Answer :
Answer:
See the below works.
Step-by-step explanation:
(c)
To find the system of equations for Mrs. Brown's test, we first determine the variables for true/false questions and multiple choice questions, then convert the restrains into mathematics equations.
Let:
- [tex]x=\texttt{number of true/false questions}[/tex]
- [tex]y=\texttt{number of multiple-choice questions}[/tex]
Then:
- "The test will have true/false questions worth 2 points each and multiple-choice questions worth 4 points each, for a total of 100 points." ⇒ 2x + 4y = 100 ... [1]
- "twice as many multiple-choice questions as true/false questions" ⇒ 2x = y ... [2]
Now, we can create the system of equations by combining the equations [1] and [2]:
[tex]\left\{\begin{aligned}2x+4y&=100\\2x&=y\end{aligned}\right.[/tex]
(d)
To find the number of each type of questions, we need to solve the system of equations. Let substitute the y in the equation [1] with equation [2]:
[tex]\begin{aligned}2x+4y&=100\\2x+4(2x)&=100\\10x&=100\\\bf x&\bf=10\end{aligned}[/tex]
Now, substitute the x with 10 in any equation to find the y:
[tex]\begin{aligned}2x&=y\\2(10)&=y\\\bf y&\bf=20\end{aligned}[/tex]
Hence, there will be 10 true/false questions and 20 multiple-choice questions.
(e)
To find out if 45 minutes will be enough, we use the (d) answer to calculate the time needed to answer all questions:
[tex]\begin{aligned}\texttt{total time}&=10\times1+1\frac{1}{2}\times20 \\&=10+30\\&=\bf 40\ minutes\end{aligned}[/tex]
Since, 40 minutes is less than 45 minutes, then they will have enough time to finish the test.
