Every formula, mnemonic, and concept behind the two subtests that decide your Quantitative and Academic Aptitude composites, built directly from your Sep 4, 2026 diagnostic results. Tap "Show answer" to check any practice problem; check a concept off once it's solid.
How to use this guide: concepts are organized by skill, not by subtest, since several (percentages, formula recall) show up in both Arithmetic Reasoning and Math Knowledge. Eight are marked PRIORITY, exactly what the diagnostic caught you missing, each with a flag explaining precisely what went wrong on that item. Six are marked LIGHT: you already have these, so they get a lighter treatment. Read the explanation, read the mnemonic until it sticks, work the example, then try the practice problems cold before revealing each answer.
The rules that decide whether a correct calculation gets read the right way: what to compute first, which "common" number a question wants, and when a negative answer has to be thrown out.
When an expression mixes several operations, work in a fixed order: parentheses/grouping first, then exponents and roots, then multiplication and division together (left to right), then addition and subtraction together (left to right). Multiplication does not outrank division, and addition does not outrank subtraction; within each pair, whichever comes first (reading left to right) goes first.
PEMDAS: Parentheses, Exponents, then MD and AS march left to right, side by side, never one before the other.
8 + 2×(5−3)²: the (5−3)=2 resolves first, then the exponent applies before anything else touches it: 2²=4. Only then multiply: 2×4=8. Add last: 8+8=16. Skipping the exponent, or adding before finishing it, is the exact trap.
Simplify 6 + 3×(4−2)³÷2 → (4−2)=2 → 2³=8 → 3×8=24, 24÷2=12 → 6+12=18
GCF (Greatest Common Factor) is the largest number that divides evenly into both numbers: list factors of each and take the biggest one they share. LCM (Least Common Multiple) is the smallest number both numbers divide evenly into: list multiples of each until one matches.
Factors shrink, multiples grow. GCF goes down into both numbers; LCM is what both numbers climb up to.
Asked for the LCM of 6 and 8, the GCF (2) was given instead: the two got swapped. If unsure which one a question wants, check whether the answer should be small and divide both (GCF), or should be at least as big as the larger number (LCM).
LCM and GCF of 12 and 18: shared factors 1,2,3,6 → GCF=6. Multiples of 12: 12,24,36… of 18: 18,36… first match 36 → LCM=36. Check: GCF×LCM = 216 = 12×18. ✓
"Percent of" means multiply by the decimal form. Percent change compares the size of the change to the original (old) value, never the new one.
"Of" means times. "Change" always divides by where you started (the OLD value).
A $40,000 salary rises to $46,000. Percent increase? (46,000−40,000)/40,000 = 0.15 = 15%
Every positive number has two square roots, one positive and one negative (√49 = 7 or −7, since both 7² and (−7)² equal 49). On the AFOQT, a variable representing something real (a length, a count, a speed) cannot be negative, so when a problem states x > 0, or the quantity is physically a size, discard the negative root.
Real things aren't negative. If x is a length, count, or the problem says x > 0, keep only the positive root.
x² = 49, x > 0: both +7 and −7 solve the equation algebraically, but the stated constraint x > 0 rules out −7. The answer given was −7, the one the constraint excludes.
Solve x² = 81, where x is a side length. x = ±9, but a side length can't be negative, so x = 9.
Your two highest-leverage clusters: every ratio-split problem and every rate problem that hides a unit conversion. Five of your six Arithmetic Reasoning misses live here.
When something is split in a ratio (like 2:3:4), treat each ratio number as one "share" of equal size. Three steps: sum the ratio numbers to find how many equal shares make up the whole; divide the whole by that sum to find the size of one share; multiply that share by whichever ratio number you need.
SUM → DIVIDE → MULTIPLY. Add the parts, split the whole, scale back up.
Item 9 (60 cadets, 5:1 pass:fail): sum=6, one share=10, failed=1×10=10. The answer given, 12, comes from dividing 60 by 5 instead of the full 6 shares. Item 10 (900 units, 2:3:4): sum=9, one share=100, largest=4×100=400. The answer given, 450, is half of 900, not 4 shares of 100.
A $60,000 budget splits 3:2:1 among vehicles, gear, and supplies. Vehicles' share? Sum=6. One share=$10,000. Vehicles (3 shares)=$30,000.
A rate problem (speed, price per unit, output per hour) only works if both quantities are in matching units before you divide. Convert first, then compute. Worth memorizing: 1 hour = 60 minutes, 1 ton = 2,000 pounds, 1 foot = 12 inches, 1 yard = 3 feet.
Convert First, Compute Second (CFCS). Never divide mismatched units.
