A student multiplied a number by 3/5 instead of 5/3. What will be the percentage error in the calculation?

Options

A. 64%
B. 65%
C. 66%
D. 67%

Answer

To solve the problem of finding the percentage error when a student multiplied a number by \( \frac{3}{5} \) instead of \( \frac{5}{3} \), we can follow these steps:

### Step 1: Define the number
Let the number be \( x \). For simplicity, we can assume \( x = 100 \).

### Step 2: Calculate the actual multiplication
The correct multiplication should be:
\[ \text{Actual Value} = x \times \frac{5}{3} = 100 \times \frac{5}{3} = \frac{500}{3} \approx 166.67 \]

### Step 3: Calculate the incorrect multiplication
The student incorrectly multiplied the number by \( \frac{3}{5} \):
\[ \text{Incorrect Value} = x \times \frac{3}{5} = 100 \times \frac{3}{5} = 60 \]

### Step 4: Determine the error
The error in the calculation is the difference between the actual value and the incorrect value:
\[ \text{Error} = \text{Actual Value} - \text{Incorrect Value} = \frac{500}{3} - 60 \]
To perform this subtraction, we need to convert 60 into a fraction:
\[ 60 = \frac{180}{3} \]
Now, we can subtract:
\[ \text{Error} = \frac{500}{3} - \frac{180}{3} = \frac{320}{3} \approx 106.67 \]

### Step 5: Calculate the percentage error
The percentage error is calculated using the formula:
\[ \text{Percentage Error} = \left( \frac{\text{Error}}{\text{Actual Value}} \right) \times 100 \]
Substituting the values we have:
\[ \text{Percentage Error} = \left( \frac{\frac{320}{3}}{\frac{500}{3}} \right) \times 100 = \left( \frac{320}{500} \right) \times 100 = \frac{32000}{500} = 64\% \]

### Final Answer
The percentage error in the calculation is \( 64\% \).

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