QuestionFind the average mass of pennies given that $10.00\%$ weigh $2.15 \mathrm{~g}$ and $90.00\%$ weigh $2.156 \mathrm{~g}$ .

Studdy Solution

STEP 1

Assumptions1.10.00% of the pennies were minted in San Francisco with an average mass of.15 g.90.00% of the pennies were minted in Philly with an average mass of.156 g3. We are asked to find the average mass of all pennies in the sample

STEP 2

To find the average mass of all pennies in the sample, we will calculate the weighted average. The weight of each group is the percentage of pennies from each location.
The formula for the weighted average is $\text{Weighted average} = \sum_{i=1}^{n} w_i \cdot x_i$ where $w_i$ is the weight of the $i$ th group and $x_i$ is the value of the $i$ th group.

STEP 3

We can plug in the given values for the weights and values of each group to calculate the weighted average.
$\text{Weighted average} =10.00\% \cdot2.15\,g +90.00\% \cdot2.156\,g$

STEP 4

Convert the percentages to decimal values.
$10.00\% =0.1$ $90.00\% =0.9$ $\text{Weighted average} =0.1 \cdot2.15\,g +0.9 \cdot2.156\,g$

STEP 5

Calculate the weighted average.
$\text{Weighted average} =0.1 \cdot2.15\,g +0.9 \cdot2.156\,g =2.155\,g$ The average mass of pennies in this sample is2.155 g. However, we need to present our answer with the correct number of significant figures.

STEP 6

The number of significant figures in our answer should be determined by the number with the least number of significant figures used in the calculation. In this case, the number2.15 g has three significant figures, which is the least.
Therefore, our final answer should also have three significant figures.

STEP 7

Round the calculated average to three significant figures.
$\text{Weighted average} =2.1556\,g \approx2.16\,g$ So, the average mass of pennies in this sample is2.16 g.

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