Math Trail 2004

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Staff: Dr. J. Beyersdorfer and Mrs. L. Vabulas
Students: A small group of Grade 3 Math Enrichment Students, the Grade 4 Math Dimensions Class, and two members of
Grade 5 Dimensions Class

Using Venn Diagrams to Solve Mathematics Problems: Solving a Circle Problem Venn diagrams allow students to visually display information to better understand the relationships between several number sentences. In the problems presented here, students must chose the calculation (addition, subtraction, multiplication) and the data carefully if they are to solve the problem correctly.
1. Read the introduction and each of the clues to identify the three major categories.
2. Label the circles by placing the first category as a name for the left (red) circle, the second category as the name for the right (blue) circle, and the third category as the name for the lowest (green) circle.
3. Remember that overlapping circles indicate shared category attributes.
4. Determine the number to be placed in each area by reading the clues and performing the necessary mathematical calculations.
	Remember you can determine a missing answer for a circle that has three of the four parts completed or if the total for the circle is known. The number of students interested in a particular category as indicated by inclusion within the red, blue, or green circle, can be found by adding the sections that make up that circle (for example: by adding sections 1, 2, 4, and 5. the number of items in the left or red circle can be determined. To find the total of all three categories add the results in all of the sections of the three circles. Do not add the totals of the categories since section totals would be counted more than once. Record the answer inside the portion of the circle that corresponds to the correct item's characteristics.

Group Problem Grade Three Students: Planning for the Ice Cream Social
	Forty-eight third graders took part in a survey to find out what type of ice cream to buy for their annual ice cream social. Note: It may be useful to solve this problem as a group using the solution notes below.
Six students wanted all three flavors.
Twenty-seven students wanted chocolate.
Thirty students wanted mint ice cream.
Nine students wanted vanilla and mint.
Five people wanted only chocolate ice cream.
Fourteen people wanted only chocolate and mint.
Solution for Ice Cream Social

Group Problem Grade Four Students: Game boards for Indoor Recess
	The students at Riley Elementary School must stay indoors at recess because of the cold winter weather. The teacher has provided several board games for them to use during this time. Their choices are Monopoly, chess, and checkers. The total number of students wanting to play the various games during recess is forty-seven (47). Note: It may be useful to solve this problem as a group using the solution notes below.
Nine students want to play any of the three games.
Four times the number that would play any game want to play Monopoly.
Five students want to play the games with identical game boards.
Twenty-four students want to play chess.
Eight students want to play either Monopoly or checkers and three times that amount want to play chess.
Solution for Indoor Recess

Small Group Problem Grade Three Students: Sports Lessons
	Fifty (50) children wanted to take three different sports lessons. They were soccer, ice skating, and swimming.
Six children wanted to take soccer and ice skating only.
Forty-four children wanted to take soccer.
Two children wanted to all three kinds of lessons.
Twelve children wanted to take ice skating.
Six wanted to take swimming and soccer lessons.
Ten children wanted to take swimming lessons.
•Two children wanted only swimming and ice skating.
Solution for Sports Lessons

Small Group Problem Grade Three Students: At the Beach
	Twenty-nine (29) third graders went on a field trip to the beach. Some students made sand castles. Some went surfing,and some decided to have a volleyball game.
Seven people did everything.
Eleven people only built sand castles.
Twenty-one people built sand castles.
Two people only wanted to build sand castles and play volleyball.
One person only surfed and built a sand castle.
Thirteen people surfed.
Fifteen people played volleyball.
Solution for At the Beach

Small Group Problem Grade Three Students: Recommending Toothpaste
	Sixty-four (64) dentists recommended brands of toothpaste to their patients. Colgate, Aqua-Fresh, and Crest were often recommended.
Forty-six dentists recommended Crest.
Eleven dentists recommended all of the toothpastes.
Forty-two dentists recommended Aqua-Fresh.
Thirty-nine dentists recommended Colgate.
Fourteen dentists recommended only Crest and Aqua-Fresh.
Ten dentists recommended Colgate and Aqua-Fresh only.
Seventeen dentists recommended Colgate and Crest.
Solution to Toothpaste Recommendations

