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Difficulty: HardFractions, Decimals, Percentages, and Approximations

A student evaluated the expression 0.0256×1.50.0064\frac{0.0256 \times 1.5}{0.0064} by first rounding each number in the expression to 11 significant figure before completing the computation. What is the percentage error in the student's result compared to the exact value?

  1. 66.67%66.67\%Answer
  2. B
    40.00%40.00\%
  3. C
    166.67%166.67\%
  4. D
    33.33%33.33\%

Answer

The percentage error in the student's result is 66.67%66.67\%.
The exact evaluation yields 0.03840.0064=6\frac{0.0384}{0.0064} = 6. Rounding each quantity to 11 significant figure gives 0.030.03, 22, and 0.0060.006, which evaluates to 0.060.006=10\frac{0.06}{0.006} = 10. The absolute error is ∣10−6∣=4|10 - 6| = 4, and dividing by the true value 66 yields 46×100%=66.67%\frac{4}{6} \times 100\% = 66.67\%.

Step-by-Step Solution

1
Calculate the exact value of the expression
Numerator: 0.0256×1.5=0.03840.0256 \times 1.5 = 0.0384. Division: 0.03840.0064=6\frac{0.0384}{0.0064} = 6.
Establishing the true reference value is required to calculate percentage error.
2
Round each number in the expression to 11 significant figure
0.0256→0.030.0256 \rightarrow 0.03, 1.5→21.5 \rightarrow 2, 0.0064→0.0060.0064 \rightarrow 0.006.
Following the approximation directive in the problem statement.
3
Calculate the estimated value using the rounded numbers
Estimated value = 0.03×20.006=0.060.006=10\frac{0.03 \times 2}{0.006} = \frac{0.06}{0.006} = 10.
Obtaining the student's evaluated result.
4
Determine the percentage error
Absolute Error = ∣10−6∣=4|10 - 6| = 4. Percentage Error = 46×100%=66.67%\frac{4}{6} \times 100\% = 66.67\%.
Percentage error is defined as ∣Estimated−True∣True×100%\frac{|\text{Estimated} - \text{True}|}{\text{True}} \times 100\%.

Key Concept

Percentage error calculation with significant figure approximations
Estimated Time:2m 0s
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Topics

Fractions, Decimals, Percentages, and Approximations

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