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The Required Mathematics and the No-Calculator Rule

Every number on the MCAT is worked by hand: log scales, slopes, significant digits, dimensional analysis, basic trig and vectors, no calculus.

10 topics · 43 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

No Calculator: Every Number Is Estimated

  1. How it worksHow do you turn an ugly division into a two-second estimate by rounding both numbers to one digit?
  2. Worked exampleFind 6.6 times 10 to the minus 34 divided by 1.6 times 10 to the minus 19 without a calculator.
  3. Worked exampleFind the square root of 150 to within a few percent using only the squares you already know.
  4. Common mistakeWhy isn't an exact answer worth 40 extra seconds when the four choices differ by a factor of ten?
  5. Going furtherWhat changes about arithmetic strategy when answer choices sit close together rather than far apart?
Your turnhard1 of 3
Without a calculator, estimate 7 point 7 times 10 to the minus 34, divided by 1.6 times 10 to the minus 19. Give the answer as a multiple of 10 to the minus 15.

Linear, Semilog and Log-Log Scales

  1. How it worksHow does a semilog plot turn constant percentage growth into a constant slope?
  2. How it worksHow does a log-log plot turn a power law into a straight line whose slope is the exponent?
  3. Worked exampleWalk me through reading a bacterial growth curve that runs from 10 to 10 million on a semilog axis.
  4. Common mistakeWhy isn't a straight line on a log axis the same as steady growth in absolute numbers?
  5. Going furtherWhat changes about reading a pH graph once you remember the scale is already logarithmic?
Your turnhard1 of 3
A bacterial culture is plotted on a semilog axis running from 10 to 10 million. The count rises by one decade every 48 minutes. How many minutes does it take to rise from 100 to 7 million?

Calculating Slopes from Figures and Tables

  1. How it worksHow do you read a slope off a figure by picking two widely spaced easy points rather than two close ones?
  2. Worked exampleFind the rate when volume climbs from 2 mL to 8 mL between the 10-second and 40-second marks.
  3. Going furtherWhat changes when a curve bends and a question asks for the slope at one single instant?
  4. Common mistakeWhy doesn't a steeper-looking line always mean a larger slope when two axes are scaled differently?
Your turneasy1 of 3
A figure shows volume climbing from 4 mL at the 14-second mark to 16 mL at the 38-second mark. What is the rate, in millilitres per second?

Significant Digits and Reasonable Estimates

  1. How it worksHow do the digits in your inputs set a hard limit on the digits you are allowed to report?
  2. Worked exampleFind how many significant digits belong in 4.2 times 1.37, and show why the extra ones are fiction.
  3. Common mistakeWhy isn't rounding at every intermediate step the same as rounding once at the end?
Your turnhard1 of 3
Multiply 6 point 6 by 1 point 43. Report the product rounded to the number of significant digits the less precise input allows.

Dimensional Analysis: Units as the Check

  1. How it worksHow does treating units as algebra cancel them until only the unit the answer needs is left standing?
  2. Worked exampleWalk me through converting 5 millimoles per litre into moles per cubic metre using units alone.
  3. Worked exampleFind which of force times distance, force over time and mass times velocity carries the units of energy.
  4. Going furtherWhat changes when a question supplies a conversion factor you have never seen before?
Your turneasy1 of 3
Convert 28 millimoles per litre into moles per cubic metre.

Algebra II Exponentials and Logarithms

  1. How it worksHow do you get the log of 5 times 10 to the minus 4 in your head, given that log 5 is about 0.7?
  2. Worked exampleFind the pH of a solution whose hydrogen ion concentration is 2 times 10 to the minus 5 molar.
  3. Worked exampleFind how many half-lives it takes for a sample to fall to one-eighth of its starting amount.
  4. Common mistakeWhy isn't the natural log the same as log base ten, and when does the difference actually matter?
  5. Going furtherWhat changes about a decay problem when you are handed a rate constant instead of a half-life?
Your turnmedium1 of 3
A solution has a hydrogen ion concentration of 7 times 10 to the minus 7 molar. What is its pH? Use log 2 = 0.30, log 3 = 0.48, log 5 = 0.70 as needed.

The Trigonometry the MCAT Requires

  1. How it worksHow do sine, cosine and tangent each pick out a different pair of sides in a right triangle?
  2. Worked exampleFind the horizontal component of a 100 N force directed 30 degrees above the horizontal.
  3. Worked exampleFind the hypotenuse of a 45-45-90 triangle whose legs are both 7, without touching a calculator.
  4. Common mistakeWhy isn't the sine of 90 degrees zero, and what breaks when the values at 0 and 90 get swapped?
Your turneasy1 of 3
A force of 151 newtons is directed 52 degrees above the horizontal. What is its horizontal component, in newtons?

Vector Addition and the Right-Hand Rule

  1. How it worksHow do you add two vectors tip to tail and read the resultant straight off the triangle?
  2. How it worksHow does the right-hand rule turn two directions into a third one perpendicular to both?
  3. Worked exampleFind the resultant of a 6 N force pointing east combined with an 8 N force pointing north.
  4. Worked exampleWalk me through the direction of the force on a positive charge moving east through a northward field.
  5. Going furtherWhat changes about a vector sum when the two vectors sit 120 degrees apart instead of perpendicular?
Your turneasy1 of 3
A force of 15 newtons points east and a force of 17 newtons points north. What is the magnitude of their resultant, in newtons?

Out of Scope: Calculus, Dot and Cross Products

  1. How it worksHow does the MCAT ask about rates of change using slopes and ratios instead of derivatives?
  2. Common mistakeWhy isn't the right-hand rule the same thing as the cross product the MCAT says you can skip?
  3. Worked exampleWalk me through a magnetic force question answered by direction and magnitude with no cross product.
  4. Going furtherWhat changes about studying physics for the MCAT once you know integrals will never be required?
Your turnmedium1 of 3
A question gives a position-against-time curve and asks for the speed at the 15-second mark, where the curve has a visible bend. Question: what does the MCAT expect?

What the Periodic Table Gives You and What It Does Not

  1. How it worksHow do you read group and row position off a periodic table to rank electronegativity and atomic radius?
  2. Common mistakeWhy doesn't the provided periodic table give you pKa values, solubility rules or any constants?
  3. Worked exampleWalk me through getting the molar mass of calcium phosphate using only the table on the screen.
  4. Going furtherWhat changes about your memorisation list once you know atomic masses are always on screen?
Your turnmedium1 of 3
Using atomic masses of Ca 40.1, P 31.0 and O 16.0 from the provided table, find the molar mass of a compound with 4 calcium atoms, 2 phosphorus atoms and 8 oxygen atoms, in grams per mole.

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