Calculating Percentage Change from Math Errors

This problem asks us to find the percentage change in the result when a number is mistakenly divided by 4 instead of being multiplied by 4. Let's break down the calculation step-by-step.

Step 1: Define the Correct and Incorrect Operations

Let the original number be represented by \(x\).

• The correct operation was to multiply the number by 4.
• The incorrect operation was to divide the number by 4.

Step 2: Calculate the Correct and Incorrect Results

• Correct Result: When the number \(x\) is multiplied by 4, the correct result is \(4 \times x = 4x\).
• Incorrect Result: When the number \(x\) is divided by 4, the incorrect result is \(x \div 4 = \frac{x}{4}\).

Step 3: Determine the Change in the Result

The change in the result is the difference between the incorrect result and the correct result. However, when calculating percentage change, we usually look at the magnitude of the difference relative to the original (correct) value. The difference in the result is:

\( \text{Difference} = \text{Correct Result} - \text{Incorrect Result} \)
\( \text{Difference} = 4x - \frac{x}{4} \)

To subtract these terms, we find a common denominator, which is 4:

\( \text{Difference} = \frac{4 \times 4x}{4} - \frac{x}{4} = \frac{16x}{4} - \frac{x}{4} = \frac{16x - x}{4} = \frac{15x}{4} \)

Step 4: Calculate the Percentage Change

The percentage change is calculated using the formula:

\( \text{Percentage Change} = \left( \frac{\text{Change in Result}}{\text{Original (Correct) Result}} \right) \times 100\% \)

In this case, the "Change in Result" is the difference we found in Step 3, and the "Original (Correct) Result" is \(4x\). Substituting the values:

\( \text{Percentage Change} = \left( \frac{\frac{15x}{4}}{4x} \right) \times 100\% \)

To simplify the fraction \( \frac{\frac{15x}{4}}{4x} \), we can write it as \( \frac{15x}{4} \div 4x \):

\( \frac{15x}{4} \div 4x = \frac{15x}{4} \times \frac{1}{4x} \)

We can cancel out \(x\) from the numerator and the denominator (assuming \(x \ne 0\), as the question implies a non-zero number):

\( \frac{15\cancel{x}}{4} \times \frac{1}{4\cancel{x}} = \frac{15}{4 \times 4} = \frac{15}{16} \)

Now, calculate the percentage change:

\( \text{Percentage Change} = \frac{15}{16} \times 100\% \)

\( \frac{15}{16} \times 100 = \frac{1500}{16} \)

Let's simplify the fraction:

• Divide 1500 by 16.
• \( 1500 \div 16 = (1600 - 100) \div 16 = 100 - \frac{100}{16} \)
• \( \frac{100}{16} = \frac{50}{8} = \frac{25}{4} = 6.25 \)
• So, \( 100 - 6.25 = 93.75 \)

Alternatively, direct calculation:

\( \frac{1500}{16} = \frac{750}{8} = \frac{375}{4} = 93.75 \)

The percentage change is 93.75%. Since the incorrect result (\( \frac{x}{4} \)) is much smaller than the correct result (\(4x\)), this represents a large percentage decrease in the result.

Revision Table: Key Concepts

Additional Information: Understanding Percentage Change Errors

Errors in calculations, like using the wrong operation (division instead of multiplication) or the wrong number, can lead to significant percentage changes in the final result. It's important to identify the intended correct value and the obtained incorrect value to accurately calculate the percentage change.

In this specific problem:

• The correct result is \(4x\).
• The incorrect result is \( \frac{x}{4} \).

Notice that the incorrect result is \( \frac{x/4}{4x} = \frac{x}{4} \times \frac{1}{4x} = \frac{1}{16} \) of the correct result. This means the incorrect result is only 6.25% of the correct result (since \( \frac{1}{16} \times 100\% = 6.25\% \)). The difference is the remaining portion: \( 100\% - 6.25\% = 93.75\% \). This confirms our calculation of the percentage change.

The percentage change is calculated relative to the correct value, quantifying how much the error affected the outcome compared to what it should have been.