In my previous post we started using weighted averages determine what your grade will be given some expected grade for your next assignment. More specifically, given your current grade, $$g_0$$, the weight of your current grade towards your final grade, $$w_0$$, the expected grade on your next assignment, $$g_1$$, and the weight of that assignment on your final grade, $$w_1$$, you can compute your new grade, $$g_2$$, as follows:

$g_2 = \frac{w_0}{w_0+w_1}g_0 + \frac{w_1}{w_0+w_1}g_1$

In this post we’ll look at using this formula as a means of setting a goal for the grade on your next assignment.

### What’s It Going To Take

Notice that this problem involves information that is accounted for by our grade formula. The catch is that the unknown isn’t $$g_2$$, your new grade, but $$g_1$$, the grade of your next assignment. That’s no problem really, we can simply plug in what we know and solve for the lone remaining variable, $$g_1$$,

$$\begin{array}{rcl} 0.85 &=& \frac{0.25}{0.25+0.15}0.81 + \frac{0.15}{0.25+0.15}g_1 \\ 0.85 &=& 0.625 * 0.81 + 0.375 g_1 \\ 0.85 &=& 0.50625 + 0.375 g_1 \\ 0.85 - 0.50625 &=& (0.50625 + 0.375 g_1) - 0.50625 \\ 0.34375 &=& 0.375 g_1 \\ 0.34375 * \frac{1}{0.375} &=& 0.375 g_1 * \frac{1}{0.375} \\ 0.9167 &\approx& g_1 \end{array}$$.

If you can get around a 92% on that paper, then you’ll have your B. Now that you know what you need to get on that paper, you can now plan your time on that paper accordingly.

All that we’ve done here is turn the question around some. Rather than focus on what your grade will be, we’re focusing on what grade you’d like to have and computing the grade you need to get on the next assignment in order to achieve that goal. This is even more proactive then thinking ahead to where you grade will be as you’re looking more specifically at the grade you need on the work that is right in front of you.

Let’s go ahead and reformulate our equation for this task. We want to isolate $$g_1$$ and then possibly reorganize the other side of the equation for clarity and insight.

$$\begin{array}{rcl} g_2 &=& \frac{w_0}{w_0+w_1}g_0 + \frac{w_1}{w_0+w_1}g_1 \\ g_2 - \frac{w_0}{w_0+w_1}g_0 &=& \frac{w_1}{w_0+w_1}g_1 \\ \frac{w_0+w_1}{w_1} \left( g_2 - \frac{w_0}{w_0+w_1}g_0 \right) &=& g_1 \\ g_1 &=& \frac{w_0+w_1}{w_1}g_2 - \frac{w_0}{w_1}g_0 \end{array}$$.

We now have an equation that let’s you compute the grade you’d need to get on your next assignment, $$g_1$$, in order to reach a target course grade, $$g_2$$, given your current grade, $$g_0$$, and all the associated weights, $$w_0$$ and $$w_1$$.

$g_1 = \frac{w_0+w_1}{w_1}g_2 - \frac{w_0}{w_1}g_0$

#### Examples

Your current grade is a 76% and carries a weight of 45% of your final grade. The next assignment is a big exam worth 15% of your final grade. You’d like to pull your grade up to a B or B-. What it would take for this exam to pull you up from 76% to 80%?

$$\begin{array}{rcl} g_1 &=& \frac{0.45+0.15}{0.15} * 0.8 - \frac{0.45}{0.15} * 0.76 \\ g_1 &=& 4 * 0.8 - 3 * 0.76 \\ g_1 &=& 3.2 - 2.28 \\ g_1 &=& 0.92 \end{array}$$.

If you achieve a 92% on your upcoming exam you’ll bump your grade to the 80% cutoff needed for a B-. How about a nice solid 85%? What would it take to reach that target? We can reuse some of our previous work and just substitute the 85% for the 80%.

$$\begin{array}{rcl} g_1 &=& 4 * 0.85 - 2.28 \\ g_1 &=& 3.2 - 2.28 \\ g_1 &=& 1.12 \end{array}$$.

It’s going to take a 112% on that exam. That’s going to require some extra credit on top of a stellar academic performance. More likely than not, you simply cannot get that grade. Does this mean you cannot get a B in the class? No. It just means you cannot get a B by way of the next exam. In a future post we’ll look at using this same kind of analysis to look at and set goals for your grade on a scale that ranges past one assignment. For now, just remember that we’re looking at what’s right in front of you, the next assignment. In the case of this example, you’ve learned that if you get at least a 92% on the exam, then you’ll be in the B- to B range and out of the C range but that nothing higher than a B is possible at this point.

$$\begin{array}{rcl} g_1 &=& \frac{0.65+0.1}{0.1} * 0.1 - \frac{0.65}{0.1} * 0.6 \\ g_1 &=& 7.5 * 0.1 - 6.5 * 0.6 \\ g_1 &=& 6.825 - 6.24 \\ g_1 &=& 0.585 \end{array}$$.

You need a 58.5% on your paper. You can get a low D- and it will only cause your grade to drop to a 91% for the class. That means any passing grade will keep you in the grade range you want to be in. Breathe a sigh of relief. Don’t let the uncertainty about your grade add to your stress. In this case, the hard work you put in to be at a 96% this far into the course out weighs whatever happens on that paper.