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Chapter 14: Problem 16

Suppose you use mass to calculate the number of beans. Which of these experimental values has a smaller percent error? (A) A calculated value of 1342 when the actual value is \(1327 .\) (B) A calculated value of 1327 when the actual value is \(1342 .\) (C) They have the same percent error. (D) There is not enough information to answer the question.

Short Answer

Expert verified
Option B has the smaller percent error (1.12%).

Step by step solution

01

Understand Percent Error Calculation

The formula for percent error is \( \text{Percent Error} = \left( \frac{\left| \text{Experimental Value} - \text{Actual Value} \right|}{\text{Actual Value}} \right) \times 100\% \). We will use this formula to calculate and compare the errors in both scenarios provided.
02

Calculate Percent Error for Option A

For Option A, use the formula: \( \text{Percent Error} = \left( \frac{|1342 - 1327|}{1327} \right) \times 100\% \). The error calculation becomes \( \frac{15}{1327} \times 100\% = 1.13\% \).
03

Calculate Percent Error for Option B

For Option B, use the formula: \( \text{Percent Error} = \left( \frac{|1327 - 1342|}{1342} \right) \times 100\% \). The error calculation becomes \( \frac{15}{1342} \times 100\% = 1.12\% \).
04

Compare Percent Errors

Option A has a percent error of 1.13% and Option B has a percent error of 1.12%. Since 1.12% is smaller than 1.13%, Option B has the smaller percent error.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Experimental Value
In science and experiments, the experimental value signifies the measurement derived from conducting tests or trials. Essentially, it's the outcome you observe or measure when performing an experiment.
When you're dealing with calculations or measurements, your experimental value is what you regard as the point of interest. For example, if you count a certain number of beans based on their mass, your calculated count, such as 1342 beans, becomes your experimental value.
The experimental value can sometimes deviate from what's considered true, real, or original, which leads to the need for careful analysis and error comparison.
Actual Value
The actual value refers to the true or known value in measurements. This is often considered the gold standard or benchmark against which the experimental value is compared. For instance, the true number of beans鈥攍et's say 1327鈥攕erves as the actual value.
  • It represents reality or the target value that researchers aim to achieve or measure.
  • The actual value is determined through precise techniques that are carefully controlled and verified.
By knowing the actual value, you can swiftly recognize how close or far your experimental measurements are, guiding you toward more accurate procedures moving forward.
Error Comparison
The comparison of errors is a critical step in analyzing measurement accuracy. It involves assessing the difference between the experimental value and the actual value. The smaller the error, the more accurate your measurements are.
This is where percent error becomes handy. It provides a standardized way to gauge how off your experimental value is relative to the actual value.
  • A smaller percent error indicates a closer match to the actual value, while a larger error suggests a significant deviation.
  • It helps you compare the reliability of various experimental methods or results.
In our exercise, we examined the errors through percent error calculations, helping decide which scenario presented more accurate results.
Mathematical Calculation
Calculating percent error is a straightforward mathematical process but requires precision. The formula used is:
\[ \text{Percent Error} = \left( \frac{|\text{Experimental Value} - \text{Actual Value}|}{\text{Actual Value}} \right) \times 100\% \]
Here, you calculate the absolute difference between the experimental and actual values first. Then, divide this by the actual value and finally multiply by 100 to convert it into a percentage.
  • This conversion to a percentage makes it easier to comprehend and compare the magnitude of errors.
  • Being meticulous with each step ensures accurate error determination.
For example, using our formula, option A showed a percent error of 1.13%, while option B demonstrated a slightly smaller error of 1.12%, revealing option B as more precise.

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