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Mathematics

Chapter 10: Mathematics

3 min read Updated Mar 26, 2026
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🎯 Diagnostic: Test Your Starting Level 24 questions (2 per section). No prior reading required, see what you already know.

Aim to answer every question before checking. Missed questions point you to the sections you need most.

1. (10.1) A reasonable estimate of 19 × 41 is:
B. Exact is 779; 800 is close enough for MCAT-style quick checks.
2. (10.1) When asked to estimate:
C. MCAT answer choices usually differ by more than rounding error.
3. (10.2) To solve ax + b = c for x:
D. Isolate x by undoing operations in reverse order.
4. (10.2) The quadratic formula for ax² + bx + c = 0 is:
A. The discriminant (b² - 4ac) tells how many real solutions: positive (two), zero (one), negative (none).
5. (10.3) log₁₀(1000) =
B. log₁₀(10ⁿ) = n. Useful in pH, decibels, and Richter scales.
6. (10.3) If log₁₀(x) = 2, then:
C. Exponentiate both sides: x = 10².
7. (10.4) sin(30°) equals:
D. Memorize sin(0°, 30°, 45°, 60°, 90°) = 0, ½, 22\frac{\sqrt{2}}{2}, 32\frac{\sqrt{3}}{2}, 1; cos follows the reverse order.
8. (10.4) cos(60°) equals:
A. cos(60°) = sin(30°) = 0.5 (complementary angles).
9. (10.5) A vector can be represented as:
B. Component form makes vector arithmetic easy.
10. (10.5) The dot product A · B equals:
C. Dot product of perpendicular vectors is zero. Used in W = F·d.
11. (10.6) To convert 36 km/h to m/s:
D. Multiply by conversion factors so the undesired units cancel.
12. (10.6) To convert 500 kg to grams:
A. 1 kg = 1000 g.
13. (10.7) Scientific notation is of the form:
B. The leading coefficient must be between 1 and 10 in standard scientific notation.
14. (10.7) 0.00045 in scientific notation is:
C. Move the decimal 4 places right; exponent is -4.
15. (10.8) If y is proportional to x², then doubling x:
D. Useful for KE, centripetal force, and many other physical relations.
16. (10.8) In an inverse-square relation F ∝ 1/r², doubling r gives:
A. Gravity and Coulomb's law both follow inverse-square.
17. (10.9) The slope of a position-time graph represents:
B. Slope of v-t gives acceleration; slope of a-t does not generally have a named quantity.
18. (10.9) The area under a velocity-time graph:
C. Because v = Δx / Δt, v × Δt = Δx; summing these strips gives total displacement.
19. (10.10) The mean of 3, 5, 7, 9 is:
D. Arithmetic mean is a simple sum divided by count.
20. (10.10) The median of 3, 5, 7, 9 is:
A. For an even number of data points, the median is the average of the two middle ones.
21. (10.11) If P(A) = 0.3 and P(B) = 0.5, and A and B are independent, then P(A AND B) equals:
B. Multiplication rule applies for independent events.
22. (10.11) For two mutually exclusive events A and B, P(A OR B) =
C. Mutually exclusive = cannot occur together (P(A ∩ B) = 0).
23. (10.12) Order-of-magnitude estimation is useful for:
D. "Fermi problems" teach order-of-magnitude reasoning.
24. (10.12) On an MCAT question where answer choices span several orders of magnitude:
A. Huge time saver. If choices are 10⁻², 10⁻¹, 10⁰, and 10¹, you do not need 3 significant figures.

You are 45 minutes into the MCAT, staring at a passage about radioactive decay. The question asks you to estimate the remaining sample after 3.2 half-lives. There is no calculator in front of you. There never will be. The MCAT is a no-calculator exam, and this is not an accident - it is a deliberate design choice. The test wants to know whether you can reason quantitatively under pressure, not whether you can punch numbers into a machine.

Here is the good news: the math on the MCAT is not hard. You will never need calculus. You will rarely see anything beyond basic algebra. But you do need to be fast, confident, and comfortable with estimation. The students who struggle with MCAT math are not the ones who lack math knowledge - they are the ones who freeze when they cannot get an exact answer and do not realize that “close enough” is the whole point.

This chapter covers every mathematical skill the MCAT tests: mental arithmetic, algebra, logarithms, trigonometry, vectors, unit conversions, scientific notation, proportional reasoning, graph interpretation, statistics, probability, and estimation strategy. Most of these topics you learned years ago. The goal here is not to teach you math from scratch - it is to sharpen your tools so you can deploy them in under 90 seconds per question with no calculator.

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