Math Techniques Every MCAT Test-Taker Should Know

The MCAT tests a wide range of skills, including your understanding of natural sciences and your critical thinking abilities. Math plays a significant role in many questions, especially in chemistry, physics, and biology sections. However, one unique challenge is that calculators are not allowed during the exam. This restriction makes it essential to develop strong mental math skills and learn strategic shortcuts to solve problems efficiently.

Efficiency in math not only saves valuable time but also reduces the cognitive load, allowing you to focus more on applying concepts rather than performing lengthy calculations. By mastering a few simple tricks, you can streamline your problem-solving process, avoid mistakes, and maintain confidence under time pressure.

In this part, we will explore foundational concepts that help build this math efficiency: recognizing important numerical patterns, memorizing key constants, and making quick estimations.

Recognizing Important Multiples and Constants

On the MCAT, certain numbers recur because of their scientific significance. Being able to quickly identify these numbers and understand what they represent can give you a major advantage.

One example is the number 22.4 liters, which represents the molar volume of an ideal gas at standard temperature and pressure (STP) in chemistry. This means one mole of gas occupies 22.4 liters under these conditions. When you encounter numbers like 44.8 or 11.2, these are multiples or fractions of 22.4. Instantly recognizing these values allows you to relate the problem to mole calculations without needing to re-derive the molar volume, speeding up your work.

Similarly, Avogadro’s number, approximately 6.02 × 10^23, is a fundamental constant representing the number of particles in one mole of a substance. Numbers that are multiples or fractions of this often indicate mole or particle count questions. Spotting these numbers quickly directs your thinking and helps you select the correct formulas or strategies.

Developing a habit of scanning numbers for these familiar constants and multiples will help you anticipate question types and answer options more effectively.

Memorizing Perfect Squares and Their Decimals

Perfect squares are the products of an integer multiplied by itself. Memorizing perfect squares is a valuable skill for the MCAT because it allows you to recognize square roots and quadratic relationships quickly.

Many test-takers remember perfect squares from 1^2 through 10^2, which include 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. Extending this knowledge up to 15^2 (225) gives you an even broader base for quick recognition.

On the MCAT, these perfect squares sometimes appear in decimal form. For instance, 1.44 is the decimal equivalent of 1.2 squared (since 1.2 × 1.2 = 1.44), and 1.69 corresponds to 1.3 squared. When you see such decimals in questions or answers, recognizing them as squares of decimals allows you to reverse-engineer the square root mentally instead of calculating it from scratch.

This trick is particularly useful in physics problems involving speed, distance, or energy, where square roots frequently appear in formulas.

Estimating Values of Pi and Square Roots

Another important constant to memorize and understand is pi (π), which equals approximately 3.14. The MCAT often includes geometry problems involving circles, where you must calculate circumferences and areas. Knowing the formulas—circumference = 2πr and area = πr²—and having a rough estimate of π lets you solve these problems quickly.

In addition to pi, estimating square roots is an essential skill. For example, if you need to find the square root of a number that is not a perfect square, you can approximate by identifying the nearest perfect squares. For instance, the square root of 17 lies between the square root of 16 (which is 4) and the square root of 25 (which is 5), but closer to 4. If answer choices include numbers like 4.1, 4.2, and 4.9, you can confidently select the one near 4.1 or 4.2 without detailed calculation.

Being comfortable with these approximations will help you avoid spending excessive time on calculations and allow you to eliminate incorrect answer choices efficiently.

MCAT Math Tricks 

Using Scientific Notation to Simplify Large and Small Numbers

The MCAT frequently involves numbers that are extremely large (like Avogadro’s number) or very small (such as atomic masses or concentrations in molarity). Scientific notation is a powerful tool to simplify these numbers and make calculations more manageable.
Scientific notation expresses numbers as a product of a decimal between 1 and 10 and a power of 10. For example, 0.000045 can be written as 4.5 × 10^-5. On the exam, recognizing when to convert to or from scientific notation helps avoid errors with zeros and decimal places.
When multiplying or dividing numbers in scientific notation, you can operate on the decimal parts and the powers of 10 separately:

• Multiply decimals and add exponents when multiplying:
  (2×103)×(3×104)=(2×3)×103+4=6×107(2 × 10^3) × (3 × 10^4) = (2 × 3) × 10^{3+4} = 6 × 10^7

• Divide decimals and subtract exponents when dividing:
  6×1072×103=62×107−3=3×104\frac{6 × 10^7}{2 × 10^3} = \frac{6}{2} × 10^{7-3} = 3 × 10^4
  Practicing these quick operations improves your speed on questions involving concentration, energy, and molecular quantities.

