Problem-Solving & Math Tactics for the MCAT

Use of Relationships and Proportionality

• Relationships typically revealed via explicit formulas, stated proportionality constants, or implied ratios in MCAT passages.
  • Direct relationship: as one variable increases, the other increases proportionally and vice-versa.
  • Expressed mathematically as $A\propto B$ or $\frac{A<em>1}{A</em>2}=\frac{B<em>1}{B</em>2}$.
  • Inverse relationship: an increase in one variable corresponds to a proportional decrease in the other.
  • Expressed as $A\propto \tfrac{1}{B}$ or $A<em>1B</em>1=A<em>2B</em>2$.
• Strategy tips
  • State the relationship verbally first, then translate to math for clarity.
  • Check whether the passage hints at hidden ratios even if no equation is printed.

Conversions

• MCAT increases difficulty by mixing units between stem data and answer choices; correct responses often hinge on correct conversions.
• Two broad scenarios require conversion:
  1. Adjusting prefixes within the same base unit (e.g., g→mg→kg).
  2. Translating between entirely different units or between SI and British systems (e.g., mi→m, Cal→J).
• Steps for reliable conversions
  1. Write every conversion factor as a fraction equal to $1$.
  2. Arrange so unwanted units cancel top & bottom.
  3. Multiply numerators, multiply denominators, then divide.
  4. Save rounding for the final step.

Metric Prefixes (Base-10 Multipliers)

• $10^{12}$ → tera (T)
• $10^{9}$ → giga (G)
• $10^{6}$ → mega (M)
• $10^{3}$ → kilo (k)
• $10^{2}$ → hecto (h)
• $10^{1}$ → deca (da)
• $10^{-1}$ → deci (d)
• $10^{-2}$ → centi (c)
• $10^{-3}$ → milli (m)
• $10^{-6}$ → micro (\mu)
• $10^{-9}$ → nano (n)
• $10^{-12}$ → pico (p)

Frequently Supplied (or Recall-Free) Conversion Factors

• $1\;\text{mi}=5280\;\text{ft}$
• $1\;\text{ft}=12\;\text{in}$
• $1\;\text{in}=2.54\;\text{cm}$
• Energy
  • $1\;\text{cal}=10^3\;\text{cal}=4.184\;\text{J}$ (MCAT sometimes gives both small-c and large-C definitions)
  • $1\;\text{eV}=1.602\times10^{-19}\;\text{J}$
• Mass
  • $1\;\text{amu}=1.661\times10^{-27}\;\text{kg}$
• Force/Weight
  • $1\;\text{lb}=4.45\;\text{N}$
• Volume
  • $1\;\text{L}=33.8\;\text{fl oz}$
• Principle: Only time factors (seconds↔minutes↔hours) must be memorized; all others are supplied on exam if needed.

Worked Example — Converting Car Speed (33 mi h⁻¹ to m s⁻¹)

• Distance chain:
  • $33\;\text{mi}\left(\frac{5280\;\text{ft}}{1\;\text{mi}}\right)\left(\frac{12\;\text{in}}{1\;\text{ft}}\right)\left(\frac{2.54\;\text{cm}}{1\;\text{in}}\right)\left(\frac{1\;\text{m}}{100\;\text{cm}}\right)\approx5.28\times10^{4}\;\text{m h}^{-1}$
• Time conversion:
  • $\frac{5.31\times10^{4}\;\text{m}}{1\;\text{h}}\left(\frac{1\;\text{h}}{3600\;\text{s}}\right)\approx1.48\times10^{1}\;\text{m s}^{-1}$
• Reported ≈$15\;\text{m s}^{-1}$ (actual $14.8\;\text{m s}^{-1}$).

Temperature Conversion Equations

• $F=\tfrac{9}{5}C+32$
• $K=C+273$
• Key distinction: unlike simple unit factors, these involve addition/subtraction, not just multiplication.

Unit (Dimensional) Analysis as a Problem-Solving Tool

• Purpose: infer correct formula or verify final units even when the exact equation is forgotten.
• Example reasoning:
  • Given $E$ in $\text{N C}^{-1}$ and $V$ in $\text{J C}^{-1}=\text{N m C}^{-1}$, to get distance $d$ (m) you must compute $V/E$ ⇒ units cancel to meters.
• Warnings
  • Dimensional analysis narrows choices but is not foolproof; variable relationships (e.g., squared or inverse) may still be missed.

Clinical Example — Ventricular Volume from Ejection Fraction

• Known: ejection fraction $=0.60$, cardiac output $=5\;\text{L min}^{-1}$, heart rate $=80\;\text{beats min}^{-1}$.
• Stroke volume (volume ejected per beat):
  • $\frac{5\;\text{L min}^{-1}}{80\;\text{beats min}^{-1}}=0.0625\;\text{L beat}^{-1}$.
• Pre-contraction ventricular volume $V_{ED}$:
  • $0.0625\;\text{L}=0.60\,V<em>{ED}$ → $V</em>{ED}=\frac{0.0625}{0.60}\;\text{L}\approx0.10\;\text{L}$ (actual $0.104\;\text{L}$).

Algebraic Systems of Equations

• Critical MCAT skill: solving linear systems, normally ≤3 variables.
• Requirement: number of independent equations ≥ number of unknowns.
  • Single equation with one unknown: simple isolation (e.g., $6-x=1 \Rightarrow x=5$).
  • One equation, two unknowns ⇒ indeterminate unless additional relation supplied.
• Three principal solution methods
  1. Substitution
  • Solve one equation for one variable, plug into the other(s).
  • Example: $5x-2y=11\Rightarrow y=\tfrac{5x-11}{2}$, insert into $3x+4y=17$, solve $x=3$ then $y=2$.
  1. Setting equations equal (special substitution)
  • Isolate same variable in each equation, set equal, solve.
  • $y=\tfrac{17-3x}{4}=\tfrac{5x-11}{2}$ ⇒ yields same $x=3$, $y=2$.
  1. Elimination (addition / subtraction method)
  • Multiply/divide to match coefficients, then add/subtract to eliminate.
  • With $3x+4y=17$ and $5x-2y=11$, doubling second gives $10x-4y=22$; adding eliminates $y$: $13x=39$ → $x=3$.
• Result convention: ordered pair $(x,y)=(3,2)$; for three variables, ordered triple $(x,y,z)$.

Key Takeaways

• Master conversions (prefixes + inter-system) to avoid hidden point losses.
• Use proportional reasoning to simplify algebra and eliminate distractors.
• Dimensional analysis is a powerful check but does not replace knowing formulas.
• Practice all three linear-system techniques; any may be fastest depending on coefficients.
• Maintain unit discipline at every algebraic step—MCAT often embeds unit traps.
• These quantitative tools support later chapters on experimental design & data analysis.