Fernando's Math Mistake: Spot The Error!

Let's dive into Fernando's math problem and pinpoint exactly where he went wrong. The original expression is:

$\frac{5(9-5)}{2}+(-2)(-5)+(-3)^2$

Fernando's solution is presented as follows:

$\frac{5(9-5)}{2}+(-2)(-5)+(-3)^2 \\$ $\frac{5(4)}{2}-10+9 \\$ $\frac{20}{2}-10+9 \\$ $10-10+9 \\$ $9$

Now, let's meticulously examine each step to identify any potential errors.

Detailed Analysis of Fernando's Solution

To effectively identify Fernando's error, we will break down each step of his solution and compare it with the correct method. This involves a thorough examination of the order of operations (PEMDAS/BODMAS) and the accurate application of arithmetic principles. By scrutinizing each transition, we can pinpoint precisely where the mistake occurred and understand the nature of the error.

Step 1: $\frac{5(9-5)}{2}+(-2)(-5)+(-3)^2$

This is the original expression. No calculations are performed here, so there's no possibility of an error in this initial step. It simply sets the stage for the subsequent calculations.

Step 2: $\frac{5(4)}{2}-10+9$

Here, two parts of the expression are simplified:

• Part 1: Inside the parentheses in the fraction, (9 - 5) is correctly evaluated to 4. So, 5(9-5) becomes 5(4).
• Part 2: The term (-2)(-5) should result in +10 because a negative times a negative is a positive. However, Fernando incorrectly wrote -10.
• Part 3: (-3)^2 which means -3 multiplied by -3, should equal 9, and it remains +9.

Therefore, the error occurred in this step. (-2)(-5) should be +10, not -10.

Step 3: $\frac{20}{2}-10+9$

In this step, the numerator of the fraction is calculated: 5(4) = 20. This part is done correctly, based on the incorrect result from Step 2. Thus, given the incorrect premise from Step 2, this calculation is accurate.

Step 4: $10-10+9$

Here, the fraction $\frac{20}{2}$ is simplified to 10. This calculation is correct. However, it still carries forward the initial error from Step 2.

Step 5: $9$

Finally, the expression 10 - 10 + 9 is evaluated. 10 - 10 = 0, and 0 + 9 = 9. The final result is correct based on all the previous steps, but because of the initial error, the entire solution leads to the correct answer through incorrect steps.

Identifying the Specific Error

The error is in the simplification of (-2)(-5). A negative number multiplied by another negative number results in a positive number. Therefore, (-2)(-5) should be +10, not -10. This error propagates through the rest of the solution, even though the final answer happens to be correct due to compensating errors.

Therefore, the correct identification of Fernando's error is that he incorrectly simplified (-2)(-5) as -10 instead of +10.

Correct Solution

Let's solve the expression correctly:

$\frac{5(9-5)}{2}+(-2)(-5)+(-3)^2 \\ = \frac{5(4)}{2} + 10 + 9 \\ = \frac{20}{2} + 10 + 9 \\ = 10 + 10 + 9 \\ = 20 + 9 \\ = 29$

The correct answer is 29, not 9.

Conclusion

Fernando's error was in the early stages of the calculation, specifically in handling the multiplication of two negative numbers. While he arrived at the answer of 9 through a series of steps, the correct answer should have been 29. This exercise highlights the importance of paying close attention to detail and adhering to the rules of arithmetic. Double-checking each step can prevent such errors and ensure accurate results. Remember, even if the final answer seems correct, the process matters, and identifying and correcting errors along the way is crucial for understanding and mastering mathematical concepts.

To further enhance your understanding of mathematical principles and error analysis, you might find valuable resources and tutorials on websites like Khan Academy. This external resource offers comprehensive lessons and practice exercises that can help solidify your grasp of arithmetic and algebra.