Cafeteria Seating Arrangement Problem
Nancy Maple
Ray Shannon
Christine Bolte Kathy Padgett
Jehan Khan
Class: Geometry.
Materials: Measuring devices and included data sheet.
Goals: To give students practice on area, problem-solving, and mathematical modeling writeups and presentations.
Time Required: Two days. One day will be needed for the students to go to the cafeteria for
measurements, and then work together on their calculations. Another day will be needed to
collect the write-ups and to hear the presentations.
Background: Prior knowledge of measuring distance, and finding the area of squares,
rectangles, and circles.
Setting: The school has decided to replace all of the cafeteria tables and chairs with new ones.
Below is a list of the prices and dimensions of the tables and chairs chosen from an office
furniture catalogue:
Furniture
Round 60" Diameter Folding Table
Cost
$208.60
Square 36" x 36" Folding Table
$96.00
Rectangular 36" x 72" Folding Table
$137.80
Bench Table 54" x 96"
$412.00
Four Chairs 193"w x 302"h x 19f"d
$194.00
Problem: The students are to decide the number of chairs and the number and style of tables
that should be purchased in order to maximize seating capacity and minimize cost. The students
should also draw a diagram or clearly explain how the tables should be arranged in the cafeteria.
Each group should submit a data sheet and a written report, and will present their results to the
class.
Evaluation: One grade will be given to the entire group. Point values will be assigned for the
data sheet, written report, and presentation. Example: data sheet/report is worth 15 points and
presentation is worth 5 points. Note: bonus points may be awarded for the best solution.
Funded in part by the National Science Foundation and Indiana University 1995
Extensions: The problem may be modified by giving the students a budget, rather than having
them find the least cost. For a more challenging project, have the students research the price list
information. Another alternative would be to require a scaled diagram. Finally, a similar
problem would be that the students are to arrange the desks/tables in their own classroom to
better suit their needs.
Teacher Notes: This project would work well for groups of three to four students. Be sure the
expectations for the written report and presentation are clear to the students before beginning the
project. In order to avoid taking the class down to the cafeteria, the teacher may provide the
dimensions for the students.
Assumptions - There are many assumptions that could be made about your specific
cafeteria. Is it important to have the maximum number of seats or are there social
considerations that your school has the luxury of being able to entertain? Do the tables
need to be used for other purposes? Is school enrollment expected to be constant?
Report - must include a description of the problem and how it was approached, a list of
assumptions, calculations, a concluding statement and/or diagram explaining the
arrangement, and the data sheet.
Presentation - should discuss everything in the report.
Funded in part by the National Science Foundation and Indiana University 1995
Cafeteria Seating Arrangement Problem
Sample Report
Project Description: Our school has decided to replace all of the cafeteria tables and chairs with
new ones. We were given a list of the prices and dimensions for the chairs and for round, square,
rectangular, and bench tables. Our problem was to decide the number of chairs and the number
and style of tables that should be purchased in order to maximize seating capacity and minimize
cost. Using a tape measure, we found the dimensions of the cafeteria to be 32' x 84'. We then
tried four different ways of arranging the tables, and determined the number of people each
arrangement would seat. Finally, we calculated the cost per student to predict the most
economical table organization.
Assumptions: The following is a list of assumptions we made during the project:
1. The number of chairs around the round tables can be determined by finding the
circumference and then allowing two feet per chair.
2. A two foot area is needed around each table to allow for chair movement.
3. We will not put two square tables together to form a 3' x 6' rectangular table, since the cost of
a rectangular table is much less than the cost of two square tables.
4. We will put two rectangular tables together and have a person seated at each end.
5. Our school has an enrollment of approximately 900 students.
6. For bench tables we will need to leave at least a two foot walkway between the tables.
7. Each bench table seats eight people.
8. The bench tables do not fold.
9. We do not need to clear the cafeteria for dances.
10. Our lunch period is divided into three periods, each serving 300 people.
Calculations: Since we measured the dimensions of the cafeteria in feet, we first converted the
dimensions of the tables to feet and then made the following calculations.
Round  60/12 = 5 (5' diameter)
Square  36/12 = 3 (3' x 3')
Rectangular  36/12 = 3, 72/12 = 6 (3' x 6')
Bench  54/12 = 2.5, 96/12 = 8 (2.5' x 8')
Round Tables: To find out how many people a round table seats, we found the circumference
and divided by two, allowing for a distance of two feet per person.