Small Group Problem Grade Four Students: Raffle Prizes
	The Riley PTO is having a raffle at the annual Riley Fun Fair. The winner will have a choice of three prizes. The choices are X-Box, Gamecube, and Playstation.
Ten people didn't care what game they received.
Thirty people only wanted Gamecube.
Double the number of people who did not care what they received only liked X-Box.
Half of the people that wanted to own only X-Box wanted to choose only Playstation.
Forty-nine people wanted to own Gamecube.
The total of people that wanted to own X-Box is 56.
Six people wanted to own X-Box and Gamecube.
Solution for Raffle Prizes

Small Group Problem Grade Four Students: Dessert Choices
	The Student Council at Riley is having a party. The Student Council Representatives want to know how many of each dessert to buy, so they are taking a survey. The choices are cheesecake, brownies, and ice cream. The total number of students being surveyed is one hundred eight (108).
Five students wanted cheesecake and brownies, not ice cream.
Three dozen students wanted only ice cream and nothing else.
A total of forty-three students wanted cheesecake.
Twenty-one fewer students wanted only ice cream and brownies than the number of students wanting only ice cream.
Twenty of the forty-three students who wanted cheesecake also liked ice cream but they did not want brownies.
Six more students wanted all of the desserts than the number of students who wanted cheesecake and brownies only.
The total number of students who wanted to have brownies as their dessert is equal to nine times seven (9 X 7 = ?)
Solution to Dessert Choices

Small Group Problem Grade Four Students: Class Pet
	A fourth grade class at Riley School wanted to get a class pet. Everyone in the class wanted a bird. The three birds they wanted to get were a Hyacinth Macaw, a cockatiel, or an Eclecticus parrot. There were thirty-one (31) students in the class.
Eight students just wanted to get a Hyacinth Macaw.
Twenty-three students wanted the Eclecticus Parrot.
Nine students are willing to have any bird except a macaw.
The students in the class wore the colors of their favorite bird; four students didn't have a favorite bird.
Multiply the number of students who just wanted to have a Hyacinth Macaw by the number of students who didn't care which bird they got. This is the number of students who wanted a cockatiel.
Nine students wanted just an Eclecticus Parrot.
Divide twenty-six by two to find the number of students who wanted a macaw.
Solution to Class Pet

Small Group Problem Grade Four Students: Growing Tropical Fruit
	The Riley Student Council is planning on growing some tropical fruit. They can only plant one kind of tropical fruit. Their choices are mangoes, bananas, and pineapples. Riley's Student Council is going to survey the students of the fourth grade. There are one hundred eighty-three (183) students to survey.
Nine students didn't care which fruit was to be planted. They didn't have a preference.
Twenty-three students only wanted the fruits where you can peel off the smooth skin.
Fourteen students wanted fruits which didn't have an enormous seed that took up about 1/3 of the fruit.
The amount of students who wanted only mangoes and pineapples was 19. Three times that amount of people wanted only mangoes.
Solve this equation to find out the number of people who wanted only to plant pineapple. (72 + 45) - 56 = number of people who wanted only pineapples planted.
One hundred three (103) students wanted to plant pineapples.
The total of the students who wanted bananas was forty-six.
Solution Growing Tropical Fruit

Small Group Problem Grade Four Students: Hot Dog Toppings
	The school had a hot dog sale on Saturday. Everybody had at least one type of topping on their hot dog. There were three toppings. They were mustard, ketchup, and tartar sauce. One hundred fifty-two people had hot dogs. How many people had each combination of hot dog toppings?
Twelve people had only tartar sauce.
The total of the people interested in having any combination of tartar sauce or tartar sauce alone was eight and a half times the number of people who only wanted tartar sauce.
Twenty-eight people had both mustard and ketchup only.
The same amount of people had mustard and tartar sauce only
Thirty-six people had tartar sauce and ketchup only.
Three times that amount (the number of people that had tartar sauce and ketchup only) ordered ketchup on their hot dogs.

