Question

A number \(x\) is mistakenly divided by 10 instead or bemg multiplied by \(10 .\) What is the percentage error in the result? (a) \(-99 \%\) (b) \(+99 \%\) (c) \(-100 \%\) (d) \(+100 \%\)

Short answer

Answer: -99%

Step-by-step solution

Calculate the actual result

Multiply the given number x by 10: Actual result = \(10x\)

Calculate the incorrect result

Divide the given number x by 10: Incorrect result = \(\frac{x}{10}\)

Calculate the difference between actual and incorrect results

Subtract the incorrect result from the actual result: Difference = Actual result - Incorrect result = \(10x - \frac{x}{10}\)

Calculate the percentage error

Divide the difference by the actual result, multiply by 100, and add the percent sign: Percentage error = \(\frac{Difference}{Actual result} \times 100\% = \frac{10x - \frac{x}{10}}{10x} \times 100\%\)

Simplify the percentage error expression

Find a common denominator and simplify the fraction: Percentage error = \(\frac{\frac{100x - x}{10}}{10x} \times 100\% = \frac{99x}{100x} \times 100\%\)

Calculate the final percentage error

Simplify the fraction and multiply by the percentage: Percentage error = \(99\%\) Since the given number x is mistakenly divided by 10 instead of being multiplied, it results in a smaller value than intended. Therefore, the percentage error is negative. Our final answer is: \(-99\%\), which corresponds to option (a).

Key concepts

Understanding Mathematical Errors

Mathematical errors often occur during problem-solving, especially when operations like division and multiplication are involved. In the given exercise, the mistake was dividing the number \(x\) by 10, instead of multiplying it by 10. This error led to a vastly different outcome. The key thing here is to understand that a small error in operation choice can result in a significant error in the final result.
This mistake transforms the correct measurement or calculation into an incorrect one, generating a percentage error. Identifying and correcting these errors is fundamental to achieving accurate results. By analyzing where the error occurred, we can use systematic methods to find the exact percentage of error.
Such errors reinforce the importance of verifying each step of a mathematical calculation, especially in more complex problems where multiple operations are performed consecutively. Correct mathematics rely heavily on patience and precision, ensuring each operation aligns with intended outcomes.

Division and Multiplication Concepts

In mathematics, division and multiplication are inverse operations. This means they undo each other's effects. Understanding their roles and relationship is crucial. If a number is multiplied, it increases, while division decreases it. In our exercise, multiplying \(x\) by 10 increases \(x\), whereas dividing \(x\) by 10 drastically reduces it.
When solving problems, it's important to choose the correct operation based on the context. A common rule of thumb is to revisit the problem's language and ensure comprehension before proceeding with calculation.
When we realize the initial mistake in operation—dividing instead of multiplying—we can see the error resulted in a much smaller number. This is easily assessed by comparing the products of multiplication and division. Mistakes in choosing the right operation are quite common, making it essential to double-check before executing any calculation.

Effective Problem-Solving in Mathematics

Successful problem-solving in mathematics involves a few key strategies. Firstly, read the problem statements carefully. Misinterpretation often leads to errors, as seen in this exercise where division was mistakenly performed instead of multiplication.
Secondly, write down each step clearly. In the solution, each computational step was outlined, helping to visualize the problem and catch any errors made. This structured approach can also make verifying the problem easier, allowing room to correct errors.
Additionally, understanding the problem context helps in selecting the correct operations and also in interpreting the results correctly. After calculations, assess whether the result is logical in the problem context; this can help spot errors. Practice combining different mathematical ideas, such as complex fractions or percents, as a comprehensive toolkit is necessary for effective problem solving.