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Question

The density of a substance is measured as $1.234 	imes 10^2$ kg/m$^3$ using instruments with precisions of 2% and 5% for mass and volume respectively.  What is the correct representation of the density including error?

NAGA SAI · Accepted Answer

## CORE INFORMATION
✓ **Answer**: A
## CONCEPT FRAMEWORK
	**Primary Concept**: Error Propagation
	**Related Topics**: Measurement Uncertainty, Significant Figures
	**Prerequisites**: Percentage, Density
## SOLUTION PATHWAY
**Given Information**:
	Density, $ho = 1.234 	imes 10^2$ kg/m³ = 123.4 kg/m³
	Percentage error in mass, $\% \Delta m = 2\%$
	Percentage error in volume, $\% \Delta V = 5\%$
**Method & Steps**:
	1. **Calculate the percentage error in density:**
		Reasoning: Since density $ho = \frac{m}{V}$, the percentage errors add.
		Formula/Rule: $\% \Delta ho = \% \Delta m + \% \Delta V$
	2. **Calculate the absolute error in density:**
		Working: $\% \Delta ho = 2\% + 5\% = 7\%$
		$\Delta ho = \frac{7}{100} 	imes 123.4$ kg/m³ $\approx 8.6$ kg/m³
		Key Point: Round the error to one significant figure, and round the value to the same decimal place as the error.
	3. **Express the density with its error:**
		Result: $ho = (120 \pm 9)$ kg/m³
		Verification: The error is in the tens place, so the density is rounded to the nearest ten.
## OPTION ANALYSIS
[A] ✓: (120 ± 9) kg/m³ - Correct.  The calculated error is approximately 9 kg/m³, and the density is rounded accordingly.
[B] :x: (123.4 ± 8.6) kg/m³ - Incorrect. The error should be rounded to one significant figure, and the density should be rounded to the same decimal place.
[C] :x: (123 ± 7) kg/m³ - Incorrect. Although the error is rounded correctly, the density isn't rounded to the same decimal place as the error.
[D] :x: (1.234 ± 0.086) x 10² kg/m³ - Incorrect. Similar to option B, the error and density are not rounded to the correct significant figures.
## LEARNING AIDS
	**Common Mistakes**:
		1. Incorrect rounding of error and value: Ensure the error is rounded to one significant figure, and the value is rounded to the same decimal place.
		2. Incorrect error propagation formula: Remember to add percentage errors for multiplication and division.
	**Quick Tips**:
		1. When adding percentage errors, consider the largest percentage error as a guide for the number of significant figures.
		2. Always round your final answer to reflect the precision of your measurements.

NAGA SAI · Answer

The correct Option is "(120 ± 9) kg/m³"

NAGA SAI · Answer

(123.4 ± 8.6) kg/m³

NAGA SAI · Answer

(123 ± 7) kg/m³

NAGA SAI · Answer

(1.234 ± 0.086) x 10² kg/m³

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