Solution to Hot Dog Toppings

Small Group Problem Grade Five: Music Class
	Scenario: At Riley School, three extra-curricular activities involving music are open to the fifth grade students. The first is Band, the second is Orchestra, and the third is Chorus. You can do one or more, but you don't have to. It just so happens that this year, the entire fifth grade wants to do one or more of these activities. Using the clues below, determine who wanted to join Band, who wanted to play in the Orchestra, and who wanted to join Chorus. P. S. The teachers want to know how many students there are total in the fifth grade, so please find that out too. Once you do, you can tell them.
Ten kids wanted to play in the Orchestra, while thirty wanted to join band.
Five kids couldn't make up their minds, as they like singing and playing musical instruments.
Fifty kids wanted to join chorus.
To find the number of kids who wanted to do band and chorus, take the number of children who want to do all three and multiply by nine then divide by three.
You now should have two answers inside the circle. To get the number of kids who want to play more than one instrument but not sing, add these two answers, divide by five, and cut the answer in half.
There is a total number of kids who want to do orchestra, and a total number of kids who want to do chorus. To get the number of children who want to do both, simply add both totals together, divide by six, and subtract eight.
The rest of the problem you can solve on your own now, so go ahead.
Solution to Music Class

Small Group Problem Grade Five: Jefferson High Reading
	Scenario:During summer break the eighth grade kids at Jefferson High were told to read at least one of three books assigned. They were Artemis Fowl, A Shingle Shard, and Harry Potter and the Philosopher's Stone.
Seventy-eight kids read the book about a criminal mastermind.
Do this equation to find out how many kids read books about magic (100 divided by 80) multiplied by (the square root of 10,000).
Only 15 kids just read a book about a boy who lives with Crane Man. You now have enough info to find out how many 8th graders there are; determine this number.
Only 73 kids read Harry Potter
Only 10% of the kids read both the book written by J. K. Rowling and the book with Tree Ear.Only 10% of the kids read both the book written by J. K. Rowling and the book with Tree Ear.
Divide the total kids in eighth grade by six (round if you need to) find out how many kids read only Artemis Fowl.
The number of people that read all three books is the same amount of students who only read both Artemis Fowl and Harry Potter.
Solution to Reading

