A Fun Mathematical Puzzle Solved with Algebra: How Many Questions Did the Boy Get Right?

Mathematics can often be approached through playful puzzles. Let's explore a fascinating example where algebraic equations help us unravel a fun scenario involving a father and his son. Solving this puzzle not only sharpens our math skills but also demonstrates the practical application of algebra in real-life situations.

The Challenge

A father offers his son $8 for every correct answer to a math problem and penalizes him with a $5 fine for every wrong answer. One day, the father gives his son a set of 26 math problems and ends up not owing any money in the end. Can you figure out how many problems the boy answered correctly?

The Solution

To solve this puzzle, we will use algebra. Let's define the variables:

x: the number of correct answers y: the number of incorrect answers We know two things:

The total number of problems answered is 26: x y 26 For each correct answer, the father pays $8, and for each incorrect answer, he charges $5, resulting in a net amount of $0: 8x - 5y 0

Now, let's translate these into equations:

x y 26 8x - 5y 0

From the first equation, we can express y in terms of x:

Method 1: Using Substitution

Step 1: Solve the first equation for y:

y 26 - x

Step 2: Substitute y in the second equation:

8x - 5(26 - x) 0

Step 3: Simplify and solve for x:

8x - 130 5x 0
13x 130
x 10

Now, substitute x 10 back into the first equation to find y:

y 26 - 10 16

Therefore, the boy answered 10 problems correctly and 16 problems incorrectly.

Method 2: Using Coincidence

We can also solve this problem by a more intuitive approach:

Step 1: Assume x is the number of correct answers, and y is the number of incorrect answers.

Step 2: From the first equation:

xy 26

Step 3: Express y in terms of x:

y 26 - x

Step 4: From the second equation:

8x - 5y 0

Step 5: Substitute the expression for y into the second equation:

8x - 5(26 - x) 0

Step 6: Simplify and solve for x:

8x - 130 5x 0
13x 130
x 10

Thus, x 10, and y 26 - 10 16. So, the boy has 10 correct answers and 16 incorrect answers.

Conclusion

The solution to this puzzle is straightforward when we use algebra. By setting up the equations and solving them step-by-step, we can determine that the boy answered 10 problems correctly. This example showcases the power of algebra in solving real-life scenarios and highlights the importance of systematic problem-solving in mathematics.

Further Reading and Practice

Interested in more puzzles and problems? Here are a few related topics for you to explore:

Solving linear equations Algebraic word problems Using algebra in real-life scenarios

Whether you're a student looking to enhance your math skills or a math enthusiast simply enjoying a challenge, puzzles like this can be both fun and educational.