Answer :
Certainly! Let's break down the problem to find out how many two-point and four-point questions are on Tiffany's test.
1. Understanding the Problem:
- The test has a total of 42 questions.
- The entire test is worth 100 points.
- There are two types of questions: two-point questions and four-point questions.
- We need to find out how many of each type of question is on the test.
2. Define the Variables:
- Let [tex]\( t \)[/tex] be the number of two-point questions.
- Let [tex]\( f \)[/tex] be the number of four-point questions.
3. Set Up the Equations:
- Since the total number of questions is 42, we can write the equation:
[tex]\[
t + f = 42
\][/tex]
- Since the total points are 100, and each two-point question contributes 2 points and each four-point question contributes 4 points, we set up the equation:
[tex]\[
2t + 4f = 100
\][/tex]
4. Check the Equations:
- The equation [tex]\( t + f = 42 \)[/tex] represents the total number of questions.
- The equation [tex]\( 2t + 4f = 100 \)[/tex] represents the total number of points.
5. Identifying Correct Equations:
- Both [tex]\( t + f = 42 \)[/tex] and [tex]\( 2t + 4f = 100 \)[/tex] are correct representations of the problem.
Let's proceed with these equations to solve for [tex]\( t \)[/tex] and [tex]\( f \)[/tex].
6. Solving the Equations:
- From the first equation, express [tex]\( t \)[/tex] in terms of [tex]\( f \)[/tex]:
[tex]\[
t = 42 - f
\][/tex]
- Substitute [tex]\( t = 42 - f \)[/tex] into the second equation:
[tex]\[
2(42 - f) + 4f = 100
\][/tex]
- Simplify and solve for [tex]\( f \)[/tex]:
[tex]\[
84 - 2f + 4f = 100
\][/tex]
[tex]\[
2f = 16
\][/tex]
[tex]\[
f = 8
\][/tex]
7. Find the Value of [tex]\( t \)[/tex]:
- Substitute [tex]\( f = 8 \)[/tex] back into the equation [tex]\( t = 42 - f \)[/tex]:
[tex]\[
t = 42 - 8
\][/tex]
[tex]\[
t = 34
\][/tex]
Therefore, there are 34 two-point questions and 8 four-point questions on the test.
1. Understanding the Problem:
- The test has a total of 42 questions.
- The entire test is worth 100 points.
- There are two types of questions: two-point questions and four-point questions.
- We need to find out how many of each type of question is on the test.
2. Define the Variables:
- Let [tex]\( t \)[/tex] be the number of two-point questions.
- Let [tex]\( f \)[/tex] be the number of four-point questions.
3. Set Up the Equations:
- Since the total number of questions is 42, we can write the equation:
[tex]\[
t + f = 42
\][/tex]
- Since the total points are 100, and each two-point question contributes 2 points and each four-point question contributes 4 points, we set up the equation:
[tex]\[
2t + 4f = 100
\][/tex]
4. Check the Equations:
- The equation [tex]\( t + f = 42 \)[/tex] represents the total number of questions.
- The equation [tex]\( 2t + 4f = 100 \)[/tex] represents the total number of points.
5. Identifying Correct Equations:
- Both [tex]\( t + f = 42 \)[/tex] and [tex]\( 2t + 4f = 100 \)[/tex] are correct representations of the problem.
Let's proceed with these equations to solve for [tex]\( t \)[/tex] and [tex]\( f \)[/tex].
6. Solving the Equations:
- From the first equation, express [tex]\( t \)[/tex] in terms of [tex]\( f \)[/tex]:
[tex]\[
t = 42 - f
\][/tex]
- Substitute [tex]\( t = 42 - f \)[/tex] into the second equation:
[tex]\[
2(42 - f) + 4f = 100
\][/tex]
- Simplify and solve for [tex]\( f \)[/tex]:
[tex]\[
84 - 2f + 4f = 100
\][/tex]
[tex]\[
2f = 16
\][/tex]
[tex]\[
f = 8
\][/tex]
7. Find the Value of [tex]\( t \)[/tex]:
- Substitute [tex]\( f = 8 \)[/tex] back into the equation [tex]\( t = 42 - f \)[/tex]:
[tex]\[
t = 42 - 8
\][/tex]
[tex]\[
t = 34
\][/tex]
Therefore, there are 34 two-point questions and 8 four-point questions on the test.