C = (3.14)(5)  16; therefore, seating capacity = 8 people
Since the diameter is five feet and we allowed a two foot clearance around the table for seating,
we used nine feet as a diameter in our calculations. To find the number of tables that could fit
lengthwise, we divided the length by nine. Similarly, we divided the width by nine.
Funded in part by the National Science Foundation and Indiana University 1995
82/9  9, 32/9  3
This means the cafeteria could be filled with nine rows of three tables. By offsetting three tables,
we found that we could fit one more row of tables, for a total of ten rows of three. In other
words, the cafeteria could hold thirty round tables. Since each table holds eight people, the
seating is 240. Because we assumed that 300 people will be served each lunch period, this
arrangement is not acceptable.
Rectangular: Since we have allowed two feet for each person and the length of the table is six
feet, three people may sit on each longer side. Also, we assumed that two rectangular tables are
pushed together and that one person may sit at each end. This means two rectangular tables
pushed together will seat a total of fourteen people. Allowing two feet on either side of the width
of the table, we divided the length of the room by seven feet.
Allowing two feet on either side of the length of the table, we divided the width of the room by
ten feet. We found that the cafeteria could hold 48 tables. Since each grouping holds fourteen
people, the total seating capacity is 336. Each package contains four chairs, so we would need to
order 84 packages.
Total Cost of Tables = Number of Tables * Cost of Table = 48(137.80) = $6,614.40
Total Cost of Chairs = Number of Packages * Cost of Package = 84(194) = $16,296.00
Total Cost = Total Cost of Tables + Total Cost of Chairs = 6,614.4 + 16,296 = $22,910
Cost Per Student = Total Cost/Number of Students = 22,910/336 = $68.19
Square: Allowing two feet for each student, we can seat four people at each table. With two feet
added around each table, we divided the length and width of the room by seven. We found that it
was possible to fit twelve rows of five tables with one table pushed against the wall in each row.
The resulting seating capacity is 216, which is not sufficient according to our assumptions.
Bench: After analyzing several situations, we found that twelve rows of two joined tables could
fit perpendicular to ten rows of two joined tables in the cafeteria for a total of forty-four tables.
Based on our assumption that eight people may sit at each table, the seating capacity is 352.
Since bench tables have built-in seating, no chairs are needed.
Total Cost of Tables = Number of Tables * Cost of Table = 44(412) = $18,128.00
Cost Per Student = Total Cost of Tables/Number of Students = 18,128/352 = $51.50
Conclusions: Although there are more possible arrangements and combinations, we have
concluded that using bench tables allows for the greatest number of students to dine at one time
Funded in part by the National Science Foundation and Indiana University 1995
for the lowest per student cost. The bench tables should be arranged in twelve rows of two
joined tables placed perpendicular to ten rows of two joined tables. Because our cafeteria is not
used for school dances, we were not required to purchase folding tables.
Funded in part by the National Science Foundation and Indiana University 1995
Cafeteria Seating Arrangement Data Sheet
Sample Solution
Team Members: Nancy Maple, Jehan Khan, Raymond Shannon

M O
diameter 60"
cost $208.60
dimensions 36" x 36"
cost $96.00
dimensions
54" x 96"
(including benches)
cost $412.00
dimensions 36" x 72"
cost $137.80
dimensions 193"x302"x19f"
cost $194.00/package
kind of table
# of tables
# of chairs
# of people
cost per pupil
total cost
round
30
240
240
$74.58
$17,898
rectangular
48
336
336
$68.19
$22,910
square
60
216
216
$75.17
$16,236
bench
44
0
352
$51.50
$18,128
Assumptions:
1. Allow 2 feet per chair around each round table.
2. A 2 foot area is needed around each table to allow for chairs.
3. Rectangular tables are pushed together and 2 people may sit at each end.
4. Two square tables will not be pushed together because of the cost.
5. The school has an enrollment of approximately 900 students.
6. Leave at least a 2 foot walkway between bench tables. Each bench table seats 8 people.
Funded in part by the National Science Foundation and Indiana University 1995
7. The cafeteria does not need to be cleared for dances.
Funded in part by the National Science Foundation and Indiana University 1995
Cafeteria Seating Arrangement Problem
Student Data Sheet
Team Members:

M O
diameter _____
dimensions ____x____
dimensions____x____
cost $______
cost $______
cost $______
dimensions
____x____
(including benches)
dimensions ____x____x____
cost $______
cost $_________
kind of table
# of tables
# of chairs
# of people
cost per pupil
Please list all of your assumptions on the back of this paper.
Funded in part by the National Science Foundation and Indiana University 1995
total cost