Item 8: 2h18m must become 2.3h (18/60=0.3) before dividing into 460 miles; using 2.5h (a common minutes-to-decimal slip) gives the wrong 184. Item 23: 2.4 tons must become 4,800 lb (×2,000) before dividing by 8 pallets. Item 25: the altitude gained (21,000−3,000=18,000 ft) has to be computed correctly before dividing by the 1,800 ft/min climb rate.
A machine bags 1.5 tons of flour in 45 minutes. Pounds per minute? Convert: 1.5×2,000=3,000 lb. Rate: 3,000÷45 = 66.7 lb/min.
Distance = Rate × Time. Cover whichever variable you're solving for to find it from the other two. Two special cases: objects moving toward each other close the gap at the sum of their speeds; one object catching up to another (same direction) closes the gap at the difference of their speeds.
Toward each other, ADD. Chasing from behind, SUBTRACT.
Two convoys start 315 miles apart, closing at 45 mph and 30 mph. Combined speed=75 mph. Time = 315÷75 = 4.2 hours.
If a job takes a certain number of "worker-days" to finish, that total stays constant no matter how the workers are split: Workers × Days = Total Work. More workers means fewer days, in inverse proportion.
More hands, less time. Multiply what you have, divide by what you want.
You missed a 3-workers/12-days problem, but solved an equivalent one correctly elsewhere on the same test (Item 19). That pattern points to an arithmetic slip under time pressure, not a concept gap: worth a couple of clean reps to lock in the steady process below.
5 machines finish an order in 8 days. Days for 4 machines? Total work=5×8=40 machine-days. 40÷4=10 days.
Clean on the diagnostic, and worth keeping that way: these are fast, mechanical points once the steps are automatic.
Whatever you do to one side of an equation, do to the other, to keep it balanced. To simplify, combine "like terms" (same variable, same power) by adding or subtracting their coefficients, and distribute any number multiplied outside parentheses across every term inside.
Do unto both sides. Same letter, same power, combine the numbers out front.
Solve 4(x+3) = 2x + 18. Distribute: 4x+12=2x+18. Subtract 2x: 2x+12=18. Subtract 12: 2x=6. Divide by 2: x=3.
Multiplying same-base powers: add the exponents. Dividing same-base powers: subtract the exponents. A power raised to a power: multiply the exponents. Difference-of-squares pattern: (a+b)(a−b) = a² − b², the middle terms always cancel.
Multiply powers, ADD exponents. Divide powers, SUBTRACT exponents. Power of a power, MULTIPLY exponents.
Simplify (3x⁴)(2x³)÷x². Multiply coefficients, add exponents: 6x⁷. Divide: 6x⁷÷x²=6x⁵.
Four of your seven Math Knowledge misses live here, and every one traces back to a formula misapplied, not a concept misunderstood. This is a formula-sheet problem: drill it fully untimed until it's automatic.
Rectangle: Perimeter = 2(length+width); Area = length × width. Triangle: Area = ½ × base × height, because any triangle is exactly half of the rectangle that encloses it (same base and height).
A triangle is half a box. Find base × height like a rectangle, then cut it in half.
Triangle, base 14, height 6: the ÷2 step was skipped, giving 14×6=84 (the rectangle's area) instead of 42 (the triangle's, half of it).
Triangular sign, base 10 in, height 7 in. Area? ½×10×7=35 sq in.
Circumference (distance around) = 2πr, or πd. Area (space inside) = πr². The common mix-up: circumference is linear (just r, doubled), area is squared (r×r).
Circumference is a belt (2πr, wraps around once). Area is a rug (πr², covers the floor, squared).
Item 20 (circumference, r=7) dropped the factor of 2: πr=22 instead of 2πr=44. Item 23 (area, r=5) used the circumference-style formula: π×r=15.7 instead of πr²=78.5, squaring the radius first.
Circular hatch, radius 4 ft (π≈3.14). Circumference=2×3.14×4=25.12 ft. Area=3.14×4²=50.24 sq ft.
In any right triangle, the two shorter sides (legs, a and b) and the longest side (hypotenuse, c, opposite the right angle) follow a²+b²=c². Square both legs, add them, then take the square root of the sum to find the hypotenuse (or rearrange to find a missing leg).
Legs squared, added, then rooted. The hypotenuse is always the longest side, across from the right angle.
Legs 6 and 8: the answer given, 48, looks like 6×8, not the theorem at all. The right move: √(6²+8²) = √100 = 10.
A ramp's base is 9 ft and it rises 12 ft. Ramp length? √(9²+12²)=√225=15 ft.
For a rectangular box, Volume = length × width × height: the area of the base "stacked up" to the given height.
Area of the floor, times how tall it stands.
Storage crate 6 ft × 4 ft × 3 ft. Volume? 6×4×3=72 cu ft.