Calculation Questions

Calculation questions are fairly ubiquitous on the MCAT and while you can solve them with a typical equation-heavy approach I find that there are much easier ways to approach them.

Back to Basics

Our calculation question approach starts with our basic question approach with some small tweaks.

As a quick recap our basic question approach consists of the following four steps:

1. Define the content area
2. Determine keywords from the question and answers
3. Triangulate passage information
4. Solve and answer the question

For calculation questions we are going to go through 4 similar steps as follows:

1. Define the variables and unit in the question stem
2. Determine the keywords from the question and answers
3. Triangulate passage variables and units
4. Define equation or set up dimensional analysis and solve

While it might seem backwards to define a bunch of units and variables then determine what equation to use it is super helpful because it can be tricky to determine which equation to use. Let’s look at an example to see what I mean.

Worked Example: Sprinkles the Mouse

How much energy did Sprinkles the mouse use when they ran on their exercise wheel for 2 minutes?

Let’s start by listing all of the energy equations we can think of:

[latexpage]
\[ KE=\frac{1}{2}mv^2\]
\[PE=mgh\]
\[Work=Fd\]
\[E=hf\]

I could keep going on if I wanted, but we don’t need an energy equation to solve this problem. Instead will need the power equation:
\[P=\frac{work}{t}\]

To see why this is the case let’s go through the four steps one by one.

Step 1: Question Units and Variables

How much energy did Sprinkles the mouse use when they ran on their exercise wheel for 2 minutes?

Here we have been given two dimensions:

Energy which has a multitude of variables including KE, PE, Work, and E with the units of joules (j).

Time which uses the variable t and SI units of seconds (s).

Now that we have defined each of the dimensions we will move on to the next step.

Note: Always make sure to check the answer choices too they usually have units and might reveal a variable you hadn’t considered.

Step 2: Identify Passage Keywords

How much energy did Sprinkles the mouse use when they ran on their exercise wheel for 2 minutes?

I know absolutely nothing about Sprinkles the mouse or mouse exercise wheels outside of what the passage tells me. This makes them great passage keywords because no matter how much I dig into my memory I am never going to bring out the right piece of information regarding Sprinkles. On the MCAT these typically take the form of acronyms, specific names, etc.

Step 3: Passage Triangulation

Now that we know what keywords we would look for any information about Sprinkles and exercise wheels that mentions variables or units. In this case, we find the following table.

Mouse Name	Running Power (W)
Sprinkles	5
Cupcake	3
Sundae	7

Again we will collect all of the dimensions present and define their variables and units. In this case, we have power which has a variable of P and units of Watts (W).

Step 4: Equation Finally!!!

Here is the information we have gathered so far organized more cleanly:

Dimension	Variable(s)	SI Unit
Energy	KE, PE, W, or E	Joules (J)
Time	t	Seconds (S)
Power	P	Watts (W)

Now we can think of an equation that incorporates a set of what we have that would allow us to solve for energy in some way.

[latexpage]
The only equation that satisfies this is the power equation
\[P = \frac{W}{t}\]
We need to rearrange the equation to solve for our energy dimension.
\[W=P\times t\]

Alternatively, we could have solved this problem using dimensional analysis or canceling out units. In my opinion, this skill is far more important than memorizing a giant list of equations, which I still think you should do.

Cancel Your Units

In order to truly master dimensional analysis, you will need to know 1) how to break units apart into their components and 2) always start with the unit or unit components you want then cancel or build from there.

[latexpage]
In this case, we need to know what a watt is equivalent to. Since power…
\[P=\frac{W}{t}\]
… we know that a watt could also be equivalent to J/s since the units of work are joules (J) and the unit of time is seconds (s).

My answers will all be in joules since it is a measure of energy therefore I will want to start with power and arrange my units to make sure the joules are in the numerator so I don’t have to invert it at a later time.

Now I can get to canceling units out. Here I need to get rid of the seconds. I know that I have a time, but it is in minutes so I will need to convert minutes to seconds then crunch the numbers.

So in this case Sprinkles produced 600 J of energy while running on their Turbo 2000 exercise wheel. You might not realize it, but we ended up using the power equation here.

Build ‘Em Up

In this situation we ended up break the joule out of power, however, sometimes we want to go the other way and build up a unit. Let’s look at another example with Sprinkles to see the difference between the two.

This time Sprinkles is out in their mouse ball zipping around your apartment at a velocity of 5 m/s. If they generate a force of 5N in order to overcome friction how much power do they produce?

First we will look at our 4-step approach here quickly

1. Given: velocity v m/s, force f N, and power P W
2. Keywords: Standalone no passage
3. Passage info: Standalone no passage
4. Time to solve!

In this case, none of our units already contains a watt (W) this means that we will need to build this unit up. Again we need to understand the units that are nested within other units. We have already seen how a watt is equal to a J/s, but the joule (J) breaks down even further into a Nm. This means that a watt is also equal to Nm/s.

At this point all of our units match up all that’s left is to arrange out units and calculate an answer.

Again we have unknowingly used a different version of the power equation (P=Fv) here, which also encompasses all of the variables we defined in the first place.

Caveats

While units are a wonderful and powerful way to solve calculation questions there are a couple of caveats to be aware of.

First, units won’t work when equation involves addition or subtraction or a non-unit component.

[latexpage]
\[
\frac{1}{f}=\frac{1}{i}+\frac{1}{o}
\]
Since all of the units have to be the same for addition or subtraction using units doesn’t help. Take the thin’s lens formula above. If you are asked to solve for f which has units of m then knowing that i and o are also measured in m doesn’t help one bit.

Another situation where using units are problematic is with equations that have non-unit components in them. For example the kinetic energy equation.

\[
KE=\frac{1}{2}mv^2
\]

Since 1/2 doesn’t have a unit we could follow through with units only to find that our answer is two times larger than the correct one.

In order for our units and variable based strategies to work we have to memorize the units and variables for each dimension that is on the MCAT. Thankfully there are a lot fewer units than variables so it is much easier to memorize them. Nonetheless, we should memorize both but if you are short on time then make sure to start with units.

Homework

• Unsuspend the unit cards in Anki