### MCAT Math

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The most important rule is speed. The math isn’t hard, you just don’t have time to let it be part of your test schedule. So, being able to power through the math in a quick way is important. This is hard to have faith in if you haven’t been practicing math lately, or rigorously, as most people just spend time reading over review materials to study for the MCAT. The problem is only compounded by the fact that most premeds are biology majors, and have jettisoned all math skills out of their minds once it was no longer a requirement. Devote time to math skills every day while you’re studying for the MCAT, divorce yourself from your cellphone calculator:

**1. Most on the MCAT are simple problems of multiplication and division and nothing more.** *The power of this concept shouldn’t be underestimated, because the rules that work for multiplication and division that you already know work for unit analysis.* (See physics example below, because it’s a long example)

**2. Learn to believe in estimation (science labs, ball park, then go for it, usually the ball park is good enough do to uncertainty).** *All values are calculable, but not all calculations have value.* Keep in mind that each round of estimation will introduction error, just keep track of the direction of the error if you need a more accurate estimation — or you can just introduce less rounds of estimation.

**3. You don’t need to master differential equations (or anything remotely close to that level), but all math concepts help.** The most important thing you will take away from calculus are slopes and areas under the curve. But, you don’t need calculus to ever find a slope or area under the curve on the MCAT, it’s just useful if you’ve been introduced to these ideas already.

**4. Become familiar with the log scale base 10, enough to estimate pH values within error of 1 pH point.** (See Chemistry example below)

**5. Any numbers written down larger than 10 should be written in scientific notation, if the number has pi, leave it till the end.** Usually the problem will rely merely on basic math and scientific notation rules. You should become a scientific notation guru — don’t be over confident, devote practice to this. Getting this down is essential to being able to estimate, or ball park huge/tiny numbers. Do your groceries in scientific notation, just make it part of your life.

**6. See relationships without doing that calculation (see 9).** For example you have to have inverse and direct proportionality down cold, as well as inverse squares.

**7. Natural decay (Aexp-t/T), half lives, be able to estimate them. ***How much there is now is based on how much there just was but a moment ago. *Patterns appear a lot in the universe, pi reflects some type of geometric relationship (or rate), Euler’s number (e) in general demonstrates another rate of growth or decline, but instead of geometries, it shows growth (or decline) compounding on itself.

For example: if an experiment is done, and it shows that the cooling of a pool of blood on the cement from a particular crime scene can be described with regard to time with the equation Aexp-t/T. If it takes the blood 10 minutes to cool to half of it’s original temperature (98F), after 20 more minutes what is the closest approximation to the temperature of the blood if left to cool? (assume the temperature obeys exponential decay)

A. 49F

B. 25F

C. 12F

D. 6F

The trick to this is understanding what a half life means. A half life is simply the time it takes for some value to be half of what it was before. So, if it takes 10 minutes to have one half life, 30 minutes would be 3 half lives. First half life 98F/2 = 49F, then the second half life was 49F/2 or 25F, the third would be 25F/2 or 12F. This idea goes for anything that follows exponential decay (Euler’s number based powers).

8.** Understand how to use (x)cos*theta or (y)sin*theta in a problem.** These show up any time there’s a vector, i.e. a magnitude and an directional. Typically, the MCAT will tell you what the cos or sin of whatever angle you need, except the typical ones you should know, so don’t spend too much time worrying about the math. If you’re not sure if you should use cos or sin for a vector question, just use this rule of thumb:

If the vector is maximum at 0 degrees (to the normal) than it’s ought to be cosine, for example work performed while pushing a block at angle theta would use cos, because the maximum value would manifest from you pushing directly on the block. While the rule doesn’t always work, it works well enough for the stuff on the MCAT. Understanding this principle will take you very far, from the MCAT, to understanding the vestibular and auditory hair system and their relationship to action potentials in medical school.

**Examples and Explanations**

**Example Mock Problem for Physics – Reference Number 1**

*Estimate the amount of time it takes to get to mars (Proving Idea’s 1 & 5)*

It has long been said that in order for humans to survive inevitable extinction we will have to travel like our forefathers. But, instead of transcribing around the Earth, we will need to eventually move the human race to other planets. At the moment, the most promising planet is Mars. The photons of light originating from the sun only take approximately 4 minutes and 40 seconds to reach Mars after passing Earth. Mars, has a much thinner atmosphere than Earth, in fact it’s atmosphere is only 0.6% that of Earth’s. Mars is frigid heavenly body, with the mean temperature hovering around -60 C. Combined with the fact that Mars has an ozone layer that is 300 times thinner than Earth’s, and has no active magnetic field, future explorers who roam the planet would be bombarded by ultra violet rays and other high energy particles we are usually shielded from on Earth.

1. There are plans to launch a probe to to orbit Mars, however the probe will only travel at 1 million times slower than light, assuming the probe has a constant velocity, how long would it take for the probe to reach Mars? (the speed of light 3 x 10^8 m/s)

You can see this question in another of ways: unit analysis, or calculate, or estimate because the answer choices are so far apart. Let’s try the first way, unit analysis:

A. ((300,000,000)(3000))/260

B. ((300,000,000)(260))/300

C. ((300,000,000)(260))(1,000,000)

D. 260/((300,000,000)(300))

A unit analysis problem usually looks like this. It’ll have a bunch of numbers, and you’ll have to figure out what the numbers represent, and that’s usually good enough to answer the problem. As you won’t get a calculator on the real test, it’s a typical question as there’s only so many math questions that are fair game. Let’s brush off our unit analysis:

The rules of unit analysis are the same for mult/div, that’s all you need to know.

