Consider the jar of jelly beans in the photo. To get an estimate of the number of beans in the jar you weigh six beans and obtain masses of 3.15, 3.12, 2.98, 3.14, 3.02, and 3.09 g. Then you weigh the jar with all the beans in it, and obtain a mass of 2082 g. The empty jar has a mass of 653 g. Based on these data, estimate the number of beans in the jar. Justify the number of significant figures you use in your estimate.

Verified step by step guidance

1. Calculate the average mass of a single jelly bean by adding the masses of the six beans and dividing by six: \( \text{Average mass} = \frac{3.15 + 3.12 + 2.98 + 3.14 + 3.02 + 3.09}{6} \).
2. Determine the total mass of the jelly beans by subtracting the mass of the empty jar from the mass of the jar with beans: \( \text{Total mass of beans} = 2082 \text{ g} - 653 \text{ g} \).
3. Estimate the number of beans in the jar by dividing the total mass of the beans by the average mass of a single bean: \( \text{Number of beans} = \frac{\text{Total mass of beans}}{\text{Average mass of a bean}} \).
4. Consider the significant figures: The average mass of a bean should be calculated to the same number of significant figures as the least precise measurement, which is two decimal places in this case.
5. The final estimate of the number of beans should be rounded to the appropriate number of significant figures based on the precision of the measurements used in the calculation.

Key Concepts

Mass and Average Calculation

To estimate the number of jelly beans, first calculate the average mass of the six beans weighed. This is done by summing the individual masses and dividing by the number of beans. The average mass provides a representative value that can be used to estimate the total number of beans in the jar based on the total mass of the beans.

Total Mass Calculation

The total mass of the jelly beans can be determined by subtracting the mass of the empty jar from the mass of the jar filled with beans. This calculation gives the total mass of the jelly beans alone, which is essential for estimating the number of beans using the average mass calculated earlier.

Significant Figures

Significant figures are important in scientific measurements as they convey the precision of the data. When performing calculations, the result should reflect the least number of significant figures from the measurements used. In this case, the average mass of the beans and the total mass of the jelly beans must be reported with the appropriate number of significant figures to ensure accuracy in the final estimate of the number of beans.