Solutions

Solution for Group Problem Grade Three Students: Ice Cream Social
Steps to the Solution
Six students wanted all three of the flavors. (This refers to section 5, place a 6 there)
Twenty-seven students wanted chocolate ice cream at the social. (This is the total of the blue circle--all students wanting chocolate ice cream but may have wanted other flavors as well.)
Thirty students wanted mint ice cream. (This is the total of the green circle--students who wanted mint but who may also have wanted other flavors as well.)
Nine students wanted vanilla and mint. (This is the number of students in the overlapping circles for vanilla and mint which are areas 4 and 5. However, we know that 6 of the 9 wanted vanilla and chocolate, leaving three students wanting mint.)
Five people wanted only chocolate ice cream and no other flavor. (This choice is indicated by area 3.)
Fourteen people wanted only chocolate and mint. (This choice is indicated by area 6--where only chocolate and mint intersect.)
Finishing the Problem:
Areas 1, 3, and 6 need to be determined. The number of students voting must be confirmed. To calculate the number of students wanting just chocolate ice cream, add the numbers in areas 2, 5, and 6. (2 + 6 +14 = 22). Subtract this number (22) from the total of students wanting chocolate (27) to find that 5 students want only chocolate ice cream. To calculate the number of students wanting just vanilla ice cream, add the numbers in areas 2, 4, and 5 (2 + 6 + 3 =11). Subtract this number (11) from the total of students wanting vanilla ice cream (22) to find that 11 students wanted just vanilla. To calculate the number of students wanting just mint ice cream, add the numbers in areas 4, 5, and 6 (3 + 6 + 14 = 23). Subtract this number (23) from the total of students wanting just mint ice cream to find that 7 students wanted just mint. Check this work by adding all the numbers inside the circles. The total of 11 + 2 + 5 + 3 + 6 + 14 + 7 = 48 This was given as the total number of students surveyed for the ice cream social.
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Solution for Group Problem Grade Four Students: Game boards for Indoor Recess
Steps to the Solution
Nine students want to play any of the three games. (This refers to section 5, place a 9 there).
Four times the number that would play any game want to play Monopoly. (4 X 9 = 36. Sections 1, 2, 4, and 5 must total 36 since these areas would indicate students interested in playing Monopoly).
Five students want to play the games with identical game boards. (Section 6 is the area that indicates both chess and checkers with identical boards as the choice.)
Twenty-four students want to play chess. (This is the total number of students willing to play chess. It would be the total of sections 2, 3, 5, and 6.)
Eight students want to play either Monopoly or checkers and three times that amount want to play chess. (Section 4 indicates 8 students wanted both Monopoly and checkers. Also, 3 X 8 = 24 which is the total number of students interested in playing chess which is the blue circle. To determine the number of students in section 2 who want to play either Monopoly or chess, add 3 + 9 + 5 =17; then subtract 24 - 17 = 7.
Finishing the Problem: Area 1 and the total number of students interested in playing games must be confirmed. So, next, solve the red circle section 1 by adding 7 + 9 + 8 = 24 then subtracting 36 - 24 = 12. To find the total of all three categories add the results in all of the sections of the three circles. Do not add the totals of the categories since section totals would be counted more than once. The total number of students interested in playing games (47) was given in the explanation. To determine the answer for section 7, it is necessary to know the total number of students who are interested in playing checkers. To find the number of interested checkers, total the students in each section: 12 + 7 + 3 + 8 + 9 + 5 + ? (missing section 7) = 47. This is 44 +? = 47; Section 7 is 47 - 44 = 3. The number of students interested in playing checkers as indicated by the green circle is 8 + 9 + 5 +3 = 25.
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Solution for Students in a Grade Enrichment Group: Sports Lessons
Six children wanted to take only soccer or ice skating. They were not interested in swimming. The area of ice skating and soccer is area 2. The number 6 is placed there.
People interested in soccer numbered 44; this is placed by the title soccer.
Two children were interested in all three sports; this is recorded in area 5.
People interested in ice skating numbered 12; this is placed by the title ice skating.
Six children wanted swimming and soccer; this concerns areas 4 and 5. Since we know that area 5 represents 2 children, 4 children are represented by area 4 (2 + 4 = 6).
