Question

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 \(3.09 \mathrm{~g}\). Then you weigh the jar with all the beans in it, and obtain a mass of \(2082 \mathrm{~g}\). The empty jar has a mass of \(653 \mathrm{~g}\). Based on these data estimate the number of beans in the jar. Justify the number of significant figures you use in your estimate. [Section 1.5]

Short answer

The estimated number of jelly beans in the jar is approximately 464, using an average mass of 3.08 g per jelly bean and a total mass of 1429 g for the jelly beans in the jar. The estimate has two significant figures, which is justified as the given data has three significant figures, and we acknowledge our answer as an approximation, not an exact value.

Step-by-step solution

Calculate the average mass of a single jelly bean

First, we need to find the average mass of a jelly bean. To do that, we will use the given masses of six jelly beans: \(3.15, 3.12, 2.98, 3.14, 3.02,\) and \(3.09 \, \mathrm{g}\). To find the average mass, we need to add up the masses and divide by the total number of beans (which is 6 in this case). Average mass = \(\frac{3.15 + 3.12 + 2.98 + 3.14 + 3.02 + 3.09}{6}\) Now, calculate the average mass: Average mass ≈ 3.08 g (rounded to two decimal places)

Estimate the number of beans in the jar

We are given that the total mass of the jar with jelly beans is 2082 g, and the empty jar's mass is 653 g. To find the mass of the jelly beans inside the jar, we will subtract the empty jar's mass from the total mass. Mass of jelly beans = Total mass - Mass of the empty jar Mass of jelly beans = 2082 g - 653 g Mass of jelly beans ≈ 1429 g (rounded to the nearest gram) Now, we can estimate the number of beans in the jar by dividing the total mass of the beans by the average mass of a single bean: Number of beans ≈ \(\frac{Mass \, of \, jelly \, beans}{Average \, mass \, of \, a \, single \, bean}\) Number of beans ≈ \(\frac{1429 \, g}{3.08 \, g}\) Number of beans ≈ 464 (rounded to the nearest whole number)

Justify the number of significant figures

In our final estimate of the total number of beans (which is 464), we used two significant figures. This is justified since the given data, like the masses of jelly beans and the mass of the jar with jelly beans, is provided with three significant figures. Since we are dealing with an estimation, it is reasonable to round our answer to a lower number of significant figures (two in our case) to acknowledge that our answer is not an exact value but rather an approximation. Therefore, based on the given data, there are approximately 464 jelly beans in the jar.

Key concepts

Mass Calculation

When estimating the number of jelly beans in a jar, one key step is calculating the mass of the jelly beans themselves. To start, we weigh the jar with the jelly beans and the empty jar separately. This helps us find the mass of just the beans by subtracting the mass of the empty jar from the mass of the jar with beans.

In this problem, the jar with jelly beans had a mass of 2082 grams and the empty jar had a mass of 653 grams. Using subtraction, we find the mass of the jelly beans:

• Mass of jelly beans = 2082 g - 653 g
• Mass of jelly beans = 1429 g

This calculation shows how we work with direct measurements to isolate what we are interested in—here, the mass of the jelly beans without the container.

Average Mass

When several measurements of an item's mass are taken, calculating the average mass is a helpful step. This average provides a basis for further calculations, such as estimating quantities.

For the jelly beans, we used the masses of six individual beans to find this average:

• Given masses: 3.15, 3.12, 2.98, 3.14, 3.02, 3.09 grams

To find the average mass, add all the values together and divide by the number of beans:

• Average mass = \(\frac{3.15 + 3.12 + 2.98 + 3.14 + 3.02 + 3.09}{6}\)
• Average mass \(\approx 3.08\) grams

This step is crucial for estimation, as it simplifies complex data by providing a single, representative value.

Estimation

Estimation is a useful technique when you need a quick, approximate number. It utilizes average values and calculations to deliver a pragmatic answer rather than an exact figure.

After calculating the total mass of the jelly beans and determining the average weight of one bean, you can use estimation to find out how many beans are present.

• Total mass of beans = 1429 grams
• Average mass of a single bean = 3.08 grams

To estimate the number of beans:

• Estimated number of beans = \(\frac{1429}{3.08}\)
• Estimated number of beans \(\approx 464\)

While not exact, this process gives us a reasonable figure fitting the context of the data we have.

Significant Figures in Measurement

Significant figures are important in measurement as they indicate the precision of the data. They tell us about the reliability of the measurement and calculations involving that data.

In this problem, all mass measurements are given to three significant figures. However, when doing estimations and rounding results, it's reasonable to use fewer significant figures to reflect the approximate nature of the calculations.

For the jelly beans, the calculated estimate of 464 uses two significant figures. This choice suitably expresses the estimate as a rounded, practical answer rather than an exact count, signifying the inherent variability in the measurement.

Understanding significant figures helps maintain consistency and clarity about the precision level in scientific measurements and calculations.