the twenty-four pies in all, and if every piece is eaten what fraction of a pie would each person eat if one person got twice as much as everyone else?

Andrew

Problem #1:
Joe organized a basketball tournament and made 10 teams. There are 5 people on each team. At the day of the big game, only 3 people showed up on each team. How many teams out of a possible 10 can Joe make? Write your answer as a fraction.

Problem #2:
There are seven people in a room and there is a plate of five cookies. The people in the room want to share the cookies equally, but two people are really hungry and quickly grabbed a cookie each and ate it. The third guy tries to do the same thing, except he stops eating after he eats only a half a cookie and puts the other half back on the plate. The four remaining guys split the remaining cookies on the plate. One of the four stopped eating after he ate only a quarter of a cookie. The last three guys split up the remaining cookies on the plate. How many pieces of cookie does each guy get?

Alex

There were three boxes of Cheerios at the grocery store. The first box had 22 Cheerios, the second box had 36 Cheerios. The last box had the amount in two boxes added together and then divided by two. There were 17 dark cheerios in the first box, 19 dark Cheerios in the second box, and 23 dark Cheerios in the third box. What is the fraction of the dark and light Cheerios in each box?

Sean

Collin had a pie. Collin cut the pie into 16 pieces. Collin had 4 friends over. Bill, Max, Bob, and Joe. One friend wasn’t very hungry. Collin had 8 pieces. Joe had as much pie as the leftover pie.
What fraction did his friends eat in all? What fraction did his least hungry friend eat?

Solutions

Christopher

Since there are 15 knives and 105 total utensils, the fraction is 15/105. I can reduce that fraction to 1/7 because 1 x 15 = 15, and 7 x 15 = 105. Since there are 30 spoons and 105 total utensils the fraction is 30/105. I can reduce that fraction to 2/7 because 2 x 15 =30, and 7 x 15 = 105. My answer is knives 15/105 = 1/7 and spoons 30/105 = 2/7.

Benjamin

I solved this problem by first making a chart. Then I divided it into sixteen columns and put the number three in each one. Then I put arcs under each one to add up until I got to ninety-six. I did this because I needed to add it up until I got to a point where I could add double what the other people got. I got to 96 by adding twenty-four four times. When I got to ninety, I added six because in the problem it said one person got twice as much as what everyone else got and everyone else got 3 so 6 is twice as much as 3. My answers are ninety-six pieces and ¾ a pie per person and one person gets 6 pieces or 1 and 1/2 of a pie.

3	3	3	3	3	3	3	3	3	3	3	3	3	3	3	3
3	3	3	3	3	3	3	3	3	3	3	3	3	3	6

Andrew

Answer 1:
Joe can make 6 teams out of a possible 10.

Explanation:
30 people showed up at the tournament. (3 X 10 = 30)
You can make 6 teams with 30 people. (30 / 5 = 6)
Out of the original 10 teams Joe can only make 6.

Answer 2:
The last three guys get ¾ of a cookie

Explanation:
This table shows what happened.

Step	Cookies	People	Comments
1	5	7	Beginning
2	3	5	2 people eat two whole cookies each
3	2 ½	4	1 person eats ½ cookie and stops
4	2 ¼	3	1 person eats ¼ cookie and stops
5	0	0	2 ¼ divided by 3 = 3/4

Alex

Explanation: First, I took the first box and I knew that there were 17 dark Cheerios in the first box and there were 22 cheerios in all. So, I knew that the fraction for the dark Cheerios for box one was 17/22. Then I had to find the light fraction for box one. I knew that there were 22 Cheerios in the box and I knew that there were 17 dark Cheerios so I subtracted 17 from 22 to equals 5 .So I knew that the fraction of light cheerios was 5/22.Next I took the second box and I knew that there were 36 Cheerios in it, I also knew that there were 19 dark Cheerios. So I knew that the fraction for the dark Cheerios for the second box was 19/36.Next I had to find the answer for the light Cheerios in box two. So I knew that there were 36 Cheerios in the box and 19 of them were dark so I subtracted 19 from 36 and got 17.I knew the fraction of light Cheerios was is 17/36. Last I had to find the sum for the last box. So, I had to find the sum of the light Cheerios. First I added 22 plus 36 and got 58. Then I divided it by to and got 29. That was my sum for the third box. I also knew there 23 dark Cheerios and 29 cheerios in all. So my answer for the dark Cheerios is 23/29. Next I had to find the sum for the light Cheerios. So, I subtracted 23 from 29 because I knew that the number of light Cheerios was what I had to subtract from the sum. I got 6 for my answer. Then I knew my answer was 6/29.

Sean

First, I made a chart and read the problem. Next, I filled in my chart and found out my fractions. Last, I found out how much extra pizza there was so I could get my answers of six sixteenths of the pie for Bill, Max, and Bob. And one sixteenth of the pie for Joe.

Collin	Bill	Max	Bob	Joe
8 pieces	4 pieces	4 pieces	4 pieces	1 piece
8/16 or 1/2	4/16 or 1/4	4/16 or 1/4	4/16 or 1/4	1/16