Ten children wanted to take swimming lessons. This is the total of the children interested in swimming and is placed by the title, Swimming.
Two children wanted only swimming and ice skating. This area is represented by
Finishing the Problem: Areas 1, 3, and 7 must be calculated. The total for the students must be confirmed. To find the number of students represented in area 1, add the numbers in areas 2, 4, and 5 which are (6 + 2 + 4 = 12); then subtract that total (12) from the total of students interested in soccer (44). The number for area one is 44 - 12 = 32. To find the number of students represented in area 3, add the numbers in areas 2, 5, and 6 which are (6 + 2 + 2 =10); then subtract that total (10) from the total of students interested in ice skating (12). The number for area three is 12 - 10 = 2 To find the number of students represented in area 7, add the numbers in areas 4, 5, and 6 which are (4 + 2 + 2 = 8); then subtract that total (8) from the total of students interested in swimming (10). The number for area seven is 10 - 8 = 2. Check this work by adding all the numbers inside the circles. The total of 32 + 6 + 2 + 2+ 2+ 2 + 4 = 50. This was given as the total number of students interested in sports lessons.
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Small Group Problem Grade Three Students: Going to the Beach
Steps to the Solution
Seven people were interested in all the activities. The 7 is recorded in area 5.
Eleven people only built sand castles. These eleven people are recorded in area 1.
Twenty-one people in all were interested in building sand castles. This total is placed by the title, Sand castles.
The two people who only wanted to build sand castles and play volleyball are recorded in area 4.
The person who built a sand castle and did some surfing is recorded in area 2.
The total of people interested in surfing is 13 and that is recorded by the title, Surfing.
Fifteen people in all were interested in playing volleyball (the green circle). This total is placed by the title, Volleyball Game
Finishing the Problem: Areas 3, 6, and 7 must be calculated. The total of students must be confirmed. To do this, determine how many students must be still be accounted for in the blue/ice skating circle (1 + 7 =8). There are a total of 13 students represented by the blue circle so 5 students must be accounted for in areas 3 and 6. This same process can be applied for the green circle. To do this, determine how many students must be still be accounted for in the green circle (2 + 7 =9). There are a total of 15 students represented by the green/volleyball circle so 6 students must be accounted for in areas 3 and 6. Consider what number combinations can be used to reach the sum of students in each circle. A combination of 3, 3, and 2 might work as well as 4, 2, and 2. Use these options to determine their location in the circles in order to produce the correct total number of students interested in each sport. Check this work by adding all the numbers inside the circles. The total of 11 + 1 + 7 + 2 + 3 + 3 + 2 = 29. This was given as the total number of students going on the field trip.
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Small Group Problem Grade Three Students: Dentists' Recommendations for Toothpaste
Steps to the Solution
The total number of dentists recommending Crest is 46. This number is placed by the title, Crest.
Eleven dentists recommended all three toothpastes which is represented in area 5.
Forty-two dentists in all recommended Aqua-Fresh. This number is placed by the title, Aqua-Fresh.
Thirty-nine dentists recommended Colgate. This number is placed by the title, Colgate.
Fourteen dentists recommended Crest and Aqua-Fresh only. This is area 6 where Aqua-Fresh and Colgate ONLY are represented in the overlapping part of the diagram.
Ten dentists recommended Colgate and Aqua-Fresh only. This is area 2 where Aqua-Fresh and Colgate ONLY are represented in the overlapping part of the diagram.
Seventeen dentists recommended Colgate and Crest only. This is represented by area 4.
Finishing the Problem: Areas 1, 3, and 7 must be calculated. The total must be confirmed. To find the number of students represented in area 1, add the numbers in areas 2, 4, and 5 which are (10 + 11 + 17 = 38); then subtract that total (38) from the total of dentists recommending Colgate (39). The number for area one is 39 - 38 = 1. To find the number of students represented in area 3, add the numbers in areas 2, 5, and 6 which are (10 + 11 + 14 = 35); then subtract that total (35) from the total of dentists recommending Aqua-Fresh (42). The number for area three is 42 - 35 = 7. To find the number of students represented in area 7, add the numbers in areas 4, 5, and 6 which are (17 + 11 + 14 = 42); then subtract that total (42) from the total of dentists recommending Crest (46). The number for area seven is 46 - 42 = 4. Check this work by adding all the numbers inside the circles. The total of 10 + 11 + 17 + 4 + 4 + 14 + 7 = 64; This was given as the total number of dentists making recommendations.