Mental Tricks for Fractions and Percentages

Many MCAT problems ask you to interpret data in terms of percentages or fractions, especially in biology and psychology sections. Being quick with conversions and estimations here saves time.

• Remember common fraction-to-percentage conversions:
  12=50%\frac{1}{2} = 50\%, 14=25%\frac{1}{4} = 25\%, 34=75%\frac{3}{4} = 75\%, 15=20%\frac{1}{5} = 20\%, and 110=10%\frac{1}{10} = 10\%.

• When estimating percentages, break complex numbers into these fractions for quick calculation. For example, to estimate 18% of 50, think: 10% of 50 is 5, 5% is 2.5, and 3% is about 1.5; adding these gives roughly 9.

• For problems requiring “percent increase” or “percent decrease,” the formula is:
  Percent Change=New Value−Original ValueOriginal Value×100%\text{Percent Change} = \frac{\text{New Value} – \text{Original Value}}{\text{Original Value}} \times 100\%

Memorizing this formula and practicing with round numbers helps you quickly assess growth or decay rates.

Approximating Logarithms on the MCAT

Logarithmic calculations often appear in chemistry and physics, especially in pH calculations and radioactive decay problems. Since exact log calculations are complex without a calculator, learning approximation strategies is helpful.

• The common log (base 10) of numbers near 1 can be approximated by noting that log⁡1=0\log 1 = 0.

• For numbers slightly above 1, use the fact that log⁡(1+x)≈x/ln⁡(10)\log (1 + x) \approx x / \ln(10), where xx is small. For example, log⁡1.2≈0.08\log 1.2 \approx 0.08.

• For pH calculations, remember that pH = −log⁡[H+]-\log[\text{H}^+]. Estimating the order of magnitude of [H+][\text{H}^+] (e.g., 10−310^{-3}) immediately gives you the pH (about 3).
  These rough estimates avoid long calculations and help eliminate improbable answer choices.

Leveraging Dimensional Analysis

Dimensional analysis is a powerful technique to check the validity of your answers and to convert between units on the fly. It involves treating units as algebraic quantities that can be canceled or converted.

• For example, converting speed from meters per second (m/s) to kilometers per hour (km/h):
  1 m/s=1×1 km1000 m×3600 s1 hr=3.6 km/hr1 \, \text{m/s} = 1 \times \frac{1 \, \text{km}}{1000 \, \text{m}} \times \frac{3600 \, \text{s}}{1 \, \text{hr}} = 3.6 \, \text{km/hr}
• Always write out units during calculations to avoid mistakes, especially when dealing with pressure, volume, or concentration units that appear in physics and chemistry.

• Using dimensional analysis as a “sanity check” can help you spot errors early and avoid wasting time on wrong paths.

MCAT Math Tricks 

Mastering Ratios and Proportions

Many MCAT problems require you to analyze or compare ratios, whether in enzyme kinetics, population genetics, or chemical concentrations. Understanding how to manipulate ratios quickly can save time and reduce errors.

• When two quantities are proportional, you can set up cross-multiplication to solve for unknowns:
  ab=cd  ⟹  a×d=b×c\frac{a}{b} = \frac{c}{d} \implies a \times d = b \times c
• For example, if a 5 mL solution contains 2 mg of a drug, how much drug is in 8 mL? Set up:
  2 mg5 mL=x8 mL  ⟹  x=2×85=3.2 mg\frac{2 \, \text{mg}}{5 \, \text{mL}} = \frac{x}{8 \, \text{mL}} \implies x = \frac{2 \times 8}{5} = 3.2 \, \text{mg}
• Recognize direct vs. inverse proportionality to choose the correct setup quickly: if one quantity increases while the other decreases, they are inversely proportional (e.g., pressure and volume in Boyle’s law).

