Math Problem Analysis
Problem Statement
Dr. Johnson has set up a unique incentive system for his son regarding math homework. He offers to pay his son $5.00 for each correct answer, while the boy must pay his father $8.00 for each incorrect answer. In total, the boy answered twenty-six math questions, and interestingly, no money was exchanged at the end of this exercise.
Objective
The task is to determine how many answers were correct and how many were incorrect.
Setting Up the Equations
Let’s define the variables:
– Let C represent the number of correct answers.
– Let I represent the number of incorrect answers.
Given the conditions of the problem, we can formulate the following equations:
1. Since no money was exchanged, the total earnings from correct answers equals the total losses from incorrect answers:
5C = 8I
2. The total number of questions answered is given by:
C + I = 26
Solving the Equations
From the second equation, we can express C in terms of I:
C = 26 – I
Substituting this expression into the first equation yields:
5(26 – I) = 8I
Expanding this equation gives:
130 – 5I = 8I
Combining like terms results in:
130 = 13I
Solving for I results in:
I = 10
Now, substituting I back into the equation for C:
C = 26 – 10, which simplifies to:
C = 16
Conclusion
The final tally reveals that the boy answered 16 questions correctly and 10 questions incorrectly.