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Small Group Problem Grade Four Students: Raffle Prizes
Steps to the Solution
Ten people didn't care what game they received. This concerns area 5 which is the overlap of all three circles. The number in area 5 is 10.
Thirty people only wanted Gamecube. This concerns area 3 where there are no circles overlapping the Gamecube circle. The number in this area is 30
Double the number of people who did not care what they received only liked X-Box. This concerns the number in area 1; the number in area 1 is 20 because 2 X 10 = 20.
Half of the people that wanted to own only X-Box wanted to choose only Playstation. This concerns area 7 where there are no overlapping circles. The number in the area 7 is 10 because 50% of 20 = 10.
Forty-nine people wanted to play Gamecube. The total number of students who wanted to receive Gamecube is 49. Place this number by the title, Gamecube.
The total of people that wanted to play X-Box is 56. The total number of students who wanted to receive X-Box is 56. Place this number by the title, X-Box.
Finishing the Problem: Areas 3, 6. and 4 must be calculated. To find the number of students represented in area 4, add the numbers in areas 1, 2, and 5 which are (6+ 20 + 10 = 36); then subtract that total (36) from the total of students wanting X-Box (59). The number for area 4 is 56 - 36 = 20. To find the number of students represented in area 6, add the numbers in areas 4,5, and 7 which are (10 + 10 + 20= 40); then subtract that total (40) from the total number of students interested in Playstation (43). The number for area 6 is 43 - 40= 3 To find the number of students represented in area 3, add the numbers in areas 2, 5, and 6 which are (10 + 6+ 3= 19); then subtract that total (19) from the total wanting Gamecube (49). The number for area three is 49 - 19= 30.
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Small Group Problem Grade Four Students: Dessert Choices
Steps to the Solution
Five students wanted cheesecake and brownies, not ice cream Five (5) would be placed in area 2 because 5 students wanted brownies and cheesecake not ice cream.
Three dozen students wanted only ice cream and nothing else Multiply 12 X 3 because a dozen is 12 and there were 3 dozen interested students. 12 X 3 = 36; this is the number that is placed in area 7 where there are no overlapping circles for ice cream.
A total of forty-three students wanted cheesecake. This is the total 43 for the red circle labeled cheesecake. The number is placed by the title.
Twenty-one fewer students wanted only ice cream and brownies than the number of students wanting only ice cream. Since 21 fewer students had brownies and ice cream only than the students that had ice cream only,
Twenty of the forty-three students who wanted cheesecake also liked ice cream. They did not want brownies. No calculation is needed.
Six more students wanted all of the desserts than the number of students who wanted cheesecake and brownies only.
The total number of students who wanted to have brownies as their dessert is equal to nine times seven (9 X 7 = ?)
Finishing the Problem Areas 1, 3 and the total of those wanting mint must be calculated. The total number of students surveyed must be confirmed. to find the number of students who wanted only cheesecake 9area 1), add the number of students in areas 2, 4, and 5 which is 5 + 20 + 11 = 36. Then subtract 36 from the total number of students who wanted cheesecake (43), to find that 7 is the answer for area 1. (43 - 36 = 7). To find the number of students in area 3 who wanted only brownies, you add the number of students in area 2, 5, and 6 which is 5 + 11 + 15 = 31. Then you subtract 31 from the total number of students ho wanted brownies (63), and get 32 as the answer (63 - 31 = 32). This is the number to be placed in area 3. to find the number of students who wanted ice cream, you add the number of students in areas 4, 5, 6, and 7 which is (20 + 11 + 15 + 36 = 82). As a result, 82 is placed by the title of the ice cream circle, as the total number of students who wanted ice cream. To find the total number os students who were surveyed, you add the number of students in each of the areas in the circles: 7 + 5 + 32 + 2 + 11 + 15 + 36 = 108
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Small Group Problem Grade Four Students: Class Pet
Steps to the Solution
