If a student multiplied a number by 7/12 instead of 11/16, then find the % error in the calculation ?

Options

A. `16.66%`
B. `15.15%`
C. `13.13%`
D. `11.11%`

Answer

To find the percentage error when a student multiplies a number by \( \frac{7}{12} \) instead of \( \frac{11}{16} \), we can follow these steps:

### Step 1: Calculate the actual value using the correct multiplier
Let’s assume the number is \( x \). The correct multiplication should be:
\[ \text{Correct Value} = x \times \frac{11}{16} \]

### Step 2: Calculate the erroneous value using the incorrect multiplier
The student multiplied the number by \( \frac{7}{12} \):
\[ \text{Erroneous Value} = x \times \frac{7}{12} \]

### Step 3: Find the difference between the correct value and the erroneous value
To find the error, we need to calculate the difference between the correct value and the erroneous value:
\[ \text{Error} = \text{Correct Value} - \text{Erroneous Value} = \left( x \times \frac{11}{16} \right) - \left( x \times \frac{7}{12} \right) \]

### Step 4: Factor out \( x \) from the error equation
Factoring out \( x \) gives us:
\[ \text{Error} = x \left( \frac{11}{16} - \frac{7}{12} \right) \]

### Step 5: Find a common denominator to simplify the expression
The least common multiple of 16 and 12 is 48. We convert the fractions:
\[ \frac{11}{16} = \frac{11 \times 3}{16 \times 3} = \frac{33}{48} \]
\[ \frac{7}{12} = \frac{7 \times 4}{12 \times 4} = \frac{28}{48} \]
Now substituting back, we have:
\[ \text{Error} = x \left( \frac{33}{48} - \frac{28}{48} \right) = x \left( \frac{5}{48} \right) \]

### Step 6: Calculate the percentage error
The percentage error is given by the formula:
\[ \text{Percentage Error} = \left( \frac{\text{Error}}{\text{Correct Value}} \right) \times 100 \]
Substituting the values we have:
\[ \text{Percentage Error} = \left( \frac{x \left( \frac{5}{48} \right)}{x \left( \frac{11}{16} \right)} \right) \times 100 \]
Canceling \( x \) from the numerator and denominator, we get:
\[ \text{Percentage Error} = \left( \frac{\frac{5}{48}}{\frac{11}{16}} \right) \times 100 \]

### Step 7: Simplify the fraction
To simplify:
\[ \frac{5}{48} \div \frac{11}{16} = \frac{5}{48} \times \frac{16}{11} = \frac{5 \times 16}{48 \times 11} = \frac{80}{528} \]
Now simplifying \( \frac{80}{528} \):
\[ \frac{80 \div 16}{528 \div 16} = \frac{5}{33} \]

### Step 8: Calculate the final percentage error
Now we multiply by 100:
\[ \text{Percentage Error} = \left( \frac{5}{33} \right) \times 100 \approx 15.15\% \]

Thus, the percentage error in the calculation is approximately **15.15%**.

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