Quick Estimation with Significant Figures

The MCAT requires you to interpret data with appropriate precision. Knowing how to round numbers according to significant figures helps you avoid errors and speed up calculations.

• Basic rules:
  • All non-zero digits are significant.

  • Zeros between non-zero digits are significant.

  • Leading zeros are not significant.

  • Trailing zeros are significant if after a decimal point.

• When multiplying or dividing, your final answer should have as many significant figures as the measurement with the fewest significant figures.

• When adding or subtracting, match the decimal place of the least precise measurement.

• In MCAT calculations, it’s often sufficient to round answers to two or three significant figures unless otherwise specified.

Using the Quadratic Formula Efficiently

Some MCAT questions require solving quadratic equations, but memorizing the quadratic formula and knowing shortcuts can save time.
The quadratic formula is:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}

• Quickly estimate the discriminant (b2−4ac)(b^2 – 4ac) to determine if the roots are real or complex without full calculation.

• For equations where b2b^2 is much larger than 4ac4ac, the roots are real and distinct.

• In special cases where the quadratic factors nicely (e.g., x2−5x+6=0x^2 – 5x + 6 = 0), try factoring first to avoid using the formula.

• For approximate solutions, focus on the dominant terms and use mental math to speed up the process.

Shortcut Tricks for Geometry and Trigonometry

The MCAT often includes questions on circles, triangles, and vectors. Some handy geometry and trig tricks include:

• Remember the Pythagorean triples (3-4-5, 5-12-13, 7-24-25) to quickly identify right triangles without calculation.

• For vectors, break them into components using sine and cosine:
  Componentx=rcos⁡θ,Componenty=rsin⁡θ\text{Component}_x = r \cos \theta, \quad \text{Component}_y = r \sin \theta
• Use unit circle values for common angles:
  sin⁡30∘=0.5,cos⁡30∘=32,sin⁡45∘=cos⁡45∘=22\sin 30^\circ = 0.5, \quad \cos 30^\circ = \frac{\sqrt{3}}{2}, \quad \sin 45^\circ = \cos 45^\circ = \frac{\sqrt{2}}{2}
• Estimate angles and distances by rounding square roots and trig values to decimals (e.g., 3≈1.73\sqrt{3} \approx 1.73, 2≈1.41\sqrt{2} \approx 1.41).

MCAT Math Tricks 

Using Unit Conversions to Your Advantage

Unit conversions are everywhere on the MCAT, from converting energy units (joules to calories) to changing between metric prefixes (milli-, micro-, kilo-). Mastering these conversions helps you avoid costly mistakes.

• Familiarize yourself with common metric prefixes and their powers of ten:
  • kilo- (k) = 10^3

  • centi- (c) = 10^-2

  • milli- (m) = 10^-3

  • micro- (μ) = 10^-6

• Convert between units by multiplying or dividing by the appropriate power of ten. For example, 5 mg = 5 × 10^-3 g.

• Remember common energy conversions:
  • 1 calorie (cal) ≈ 4.184 joules (J)

  • 1 kilocalorie (kcal) = 1000 cal

• Use dimensional analysis as a check: your final units should match the units asked for in the problem.

Simplifying Complex Fractions and Expressions

Complex fractions and algebraic expressions often appear daunting at first glance, especially during timed exams like the MCAT. These problems can become even more stressful when they involve multiple variables, polynomials, or intricate fractions within fractions. However, with a systematic approach, these seemingly complicated problems can be broken down and simplified, saving valuable time and minimizing errors. In this guide, we will explore methods to simplify complex fractions and expressions efficiently, providing a step-by-step breakdown that will make these problems more manageable.

Understanding Complex Fractions

A complex fraction is essentially a fraction where either the numerator, denominator, or both contain one or more fractions. They often appear in the form:

abcd\frac{\frac{a}{b}}{\frac{c}{d}}

Such fractions may look intimidating, but with the right approach, they can be simplified efficiently. The key is to rewrite the complex fraction as a division problem between two simple fractions.

Rewriting Complex Fractions

To simplify a complex fraction, follow these basic steps:

1. Rewrite as Division:
   Start by interpreting the complex fraction as the division of two fractions:
   abcd=ab×dc=adbc\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}
   Here, the division of two fractions is transformed into multiplication by the reciprocal of the denominator fraction.

