To find the weighted average of the mass of a bean in the given sample data, you need to take into account both the abundance and the mass of each type of bean. Here is the step-by-step solution for calculating the weighted average:
1. Understand the given data:
- Black Eyed Peas:
- Percent Abundance: 82%
- Mass of one bean: 1.32 grams
- Cannellini Beans:
- Percent Abundance: 18%
- Mass of one bean: 2.94 grams
2. Formula for weighted average:
The weighted average [tex]\( \text{W.A.} \)[/tex] is given by:
[tex]\[
\text{W.A.} = \frac{(P_1 \times M_1) + (P_2 \times M_2)}{P_1 + P_2}
\][/tex]
where [tex]\( P_1, P_2 \)[/tex] are the percent abundances and [tex]\( M_1, M_2 \)[/tex] are the masses of each type of bean.
3. Substitute the given values into the formula:
[tex]\[
P_1 = 82, \quad M_1 = 1.32 \, \text{grams}
\][/tex]
[tex]\[
P_2 = 18, \quad M_2 = 2.94 \, \text{grams}
\][/tex]
Plug these values into the formula:
[tex]\[
\text{W.A.} = \frac{(82 \times 1.32) + (18 \times 2.94)}{82 + 18}
\][/tex]
4. Calculate the weighted contributions:
[tex]\[
82 \times 1.32 = 108.24
\][/tex]
[tex]\[
18 \times 2.94 = 52.92
\][/tex]
5. Add the weighted contributions:
[tex]\[
108.24 + 52.92 = 161.16
\][/tex]
6. Sum of the abundances:
[tex]\[
82 + 18 = 100
\][/tex]
7. Divide the weighted contributions by the total abundance:
[tex]\[
\text{W.A.} = \frac{161.16}{100} = 1.6116
\][/tex]
8. Final weighted average:
[tex]\[
\boxed{1.6116 \, \text{grams}}
\][/tex]
So, the weighted average mass of a bean in this sample data is 1.6116 grams.