Eight students just wanted to get a Hyacinth Macaw. This refers to area 1 where there are no overlapping circles for macaw. Place an 8 in area 7.
Twenty-three students wanted the Eclecticus Parrot. This number is placed by the title, Eclecticus Parrot to show that the total number of students who wanted a parrot was 23.
Nine students are willing to have any bird except a macaw. Nine students wanted any bird but (except) a macaw, so put a 9 in area 2.
The students in the class wore the colors of their favorite bird; four students didn't have a favorite bird. Because 4 students did not have a favorite bird (any of the three were acceptable) the number 4 goes in area 5.
Multiply the number of students who just wanted to have a Hyacinth Macaw by the number of students who didn't care which bird they got. This is the number of students who wanted a cockatiel. 32 students wanted cockatiels because 8 X 4 = 32.
Nine students wanted just an Eclecticus parrot. This refers to area 3 where there are no overlapping circles. Place a 9 in area 3.
Divide twenty-six by two to find the number of students who wanted a macaw. 50% of 26 is 13. This is placed outside the circle by the title, Macaw.
Finishing the Problem: Areas 1, 4, and 6 are still unknown. The total of those surveyed must be confirmed. To find the number of students represented in area 6, add the numbers in areas 2, 3, and 4 which are (9 + 9 + 4 = 22); then subtract that total (22) from the total number of students interested in the parrot (23). The number for area 6 is 23 - 22= 1 To find the number of students represented in area 4, add the numbers in areas 5, 6, and 7 which are (4 + 1 + 8 = 13); then subtract that total (13) from the total of students wanting a macaw (13). The number for area 4 is 13 - 13= 0. Place a zero in area 4. To find the number of students represented in area 1, add the numbers in areas 2, 4, and 5 which are 9 + 4 + 0 = 13; then subtract that total (13) from the total of students wanting a cockatiel (32). The number for area one is 32 - 13 = 19. To find the total number of students participating in the survey add the numbers for each of the areas: 19 + 9 + 9 + 0 + 4 + 1 + 8 = 31
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Small Group Problem Grade Four Students: Growing Tropical Fruit
Steps to the Solution
Nine students didn't care which fruit was to be planted. They didn't have a preference, any of the three fruits would have been acceptable to plant. This is recorded in area 5.
Twenty-three students only wanted the fruits where you can peel off the smooth skin. The skins of mangoes and bananas are smooth and they peel off; this refers to area 2. Place 23 in area 2.
Fourteen students wanted a fruit which didn't have an enormous seed that took up about 1/3 of the fruit. Since mangoes have an enormous seed, and bananas and pineapples don't, then that refers to area 6. Record 14 in area 6.
Three times the amount of students who wanted only mangoes and pineapples, wanted only mangoes. Since 19 people wanted only mangoes and pineapples, three times this amount is 57. Place that number in area 1.
Solve this equation to find out the number of people who wanted only to plant pineapple. (45 + 72) - 56 = # of people who wanted pineapple. This means: 117 - 56 = 61. Place 61 in area 7.
One hundred three (103) students wanted to plant pineapples. Put this number under the title pineapple to indicate that total.
The total of the students who wanted bananas was forty-six. Place that under the title banana to indicate the total for banana.
Finishing the Problem: Area 3 and the total for mangoes are unknown. Now you can solve for the total for mangoes by adding the numbers in areas 1 , 2, 4, 5 . This is (57 + 23 + 9 + 19 = 108). Place this number by the title, Mangoes. Solve for area 3 by adding 23 + 9 + 14 = 46. Then subtract this total (46) from the total of students wanting to plant bananas (46). 46 - 46 = 0. This is the number of students who wanted to plant only bananas. The number is placed in area 3. To determine the number of students voting for the tropical fruits to be planted, add all the numbers in areas 1 through 7. (57 + 23 + 0 + 19 + 9 + 14 + 61 = 183). This the total for the class members.
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Individual Problem Grade Four: Hot Dog Toppings
Steps to the Solution
Twelve people had only tartar sauce. This refers to area 7 where there are no overlapping circles.
Eight and a half times the amount of people that had tarter sauce only had tartar sauce total. 8.5 X 12 = 102. Place this number by the title, tartar sauce, since it refers to the total of areas 4, 5, 6, and 7.
Twenty-eight people had both mustard and ketchup only. This refers to area 2, where the mustard and ketchup only areas overlap. Place a 28 in area 2.