2. Multiply Across:
   Multiply the numerators together and the denominators together:
   a×db×c=adbc\frac{a \times d}{b \times c} = \frac{ad}{bc}
3. Simplify the Result:
   If possible, reduce the fraction by canceling out any common factors between the numerator and denominator.

Example 1: Basic Complex Fraction

Consider the following problem:

3425\frac{\frac{3}{4}}{\frac{2}{5}}

Step 1: Rewrite as a division problem:

34÷25\frac{3}{4} \div \frac{2}{5}

Step 2: Multiply by the reciprocal of the divisor:

34×52=158\frac{3}{4} \times \frac{5}{2} = \frac{15}{8}

The result is a simplified fraction: 158\frac{15}{8}.

Dealing with Multiple Fractions

Sometimes, complex fractions contain sums or differences both in the numerator and the denominator. In these cases, it’s essential to simplify each part separately before combining the results.

Example 2: Nested Complex Fractions

1x+2y3z−4w\frac{\frac{1}{x} + \frac{2}{y}}{\frac{3}{z} – \frac{4}{w}}

Step 1: Find a common denominator for the numerator and denominator separately:

Numerator:

1x+2y=y+2xxy\frac{1}{x} + \frac{2}{y} = \frac{y + 2x}{xy}

Denominator:

3z−4w=3w−4zzw\frac{3}{z} – \frac{4}{w} = \frac{3w – 4z}{zw}

Step 2: Rewrite the entire expression:

y+2xxy3w−4zzw\frac{\frac{y + 2x}{xy}}{\frac{3w – 4z}{zw}}

Step 3: Multiply by the reciprocal:

y+2xxy×zw3w−4z=(y+2x)×zwxy×(3w−4z)\frac{y + 2x}{xy} \times \frac{zw}{3w – 4z} = \frac{(y + 2x) \times zw}{xy \times (3w – 4z)}

The final simplified form is:

zw(y+2x)xy(3w−4z)\frac{zw(y + 2x)}{xy(3w – 4z)}

Simplifying Complex Expressions

When simplifying algebraic expressions that contain multiple variables, fractions, or polynomials, it is crucial to follow a structured approach. Let’s break it down:

1. Factorize Wherever Possible:
   Always start by factoring common terms in both the numerator and the denominator.

2. Use the Distributive Property:
   Expand products and then group like terms for easier cancellation.

3. Reduce the Expression:
   After factoring, cancel out any common terms between the numerator and denominator.

Example 3: Polynomial Complex Fraction

x2−4×2+2xx−2x+1\frac{\frac{x^2 – 4}{x^2 + 2x}}{\frac{x – 2}{x + 1}}

Step 1: Factorize both the numerator and denominator:

Numerator:

x2−4=(x−2)(x+2)x^2 – 4 = (x – 2)(x + 2) x2+2x=x(x+2)x^2 + 2x = x(x + 2)

Denominator:

x−2x+1\frac{x – 2}{x + 1}

Step 2: Substitute the factored forms:

(x−2)(x+2)x(x+2)x−2x+1\frac{\frac{(x – 2)(x + 2)}{x(x + 2)}}{\frac{x – 2}{x + 1}}

Step 3: Multiply by the reciprocal:

(x−2)(x+2)x(x+2)×x+1x−2\frac{(x – 2)(x + 2)}{x(x + 2)} \times \frac{x + 1}{x – 2}

Step 4: Cancel common terms:

(x+2)(x+1)x\frac{(x + 2)(x + 1)}{x}

Managing Complex Expressions with Multiple Variables

Handling expressions with multiple variables often requires careful organization. Writing down intermediate steps helps keep track of the work and minimizes errors.

