Calculating the Percentage Error in a Math Calculation: A Step-by-Step Guide

Mathematics is the language of precision, but even students can make mistakes. In this guide, we will explore a common error made by a student and calculate the percentage error due to this mistake. Understanding how to find the percentage error is crucial for improving accuracy in mathematical calculations.

Understanding the Problem

A student multiplied a number by 3/5 instead of 5/3. We will denote the original number as x to explore this scenario in detail.

Correct Calculation

The correct operation is to multiply by 5/3.

Correct Result: $$ x times frac{5}{3} frac{5x}{3} $$

Incorrect Calculation

The student multiplied by 3/5.

Incorrect Result: $$ x times frac{3}{5} frac{3x}{5} $$

Difference Between Correct and Incorrect Results

To find the difference between the correct and incorrect results, we need to subtract the incorrect result from the correct result:

Difference: $$ frac{5x}{3} - frac{3x}{5} $$

Both fractions need a common denominator. The least common multiple of 3 and 5 is 15. Let's rewrite the fractions:

$$ frac{5x}{3} frac{25x}{15} $$ $$ frac{3x}{5} frac{9x}{15} $$

Now, we can subtract the fractions:

$$ frac{25x}{15} - frac{9x}{15} frac{16x}{15} $$

Calculating the Percentage Error

The percentage error is calculated as the absolute difference divided by the correct result, multiplied by 100:

Percentage Error: $$ left( frac{frac{16x}{15}}{frac{5x}{3}} right) times 100 $$

Simplifying the fraction:

$$ left( frac{16x}{15} times frac{3}{5x} right) times 100 left( frac{16 times 3}{15 times 5} right) times 100 $$

$$ left( frac{48}{75} right) times 100 64 $$

Thus, the percentage error in the calculation is 64%.

Additional Examples

Another Approach to the Problem

Let the number be x.

From the correct value:

A 2/5 x

From the incorrect value:

A 3/5 y

Absolute error y - x 3x/5 - 2x/5 x/5

Relative error x - y / x x/5 / 2x/5 1/2 0.5 (50%)

Another Example with Different Numbers

Let the number being multiplied be A.

So, A 2/5 X (correct value).

A 3/5 Y (value with error).

Absolute error: Y - X A 3/5 - 2/5 A/5

Relative error: X - Y / X A 1/5 / A2/5 1/2 0.5 (50%)

Percentage error: 50%

Conclusion

Understanding and calculating percentage error is essential for improving the accuracy of mathematical calculations. By following these steps, you can identify and correct potential errors in your calculations and ensure that your results are as accurate as possible. If this information has been helpful to you, please upvote this guide to support our efforts in providing valuable educational content.