Answer:
30 multiple choice questions and 8 word problems
Step-by-step explanation:
Let m = number of multiple choice questions.
Let w = number of word problems.
There are 38 problems altogether, so we have the first equation concerning the number of problems.
m + w = 38
A multiple choice question is worth 2 points, so m multiple choice questions are worth 2m points. A word problem is worth 5 points, so w word problems are worth 5w points. Altogether, both types of problems are worth 2m + 5w points. We are told all problems combined are worth 100 points. We can now write our second equation concerning the number of points.
2m + 5w = 100
We have a system of two equations in two unknowns.
m + w = 38
2m + 5w = 100
We can use the substitution method to solve the system of equations.
Solve the first equation for m.
m + w = 38
Subtract w from both sides.
m = 38 - w
Now where we see m in the second original equation, we substitute it with 38 - w.
The second original equation is
2m + 5w = 100
Substitute 38 - w for m in the equation.
2(38 - w) + 5w = 100
We have a single equation in a single variable, so we can solve it for the variable.
Distribute the 2.
76 - 2w + 5w = 100
Combine like terms on the left side.
76 + 3w = 100
Subtract 76 from both sides.
3w = 24
Divide both sides by 3.
w = 8
There are 8 word problems.
Now we use the first original equation and substitute 8 for w and solve for m.
m + w = 38
m + 8 = 38
Subtract 8 from both sides.
m = 30
There are 30 multiple choice questions.
Answer: 30 multiple choice questions and 8 word problems
Check:
30 questions + 30 problems = 38 problems in total
30 questions @ 2 pts each = 2 * 30 pts = 60 pts
8 problems @ 5 pts each = 5 * 8 pts = 40 pts
60 pts + 40 pts = 100 pts
The total is 38 problems and 100 points, so our answer is correct.