Example 4: Multi-Variable Complex Fraction

a2−b2abb−aa2\frac{\frac{a^2 – b^2}{ab}}{\frac{b – a}{a^2}}

Step 1: Factorize the numerator and denominator:

Numerator:

a2−b2=(a−b)(a+b)a^2 – b^2 = (a – b)(a + b) ab=abab = ab

Denominator:

b−a=−(a−b)b – a = -(a – b) a2=a×aa^2 = a \times a

Step 2: Substitute the factored forms:

(a−b)(a+b)ab−(a−b)a2\frac{\frac{(a – b)(a + b)}{ab}}{\frac{-(a – b)}{a^2}}

Step 3: Multiply by the reciprocal:

(a−b)(a+b)ab×a2−(a−b)\frac{(a – b)(a + b)}{ab} \times \frac{a^2}{-(a – b)}

Step 4: Cancel common factors:

(a+b)×a2−ab=a(a+b)−b\frac{(a + b) \times a^2}{-ab} = \frac{a(a + b)}{-b}

Tips for Efficient Problem Solving

• Practice Common Factoring: Quickly recognizing common factors saves time.

• Organize Your Work: Write intermediate steps clearly to avoid confusion.

• Avoid Skipping Steps: Even when confident, skipping steps can lead to mistakes.

• Check for Extraneous Solutions: After simplification, make sure your answer is valid within the context of the problem.

Handling Tricky Expressions

Some problems involve more complicated algebraic manipulations, like dealing with radical expressions or higher-degree polynomials. In these cases:

• Rationalize Denominators: Multiply numerator and denominator by the conjugate if needed.

• Simplify Radicals: Combine like terms and reduce where possible.

• Watch for Domain Restrictions: Identify values that make the denominator zero, as they must be excluded.

Simplifying complex fractions and algebraic expressions is a crucial skill for the MCAT and other standardized tests. The goal is to methodically break down the problem into manageable parts, factorize where possible, and systematically reduce the expression. Practice is essential, as familiarity with common patterns and factoring techniques will increase speed and accuracy. Keeping expressions neat and well-organized during calculations not only reduces errors but also makes it easier to check your work. Developing these strategies will help you confidently tackle even the most complicated algebraic challenges on exam day.

Speedy Calculations for Exponents and Roots

Calculating powers and roots can be time-consuming, but knowing certain patterns helps:

• Square numbers ending in 5 always end with 25. For example, 252=62525^2 = 625, 352=122535^2 = 1225.

• To square a number near 100, use the formula:
  (100+x)2=1002+2×100×x+x2(100 + x)^2 = 100^2 + 2 \times 100 \times x + x^2
  For example, 1022=10000+400+4=10404102^2 = 10000 + 400 + 4 = 10404.

• Cube roots of perfect cubes (like 8, 27, 64) are often tested; memorizing these helps.

• For roots not perfect squares, approximate by finding the closest perfect squares and interpolating.

Handling Probability and Statistics Quickly

Probability questions are common, especially in biology and psychology sections. Here are some tricks:

• For independent events, multiply their probabilities.

• For mutually exclusive events, add their probabilities.

• Use the complement rule to find the probability that an event does NOT occur:
  P(not A)=1−P(A)P(\text{not A}) = 1 – P(A)
• When dealing with combinations or permutations, focus on the logic rather than memorizing formulas, unless you are comfortable with factorials and nCrnCr calculations.

• For normal distribution problems, rough estimates based on the empirical rule (68-95-99.7) can be very helpful:
  • ~68% of data falls within 1 standard deviation of the mean

  • ~95% within 2 standard deviations

  • ~99.7% within 3 standard deviations

MCAT Math Tricks 

Final Thoughts

Math efficiency on the MCAT is a skill developed over time. Regular practice not only helps you memorize key constants and formulas but also builds your confidence to tackle questions quickly and accurately. Use practice passages and timed drills to simulate test conditions and identify which tricks work best for you.

While shortcuts and tricks save time, a strong grasp of the underlying concepts ensures you know when and how to apply them. Don’t just memorize formulas—understand why they work. This mindset will help you adapt tricks to unfamiliar problems you encounter on the exam.

Speed is important, but accuracy is crucial. Rushing through calculations can lead to careless mistakes. Use these math tricks to reduce calculation time, so you have more mental bandwidth to analyze questions carefully and double-check your work when possible.

Estimations help eliminate obviously wrong answer choices and avoid getting bogged down in complex math. However, always assess whether a problem calls for a precise answer or if an approximation is sufficient.

Math anxiety can slow you down. Remind yourself that the MCAT math is designed to be manageable without a calculator. By mastering these tricks, you’ll reduce stress and improve your overall test performance.