Understanding and Calculating Percentage Error in Mathematical Operations

In this article, we will delve into a specific scenario involving a common mathematical mistake made by students. We will calculate the percentage error when a number was multiplied by the wrong fraction. By going through the detailed steps involved, we aim to provide a clear understanding of how to approach similar problems.

Scenario: Multiplying by the Wrong Fraction

Sometimes, a student might accidentally perform a multiplication with the wrong fraction instead of the intended one. This can lead to inaccuracies in calculations, affecting not only the numeric value but also the overall understanding of the problem. Let us explore the scenario where a student multiplies a number by 2/3 instead of 3/2.

Calculations and Analysis

Let the original number be x.

Intended Result

The student should have multiplied x by 3/2 to get the intended result:

IntendedResult x * frac{3}{2} frac{3x}{2}

Actual Result

However, the student mistakenly multiplied x by 2/3 to get the actual result:

ActualResult x * frac{2}{3} frac{2x}{3}

Calculating the Error

The error in the calculation is the difference between the intended and actual results:

Error IntendedResult - ActualResult frac{3x}{2} - frac{2x}{3}

To simplify this, we find a common denominator (which is 6) and rewrite the fractions:

(frac{3x}{2} frac{9x}{6}) and (frac{2x}{3} frac{4x}{6})

Thus,

Error frac{9x}{6} - frac{4x}{6} frac{5x}{6}

Calculating the Percentage Error

The percentage error is given by the ratio of the error to the intended result, multiplied by 100:

PercentageError left(frac{Error}{IntendedResult}right) times 100 left(frac{frac{5x}{6}}{frac{3x}{2}}right) times 100)

Simplifying this expression:

PercentageError left(frac{5x}{6} times frac{2}{3x}right) times 100 left(frac{10}{18}right) times 100 frac{5}{9} times 100 approx 55.56%)

Therefore, the percentage error in the calculation is approximately 55.56%.

Practical Examples and Applications

Let's further explore the concept with some practical examples. We will first consider a specific case where the number is 100.

Example 1: Multiplying by 2/5 and 5/2

If the number to be multiplied by 5/2 is 100:

(2/5 times 100 40)

(5/2 times 100 250)

Error: The difference between the correct and incorrect results is (250 - 40 210)

Percentage Error: ( frac{210}{40} times 100 525% )

Example 2: Multiplying by 53 and 35

Let the number to be multiplied by 53 (nonzero) be x.

Correct value: (53x)

Value with error: (35x)

Error: (53x - 35x 18x)

Percentage Error: (frac{18x}{53x} times 100 frac{1800}{53} approx 33.96%)

Example 3: A General Case

Assume the number you take is 100 (for ease of further calculations):

(frac{1}{5} times 100 20)

(frac{7}{9} times 100 approx 77.77)

Error: The difference between the correct and incorrect results is (77.77 - 20 57.77)

Percentage Error: (57.77%)

In summary, understanding and calculating the percentage error is crucial for ensuring accurate mathematical computations. It helps identify and correct mistakes, improving the reliability and precision of results.