First let’s recognize what each number was representing, 300,000,000 was the speed of light, with units of m/s because it’s a velocity. The question stem told us that the speed of light is 1,000,000 times faster than our speed, so that was just scientific notation. That is, since 300,000,000/x = 1,000,000, is that equals (3 x 10^8)/(3 x10^2) = 1 x 10^6, i.e 3 x 10^2 is “x”. Our 300 number is another velocity, so the units are m/s. The last bit is 260, I’ll save that for the last. Let’s see what we have so far:

3,000,000 number will have **m/s**

300 will have m/s

The final answer will be in seconds, and the m will magically disappear so if we take our units (I bolded it so its easier to track):

**(m/s)**/(m/s) = (m/s) x (s/m) = everything cancels

Yay, our units canceled out, but what about that 260 number? Remember the answer was asking about time, so as everything else canceled out we know 260 has to have the units of time, and since the answer is X secs, we know that 260 most be on top:

(**(m/s)**/(m/s))(s) = seconds

In this problem it was enough to know this much because the answer has to be B according to unit analysis. If the answer choices were more similar, then you’ only need to go one step further and remember that **(m/s)**/(m/s) doesn’t equal (m/s)**(m/s)**. In other words don’t let unit analysis make you think that **300,000,000**/300 is equal to 300/**300,000,000**. So, you would simply have to pick which goes on the top or the bottom. Alternatively, you could of sat down and calculated the real number, but that would of taken longer, and wouldn’t be a choice selection anyways. Physics and chemistry problems are generally solvable with unit analysis and basic math. On the actual MCAT it’s more likely that they’ll ask you to convert the seconds into days, just to torture you, but the work is the same.

**Example of Chemistry Problem – Reference Number 4**

In human blood the pH is buffered by the interaction of carbonic acid with carbon dioxide. The hydration of CO2 with H2O is the rate limiting step, and the reaction can be described as:

[CO2] + [H20 ] <-> [H2CO3] <-> [H+] + [HCO3-]

[CO2] + [H2O] <-> [H+] +[HCO3-] (mediated by the enzyme carbonic anhydrase)

The buffer system will try to maintain an equilibrium described by:

K = [H+][HCO3-]/[CO2]

In logarithmic form it becomes:

pH = pKa + log [HCO3-]/[CO2]

Question 1.

Assuming if the pKA of HCO3- is shifted to 6.1 because of enzyme activity and if the concentration of HCO3- is 24 mM and the soluble amount of CO2 in the blood is 1 mM, what is the blood pH?

A. 6.97

B. 7.10

C. 7.48

D. 8.10

Now, the proper way to get to the answer would be to pick up a calculator. Type in…6.1 + log(24) = something. Of course you don’t get a calculator. Is this a memory question, should you remember the exact pH of blood (probably not, but knowing the range would be helpful, but let’s ignore that fact)? The question is actually a lot easier than it looks, you have to first understand logs, and be okay with estimating. First, we are dealing with log base 10, so every factor of 10 represents a change in one pH value:

log (.01) = -2

log (.1) = -1

log(0) = meaningless

log (10) = 1

log (100) = 2

log (1000) = 3

Looking back at the original problem, 6.1 + log (24), well we have a problem, because you an I both don’t know what the log of 24 is according to log values above. But, let’s just estimate for now, we’ll prove way this was okay later. So, let’s use a number we know from our values I listed, 10. So, if we put in 6.1 + log (10) we get 7.1, this is much easier than figuring out what the log of 24 was. Now, we just have to think logically, 7.1 can’t be the answer because the concentration was actually 24 mM and not 10 mM. So, that eliminates B, an the answer can’t be A, because that answer is even lower than B. The answer can’t be D either, because 6.1 and 8.1 would mean that there is a 100 times more carbonic acid conguate [HCO3-] than acid [CO2]. In other words, 6.1 + log (100) = 8.1, so this is a gross overshoot. **The answer can only be C.**

We can use calculus to prove that the pH will change in minute amounts with the equation:

dpH = 0.4343/[initial acid concentration] x d[H+]

What this means is that small fluctuations in the concentration of the protons added to solution will change the pH at a rate of (0.4343/initial concentration). So, let’s say we had an initial concentration of 1M, an you added .01M more of [H+ ]you expect the pH to chance to .004343. So, don’t expect the pH change from point to point to be intuitive, instead know how to use a less refined estimation for the MCAT.

Here’s the math behind that, which you’ ll never need to know. Just take it as a lesson that you should remember the log rule base 10, as opposed to expect to calculate it exactly on the MCAT:

[*correction 2/24: the last value in the pic above should be dy = dpH, I’ll fix it later * fixed 2/26]

Well, that’s all I have for now. I made up the questions, so if you see any mistakes let me know! Anyways, good luck and study hard, the math skills you pick up now will stick with you!

March 8, 2014 at 11:08 pm

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March 13, 2014 at 5:35 pm

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October 13, 2014 at 9:48 pm

i don’t know where you are getting the 6.1 PKa in the pH problem?

October 13, 2014 at 10:11 pm

good question, the pKA of carboxylic acid is just a good number to have in your memory. Officially it’s taught as 6.1- 6.3, I just chose the lower of the two arbitrarily. But, seeing that it’s vague, I added a reason within the problem.