Math Problem: Dr. Johnson’s Homework Payment Scheme
Problem Overview
Dr. Johnson offers his son a financial incentive for completing his math homework. He promises to give his son $5.00 for each correct answer. Conversely, for each incorrect answer, the son must pay his father $8.00. The boy answered a total of twenty-six math questions, and ultimately, no money was exchanged.
Objective
The goal is to determine how many answers the boy got correct and how many were incorrect.
Solution Approach
Let:
– C represent the number of correct answers.
– I represent the number of incorrect answers.
From the problem, we know that no money was exchanged, leading to the equation:
5C = 8I
Additionally, since the total number of questions answered is 26, we have:
C + I = 26
Substituting and Solving
We can express C in terms of I:
C = 26 – I
Substituting this expression for C into the first equation gives:
5(26 – I) = 8I
Expanding this equation results in:
130 – 5I = 8I
Combining like terms leads to:
130 = 13I
Final Calculations
Solving for I:
I = 10
Substituting back to find C:
C = 26 – I = 26 – 10 = 16
Conclusion
The boy answered 16 questions correctly and 10 questions incorrectly.