The same amount of people that had mustard and ketchup only had mustard and tartar sauce only. This refers to area 4, where the mustard and tartar sauce circles only overlap. Place a 28 in area 4.
Thirty-six people had tartar sauce and ketchup only. This refers to area 6 where the two circles representing tartar sauce and ketchup overlap.
Three times the amount of people who had tarter sauce and ketchup only ordered ketchup on their hot dogs. (3 X 36 = 108). This number is placed next to the title, ketchup.
Finishing the Problem: Areas 1, 3, 5, and the total for mustard must be calculated. The total of hot dog eaters must be confirmed. To find the number of students represented in area 5, add the numbers in areas 4, 6, and 7 which are 12 + 28 + 36 = 76; then subtract the total (76) from the total number of students wanting tartar sauce (102). the number for area 5 is 102 - 76 = 26. To find the number of students represented in area 3, add the numbers in areas 2, 5, and 6 (28 + 26 + 36 = 90). Subtract this total (90) from the total of those wanting ketchup 108 (108 - 90 = 18). Place this number in area 1. To find the number of students wanting only mustard, add the numbers in areas 2, 3, 4, 5, 6, and 7 (28 + 18 + 28 + 26 + 26 + 12 = 128. Subtract this number from the total of students eating hot dogs to find the number of mustard-only hot dog eaters. (152 - 128 = 24). To find the total of mustard eaters of all combinations (the total by the title) add all the numbers in the mustard circle, that is areas 1, 2, 4, and 5. (24 + 28 + 26 + 28 = 106).
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Individual Problem Grade Five: Music Class
Steps to the Solution
Step One: Since you know that 10 kids want to participate Orchestra, you would put 10 outside of the orchestra circle, and place a 30 outside the band circle.
Step Two: It tells you that there are 5 kids who want to do everything, so you would put the number 5 in area five.
Step Three: Now you know that 50 kids want to sing in the Chorus, so you would write 50 outside of the Chorus circle.
Step Four: You now know that five kids want to do everything, so you would take 5, and multiply by 9 to get 45, which you divide by 3 to get 15, which you put in area four.
Step Five: Since five and 15 are your two inside-the-circle numbers, you add five and 15, to get 20, which is divided by five to get four, which is cut in half to get two, which is put in area number three.
Step Six: The total number of Chorus children is 50, while the total number of Orchestra children is 10. You add these 2 numbers together and get 60, which you divide by 6 to get 10, and subtract 8 to get 2. This number goes in area six.
Step Seven: You know that 30 kids want to play in the band. Add up the total number of band students also doing something else (2+5+15=22) and subtract from your total (30-22=8) which gives you the number of students (8) who only want to do band. This number goes in the band circle (area 1). Next, you have to determine the number of Orchestra students. Add the total number of Orchestra students also doing something else (2+5+2=9) and subtract from the total (10-9=1) to get the answer (1) which goes in the Orchestra circle area 2. Now you need to do the Chorus. Add all of the students in chorus yet doing another activity (15+5+2=22) and subtract this from the total (50-22=28) to get the answer (28) which goes in the Chorus circle (area 3).
Good Job!! You got through that impossible step! Now all you have to do is find out how many fifth graders there are total. Add all of your answers within the diagram. (8+2+1+15+5+2+28=68) and there's the answer! Now if only the fifth grade teachers knew that.
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Individual Problem Grade Five: Jefferson High Reading
Steps to the Solution
Step One: You now know that 78 people read Artemis Fowl so write 78 next to the Artemis Fowl title.

Step Two: The answer is 125 so you know now 125 people read in one of these areas of the circle 1 ,2 ,4, 5, 6 or 7
Step Three: Put 15 in area 3. To find the total, add 15 and 125 which will equal 140.
Step Four: Put 73 by Harry Potter, the title.
Step Five: 10% of 140 is 14 so put 14 in area 6
Step Six: Next subtract (14 + 78 = 92 ). Then subtract 125 - 92 = 33. Place 33 in area 7.
Step Seven: Then add 33 + 14 = 47. Subtract 73 - 47 = 26 so you now know that areas 4 and 5 must equal 26.
Step Eight: That equals 23.3 so round down to 23 and put that number in area one.
Step Nine: You know that area 4 (13) and area 5 (13) total 26. To find the number for area two, add (13 + 13 = 28) + 23 = 49. Next, subtract from the total of Artemis Fowl readers to find area 2's number. 78- 49 = 29
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Reviewed/Revised March, 2004
Page Designers:Emily B. and Jake M.
Instructor: J. Beyersdorfer