MCAT Physics and Math Review
Chapter 10: Mathematics
10.4 Problem-Solving
Now that we’ve examined some individual mathematical skills, let’s explore some common problem-solving strategies to attack MCAT questions. The use of relationships and proportionality is especially important in passage-based questions, while unit analysis can help determine which formulas are appropriate for a given question. The use of conversion factors is ubiquitous on the MCAT, as answer choices are often given in different units than the information presented. Algebraic systems are less often required, but may underlie passage interpretation and the approach to some questions.
USE OF RELATIONSHIPS
Relationships are generally indicated in MCAT passages by formulas or the use of proportionality constants. In other cases they may be implied and require a bit more work on our part to calculate ratios. Calculations of this type are based on multiplication and division, but explaining the relationship in words—rather than math—may make it challenging to decode the connections between the variables. In direct relationships, increasing one variable proportionately increases the other; as one decreases, the other decreases by the same proportion. In inverse relationships, an increase in one variable is associated with a proportional decrease in the other.
BRIDGE
According to Boyle’s law, pressure and volume have an inverse relationship: as one is doubled, the other is cut in half (keeping all else constant). On the other hand, according to Gay-Lussac’s law, pressure and temperature have a direct relationship: as one is doubled, so is the other (keeping all else constant).
CONVERSIONS
The MCAT frequently increases the difficulty of a question by requiring the use of conversion factors. Equations may require that variables be in certain formats, or answer choices may differ in units from those given in the question stem. In both cases, it is necessary to convert units. The simplest conversions to perform maintain the same base unit. For example, conversion between grams, kilograms, and milligrams only requires multiplication by an appropriate power of ten. Metric prefixes and their associated powers of ten are found in Table 10.4.
Factor |
Prefix |
Prefix Abbreviation |
10^{12} |
tera– |
T |
10^{9} |
giga– |
G |
10^{6} |
mega– |
M |
10^{3} |
kilo– |
k |
10^{2} |
hecto– |
h |
10^{1} |
deka– |
da |
10^{−1} |
deci– |
d |
10^{−2} |
centi– |
c |
10^{−3} |
milli– |
m |
10^{−6} |
micro– |
μ |
10^{−9} |
nano– |
n |
10^{−12} |
pico– |
p |
Table 10.4. Metric Prefixes |
In addition to the conversions that are necessary for changes in prefixes, we must often convert between units, particularly between the British system and SI units. Table 10.5 shows several important conversion factors to recognize on Test Day. Conversion factors (except those for time) should not be memorized; the MCAT will provide them as necessary.
Base Unit |
Equivalent Units |
1 mile |
5280 feet (ft) |
1 ft |
12 inches (in) |
1 inch (in) |
2.54 cm |
1 Calorie (Cal) |
1000 cal |
1 calorie (cal) |
4.184 J |
1 electron–volt (eV) |
1.602 × 10^{−19} J |
1 L |
33.8 ounces (oz) |
1 pound (lb) |
4.45 N |
1 atomic mass unit (amu) |
1.661 × 10^{−27} kg |
Table 10.5. Common Conversion Factors on the MCAT |
Example:
A car’s speedometer registers a speed of 35 miles per hour. What was its speed in meters per second?
Solution:
First convert distance measurements, being careful to cancel them out by arranging numerators and denominators.
Then, repeat the procedure with the time measurements.
One special case of conversions occurs with temperature. Rather than simply multiply by a conversion factor, there is also a component of addition or subtraction. The following formulas relate the Fahrenheit, Celsius, and Kelvin systems:
Equation 10.20
where F, C, and K are the temperatures in degrees Fahrenheit, degrees Celsius, and kelvins, respectively.
REAL WORLD
It is important to be able to convert between Fahrenheit and Celsius scales in medicine as different hospitals (and different medical records) may use different units. Body temperature is 98.6°F or 37°C. A fever is usually defined as a temperature above 100.4°F or 38°C. Hypothermia is usually defined as a temperature below 95.0°F or 35°C.
UNIT ANALYSIS
Unit analysis, also called dimensional analysis, may help determine the correct answer even if you forget a relevant formula on Test Day. It can also serve as a double check on one’s calculations because the units of the calculated answer must match the units of the answer choices. For example, consider a question in which we are given two quantities: one in and the other in volts. The answer choices for the question are all in meters. Even without remembering the equation V = Ed, we can infer that we must divide the voltage by the electric field to get a distance in meters:
Dimensional analysis is not a foolproof strategy; it is always better to know the true relationships between variables than to infer them based on units. Still, this strategy can be effective for narrowing down (or even choosing) answer choices on Test Day.
Example:
The ejection fraction is the proportion of the left ventricular volume expelled with each contraction of the heart. A patient is known to have an ejection fraction of 0.7, a cardiac output of and a heart rate of What is the volume of the left ventricle in this person?
Solution:
A formula was not provided in this question, but we can recognize that the desired answer is a volume. We can start with the cardiac output and heart rate terms to determine the volume ejected per beat.
The question also explains that only 70 percent of the volume of the left ventricle is expelled per heartbeat. From this, we can determine the volume of the ventricle.
ALGEBRAIC SYSTEMS
The last key mathematical skill for Test Day is the ability to solve systems of linear equations. In order to solve a system of equations, there must be at least as many equations as there are variables. Where there is only one variable (which does not truly constitute a system), only one equation is necessary; for example, 6 − x = 1 reduces to x = 5. In contrast, with an equation like 3x + 4y = 17, there is insufficient data to solve for either variable with only the one equation. If a second equation is introduced, such as 5x − 2y = 11, then we can solve for both variables using one of three methods: substituting one variable in terms of the other, setting equations equal to each other, or manipulating the equations to eliminate one of the variables.
Substitution
In substitution, we solve for one variable in one of the equations, and then insert this term into the other equation. The steps of this method are listed below.
· Solve for one of the variables in one of the equations:
· Insert the expression into the other equation:
· Isolate the variable and solve the resulting equation:
· Solve for the other variable using this value:
Setting Equations Equal
Setting equations equal to one another is a specialized case of substitution. In this method, we solve for the same variable in both equations and then set the two equations equal to each other. The steps of this method are listed below.
· Solve for the same variable in both equations:
· Set the equations equal to each other, isolate the variable, and solve for the variable:
· Solve for the other variable using this value:
Elimination
In elimination, multiply or divide one (or both) of the equations to get the same coefficient in front of one of the variables in both equations. Then, add or subtract the equations as necessary to eliminate one of the variables. The steps of this method are listed below.
· Multiply or divide one (or both) of the equations by a constant so that the coefficient in front of one of the variables in both equations is the same:
· If the sign of both coefficients is the same, subtract one equation from the other. If the sign is opposite, add the two equations together:
· Solve for the other variable using this value:
Note that each method results in the same answer despite slight differences in the steps taken. As a matter of convention, the answers for systems of equations with the variables x and y are reported as coordinates on the Cartesian plane (x,y); thus, our answer for this system would be (3,2). Systems of equations can have many variables, but it is unlikely that you will encounter a system with more than three variables (x,y,z) on the MCAT.
MCAT Concept Check 10.4:
Before you move on, assess your understanding of the material with these questions.
1. How are conversions between metric prefixes accomplished?
2. What does it mean for two variables to have a direct relationship? An inverse relationship?
· Direct:
· Inverse:
3. Each of the three methods for solving systems of equations discussed in this chapter solve for one variable, and then use this value to solve for the other. How does each method solve for the first variable?
· Substitution:
· Setting equations equal:
· Elimination: