Video Transcript
A teacher gave his students 100
minutes to solve a test that has two sections, section A and section B. The students had to answer at least
four questions from section A and at least six questions from section B and answer
at least 11 questions in total. If a girl answered each question in
section A in three minutes and each question in section B in six minutes, derive the
system of inequalities that would help to know how many questions she tried to solve
in each section. Use 𝑥 to represent the number of
questions answered from section A and 𝑦 to represent the number from section B.
In this question, we are asked to
derive a system of inequalities. We have two variables 𝑥 and 𝑦
where 𝑥 is the number of questions answered from section A and 𝑦 the number of
questions from section B. We are told that the students had
to answer at least four questions from section A. This corresponds to the inequality
𝑥 is greater than or equal to four. The students also had to answer at
least six questions from section B. This means that 𝑦 must be greater
than or equal to six.
Next, we are told that the students
had to answer at least 11 questions in total. The total number of questions
answered corresponds to the expression 𝑥 plus 𝑦. And since a student has to answer
at least 11 questions, this must be greater than or equal to 11. Finally, we have a time constraint
as the teacher gave the students 100 minutes to solve the test. We are told that one girl answered
each question in section A in three minutes and she answered each question in
section B in six minutes. This means that the total time she
spent answering questions in section A is three multiplied by 𝑥, or three 𝑥. And the total time she spent
answering questions in section B is six 𝑦.
The total time that she spent
answering questions is therefore equal to three 𝑥 plus six 𝑦. And since the test was 100 minutes
long, this must be less than or equal to 100. We now have a system of four
inequalities that could help us calculate how many questions the girl tried to solve
in each question. They are 𝑥 is greater than or
equal to four, 𝑦 is greater than or equal to six, 𝑥 plus 𝑦 is greater than or
equal to 11, and three 𝑥 plus six 𝑦 is less than or equal to 100. Whilst it is not required in this
question, this is an example of a linear programming problem that we could solve to
find the optimal solution. This could be done either
graphically or algebraically.
