Auditorium Problem
6.RP - Understand ratio concepts and use
ratio reasoning to solve problems.
7.RP - Analyze proportional relationships
and use them to solve real-world and
mathematical problems.
8.EE - Understand the connections between
proportional relationships, lines, and linear
equations.
5/7 of the students seated in an
auditorium were girls. There are 48 more
girls than boys. How many students are
seated in the auditorium?
Use Proportional Reasoning- Method 1
5/7 of the students seated in an auditorium were girls.
There are 48 more girls than boys.
How many students are seated in the auditorium?
girls
5 x + 48
= =
boys
2
x
5x = 2(x + 48)
5x = 2x + 96
3x = 96
What does 32 represent?
x = 32
Use Proportional Reasoning - Method 1
5/7 of the students seated in an auditorium were girls.
There are 48 more girls than boys.
How many students are seated in the auditorium?
x = 32
girls
5 x + 48 32 + 48 80
=
=
= =
boys
x
32
32
2
Total # of Students = 80 + 32
Total # of Students = 112
Use Proportional Reasoning – Method 2
5/7 of the students seated in an auditorium were girls.
There are 48 more girls than boys.
How many students are seated in the auditorium?
5 x + 48
girls
= =
7 2x + 48
total
5(2x + 48) = 7(x + 48)
10x + 240 = 7x + 336
3x = 96
x = 32
Use Proportional Reasoning – Method 2
5/7 of the students seated in an auditorium were girls.
There are 48 more girls than boys.
How many students are seated in the auditorium?
x = 32
girls 5 x + 48
= =
total 7 2x + 48
Total # of Students = 2x + 48
Total # of Students =112
Use logical Reasoning- Method 3
5/7 of the students seated in an auditorium were girls.
There are 48 more girls than boys.
How many students are seated in the auditorium?
If 5/7 of the students are girls, then 2/7 of
the students have to be boys. Therefore
the difference between girls and boys is
3/7 of the students and since there are 48
more girls than boys, then 3/7 of the
students must be equal to 48.
Use logical Reasoning- Method 3
If 5/7 of the students are girls, then 2/7 of the
students have to be boys. Therefore the difference
between girls and boys is 3/7 of the students and
since there are 48 more girls than boys,
then 3/7 of the students must be equal to 48.
7
7 3
· x = 48·
3
3 7
x =112
Use Model Drawings – method 4
5/7 of the students seated in an auditorium were girls.
There are 48 more girls than boys. How many students
are seated in the auditorium?
Girls
16 16 16 16 16
48¸ 3 =16
Each box represents 16
48
Boys
16 16
Total # of Students =16 ´ 7 =112
Use Systems of Equations – Method 5
5/7 of the students seated in an auditorium were girls.
There are 48 more girls than boys. How many students
are seated in the auditorium?
5
( b + g) = g
7
g = b + 48
Let g = # of girls
Let b = # of boys
Total # of students = b + g
Use Systems of Equations – Method 5
5
( b + g) = g
7
é5
ù
7 ê ( b + g ) = gú
ë7
û
5 ( b + g) = 7g
5b + 5g = 7g
5b - 2g = 0
g = b + 48
5b - 2g = 0
5b - 2(b + 48) = 0
5b - 2b - 96 = 0
3b = 96
b = 32
Use Systems of Equations – Method 5
5
( b + g) = g
7
g = b + 48
if b = 32
then g = 32 + 48 = 80
How many students are seated in the auditorium?
Total # of Students =
b + g = 32 + 80 =112
Use Systems of Equations – Method 6
5
( b + g) = g
7
[
7
]
g = b + 48
2
g = g + 48
5
5b + 5g = 7g
-5g -5g
æ2 ö æ2 ö
-ç g÷ -ç g÷
è5 ø è5 ø
5b = 2g
5 5
2
b= g
5
g = b + 48
80 = b + 48
b = 32
æ 5ö 3
æ 5ö
ç ÷ g = 48 ç ÷
è 3ø 5
è 3ø
g = 80
Use Systems of Equations- Method 6
5/7 of the students seated in an auditorium were girls.
There are 48 more girls than boys. How many students are seated in the auditorium?
Let g = # of girls
Let b = # of boys
Total # of students = b + g =112
g = 80
b = 32
Discussion
• How are the solution methods
similar?
• How are the solution methods
different?
• Identify correspondences between
different solution methods.
What is the Error?
girls
5 x + 48
= =
boys
2
x
2x + 48 = 5x
3x = 48
x =16
Total = x + x + 48 = 80
What is the Error?
girls
5
x
= =
boys
2
x + 48
5(x + 48) = 2x
5x + 240 = 2x
-3x = 240
x = -80
What is the Error?
girls
5
x + 48
= =
boys
7
x
5x = 7(x + 48)
5x = 7x + 336
2x = 336
x =168
Total = x + x + 48 = 384