Calculating weighted averages is useful in several places – for example, the atomic mass that is reported in the periodic table is the average mass of all isotopes, weighted by their natural abundance.

### Calculating an average atmoic mass using relative abundances

There are two naturally occurring isotopes of bromine: $^{79} Br$ (mass of 78.91338 u) and $^{81}Br$ (mass of 80.916291 u). Bromine-79 accounts for 50.541% of all bromine found naturally, and bromine-81 has a percent isotopic abundance of 49.459%. What is the weighted average atomic mass of bromine?

#### Solution

To calculate a “regular” or *mean* average of *n* values, we would use:
$$\text{mean value} = \frac{\text{value(1) + value(2) + … value(n)}}{n}$$
In this formula, each value “counts” equally in the sum, and we divide by the number of items as our total.

Since we want a **weighted** average in this calculation, we will modify this formula to account for the different weights, and the fact that our “total” = 100% (all the bromine in the world):
$$\text{average mass} = \frac{(\%\, ^{79} Br)(\text{mass}\,^{79} Br)+(\%\, ^{81} Br)(\text{mass}\,^{81} Br)}{100\%} \\
\text{average mass} = \frac{(50.541\%)(78.91338 u)+(49.459\%)(80.916291 u)}{100\%} \\
\text{average mass} = \mathbf{79.904\, \text{u}}$$

This aligns with the mass that is reported on the periodic table (hooray!)

### Check Your Learning

Silicon has three naturally occurring isotopes. Their masses and relative abundances are:

- $^{28}Si$: 27.97693 u; 92.223%
- $^{29}Si$: 28.97649 u; 4.685%
- $^{30}Si$: 29.97337 u; 3.092%

## Answer

28.08 u

Another place where you will see weighted averages is in calculating your course grade! You can use a weighted average calculation to figure out your current grade, or calculate the score you need on an assignment to achieve your target grade.

### Calculating a course grade

In a chemistry course, the grade breakdown (from the course outline) is:

- Labs – 25%
- Tutorials – 25%
- Midterm – 15%
- Final – 35%

If Jessie has earned an average grade of 82.0% in the labs, 94.0% in the tutorials, and 76.0% on the midterm exam, what is Jessie’s grade going into the final exam?

#### Solution

What we need to calculate here is the weighted average of all the grades *except* the final exam (which we don’t have yet). We can use a similar form to the way the bromine mass was calculated above:
$$\text{average grade} = \frac{(\text{% for labs})(\text{grade on labs})+(\text{% for tutorials})(\text{grade on tutorials})+(\text{% for midterm})(\text{grade on midterm})}{{\text{ lab + tutorial + midterm}}} \\
\text{average grade} = \frac{(\text{25%})(\text{82})+(\text{25%})(\text{94})+(\text{15%})(\text{76})}{{\text{ 25 + 25 + 15}}} \\
\text{average grade} = 85.2% $$

### Check Your Learning

Hollis, a student in the same chemistry class, would also like to predict their grade coming into the final. Hollis’ grades so far are: 67.0% in the labs, 81.0% in the tutorials, and 73.0% on the midterm exam.

## Answer

73.8%

### Check Your Learning

If Hollis wants to get a grade of A- on the course (a minimum overall grade of 82.0%), what minimum grade do they need on the final exam?

*Hint: You will need to extend the formula to include the final exam. Try assigning a grade of “x” to the final, then solve for “x”.*

## Answer

97.3%

Use this activity from the King’s Centre for Visualization in Science below to explore how the average masses of isotopes are calculated: