ASVAB Mathematics Knowledge study guide

What this subtest actually asks of you

Mathematics Knowledge is 15 of the 135 scored questions on the computer ASVAB — 11% of the scored test — and it is one of the four subtests that produce your AFQT score (how the ASVAB is scored).

Where Arithmetic Reasoning hands you a sentence to turn into arithmetic, this subtest hands you symbols and figures: an equation to solve, an exponent to simplify, a shape to measure. There is nothing to translate. There is a procedure, and the question is whether you know it — and the procedure set is small and finite. All twenty-one are listed below, and every one has a worked example after it, and every formula on them is collected on one table at the end. Work through them once with a pencil and you have covered the arithmetic this subtest can ask for.

Two things worth knowing before you start:

• Name the type before you touch the problem. Nearly every question announces what it is ("solve for x", "what is the area of"). Deciding that first is what stops you applying the perimeter formula to an area question.
• Almost every answer can be checked. An equation's answer goes back into the original equation. A geometry answer's unit tells you whether you used the right formula. That is step 4 of the method below, and it is where most of the recoverable points are.
• The formulas are collected on one table at the end of this page. Copy it onto the top of your scratch paper before you answer anything. This subtest tests whether you can use the formula, not whether you can recall it — so there is no reason to test your memory.

The twenty-one procedures

Every Mathematics Knowledge question is one of these twenty-one. Each one is worked out below — the rule, the steps, and the trap that costs points on it.

1. Order of operations (PEMDAS). Parentheses first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right. → example 1
2. Signed numbers. Subtracting a negative is the same as adding. When the signs differ, subtract the smaller magnitude from the larger and keep the sign of the larger number. → example 2
3. Solving a linear equation for x. Undo the operations in reverse order, doing the same thing to both sides. Divide out parentheses first, and when x appears on both sides, collect the x terms on one side. → example 3
4. Substituting values into an expression. Replace each letter with its value, put negative values in parentheses, then evaluate using the order of operations. → example 4
5. Fractions, decimals and percents. Converting between the three, reducing to lowest terms, adding fractions by finding a common denominator first, and the two percent questions — "what is x% of y" and "what percent of y is x". → example 5
6. Exponents and square roots. Multiplying powers of the same base means adding the exponents; a square root undoes a square; and if x² = 49 with x positive, then x = 7. → example 6
7. Factoring. The difference of squares, and expanding two brackets back out — which is also how you check a factoring answer. → example 7
8. Perimeter and area. The perimeter of a rectangle, the area of a rectangle, a triangle and a circle. A perimeter is a length; an area is in square units, and that unit is often the only thing separating two choices. → example 8
9. Volume. A box is length × width × height; a cube is the edge length cubed. → example 9
10. The angle facts. A straight line is 180 degrees, and the three interior angles of any triangle add up to 180 degrees. → example 10
11. The Pythagorean theorem. a² + b² = c², where c is the hypotenuse — the side opposite the right angle. A rectangle's diagonal is a hypotenuse. → example 11
12. Averages. The average is the total divided by the count, so the total is the average × the count, and a missing value is that total minus the numbers you already have. → example 12
13. Factoring a trinomial, and solving a quadratic equation. Two numbers that multiply to the constant and add to the middle coefficient give the factors; setting each factor to zero gives the two solutions. Rearrange first — a quadratic must equal zero before it can be factored. → example 13
14. Solving a system of two equations. Add or subtract the equations to cancel one variable, or substitute one equation into the other, then solve for the value the question asks for. The answer is the pair that satisfies both equations. → example 14
15. Solving an inequality. Work it exactly like an equation, with one difference: multiplying or dividing both sides by a negative number reverses the direction of the sign. → example 15
16. Absolute value. |x| is the distance from zero, so it is never negative and the bars do not cancel a minus sign. An equation like |x - 4| = 6 splits into two cases — what is inside the bars equals 6, or it equals -6. → example 16
17. Slope, midpoint and distance on the coordinate plane. Slope is the rise over the run. The midpoint averages the two coordinates. The distance between two points is the Pythagorean theorem on the differences — the same 3-4-5 triangle, seen on a grid. → example 17
18. Primes, the greatest common factor and the least common multiple. A prime number has no factor but 1 and itself. The greatest common factor is the largest number that divides both; the least common multiple is the smallest number both divide into. → example 18
19. Scientific notation. One digit before the point, times a power of ten: 45,000 is 4.5 × 10⁴. Multiplying adds the exponents; dividing subtracts them. → example 19
20. Parallel lines and the angles of a polygon. Angles on the same side of a transversal add to 180 degrees; the alternating ones are equal. A polygon with n sides splits into n - 2 triangles, so its interior angles total (n - 2) × 180. → example 20
21. Circumference, cylinders and surface area. Circumference is pi × diameter. A cylinder's volume is the base area × the height. A surface area is the area of every face added up. → example 21

A four-step method

1. Name the type. Out loud: "this is a linear equation", "this is an area question". Step 1 is the whole defense against the error this subtest is built to catch — using the right procedure on the wrong thing.

2. Write the procedure for that type before you substitute anything: the formula, or the list of undoing steps. A procedure written down can be checked; one done in your head cannot.

3. Substitute and compute, one operation at a time. Show the work, even though the scratch paper is not graded. Almost every MK error is a step done invisibly and wrongly.

4. Check it two ways. Substitute your answer back into the original — does 3(2 × 4 − 1) really equal 21? — and check the unit and the magnitude: an area is in square units; a hypotenuse is longer than either leg; a cube's volume is larger than its edge. Two seconds, and it catches the answer of the wrong type.

The twenty-one worked examples

Each example is one of the twenty-one procedures above: the rule in action, the steps, and the trap that example is built to catch.

1. Order of operations (PEMDAS)

What is 6 + 4 × 3? And what is 5 × (3 + 4)?

• Multiplication before addition: 4 × 3 = 12.
• Then add: 6 + 12 = 18.
• Parentheses first when they are there: 3 + 4 = 7, then 5 × 7 = 35.
• Check the size: 18 is bigger than either number, but far smaller than 6 × 4 × 3 = 72.
• The trap: 30 — that is 6 + 4 = 10, then × 3. Adding before multiplying is the classic arithmetic slip, and the whole reason the order of operations exists.

2. Signed numbers: subtracting a negative

What is 5 - (-3)?

• Two minus signs together make a plus: 5 - (-3) = 5 + 3.
• Add: 5 + 3 = 8.
• The rule when the signs differ: -7 + 3 gives 7 - 3 = 4, and since the larger number (7) is negative, the answer is -4.
• Check the direction: subtracting a negative has to make the answer bigger than 5, and 8 is.
• The trap: 2 — that is 5 - 3, subtracting 3 instead of subtracting negative 3.

3. A linear equation, including parentheses and x on both sides

Solve for x: 3(2x − 1) = 21

• Divide both sides by 3 first: 2x − 1 = 7.
• Add 1: 2x = 8.
• Divide by 2: x = 4.
• Check it: 3(2 × 4 − 1) = 3 × 7 = 21.
• The trap: 7 — the value of 2x − 1, one step short of x. When x appears on both sides (3x + 2 = 2x + 9), subtract 2x from both sides first: x + 2 = 9, so x = 7.

4. Substituting values into an expression

If a = 2 and b = -3, what is the value of a - b?

• Substitute, with parentheses around the negative: a - b = 2 - (-3).
• Two minus signs make a plus: 2 + 3 = 5.
• The same move with a product: y = 3x at x = 4 is 3 × 4 = 12.
• Check the sign by saying it out loud: "2 minus negative 3" has to be larger than 2.
• The trap: -1 — that is 2 - 3, dropping the parentheses around the negative.

5. A fraction as a percent, and adding fractions

What is 3/8 written as a percent?

• Divide the top by the bottom: 3 / 8 = 0.375.
• Move the decimal point two places: 37.5%.
• To add fractions, find a common denominator first: 4/5 + 1/10 = 8/10 + 1/10 = 9/10.
• The reverse question — what percent of 48 is 12 — is part divided by whole: 12 / 48 = 0.25 = 25%.
• The trap: 38% (rounded past the answer) and 26.7% (8 / 3 — the division done upside down).

6. Multiplying powers and taking a square root

Simplify (x³)(x⁴). And what is the square root of 144?

• Multiplying powers of the same base adds the exponents: x³ × x⁴ = x³⁺⁴ = x⁷.
• Prove it with numbers: 2³ × 2² = 8 × 4 = 32, and 2³⁺² = 2⁵ = 32.
• A square root undoes a square: the square root of 144 is 12, because 12 × 12 = 144.
• Solving x² = 49 is the same move: x = 7 when x is positive.
• The trap: multiplying the exponents instead of adding them (x¹²) — that is the rule for a power of a power, not for a product.

7. Factoring a difference of squares

Factor completely: x² − 9

• The pattern is a² − b² = (a − b)(a + b), and here a = x and b = 3.
• So x² − 9 = (x − 3)(x + 3).
• Check it by expanding: x² + 3x − 3x − 9 = x² − 9.
• The trap: (x − 3)(x − 3), which expands to x² − 6x + 9. When a factoring choice looks plausible, expand it — that is the check step.

8. Area or perimeter? (and the right circle formula)

A rectangle is 8 feet long and 5 feet wide. What is its area, and what is its perimeter?

• Name the type first: "area" wants square units, "perimeter" wants a length.
• Area: length × width = 8 × 5 = 40 square feet.
• Perimeter: 2(8 + 5) = 2 × 13 = 26 feet.
• A triangle is half a rectangle: 1/2 × base × height = 1/2 × 10 × 6 = 30.
• A circle's area is pi × r² = 3.14 × 3² = 3.14 × 9 = 28.26; its perimeter (the circumference) is 2 × pi × r = 2 × 3.14 × 3 = 18.84.
• The trap: answering 26 when the question asked for area. The unit settles it — 26 feet is a length, not an area.

9. The volume of a box or a cube

What is the volume of a cube with an edge length of 4?

• A cube is the edge cubed: 4³ = 4 × 4 × 4 = 64 cubic units.
• A box is length × width × height — the same rule with three different measurements.
• Check the unit and the size: volume is in cubic units, and 64 is larger than the edge, which it has to be.
• The trap: 16 — that is the area of one face (4²), not the volume, and the word "square" in the unit gives it away.

10. The angle facts

How many degrees are in a straight angle, and what do the three angles of a triangle add up to?

• A straight line is 180 degrees — that is what "straight angle" means.
• The three interior angles of any triangle also add up to 180 degrees.
• So a triangle with angles of 60 and 70 has a third angle of 180 − 60 − 70 = 50 degrees.
• Check by size: the three angles together are less than the 360 degrees of a full turn.
• The trap: remembering the 180 and applying it to the wrong shape — a rectangle's four angles add up to 360, not 180.

11. A diagonal is a right triangle in disguise

The rectangle shown is 12 cm long and 5 cm wide. What is the length of its diagonal?

• Name the type: this is the Pythagorean theorem. The diagonal and two sides of the rectangle make a right triangle, so the diagonal is its hypotenuse.
• Write the procedure: a² + b² = c², where c is the diagonal.
• Compute: 5² + 12² = 25 + 144 = 169, and the square root of 169 is 13, so the diagonal is 13 cm.
• Check the magnitude: the diagonal is longer than either side (13 > 12) and shorter than the two sides added together (13 < 17).
• The trap: 17 cm — the two sides added. Step 4 kills it in a second, because no diagonal can be as long as the two sides laid end to end.

12. An average, and finding a missing value

The average of four numbers is 15. Three of them are 12, 18 and 10. What is the fourth?

• Total = average × count = 15 × 4 = 60.
• Add what you have: 12 + 18 + 10 = 40.
• Subtract: 60 − 40 = 20.
• Check it: 12 + 18 + 10 + 20 = 60, and 60 / 4 = 15.
• The trap: using the wrong count. The total is the average × the number of items, not x the number you were given.

13. Factoring a trinomial, and solving a quadratic equation

Solve for x: x² - 9x + 20 = 0

• The equation already equals zero, so factor the left side.
• Two numbers that multiply to 20 and add to -9 are -4 and -5: x² - 9x + 20 = (x - 4)(x - 5).
• A product is zero only when a factor is zero, so x - 4 = 0 or x - 5 = 0: x = 4 or x = 5.
• Check both: 16 - 36 + 20 = 0 ✓ and 25 - 45 + 20 = 0 ✓.
• Rearranging comes first when the equation does not already equal zero: x² + 2x = 15 becomes x² + 2x - 15 = 0 = (x + 5)(x - 3), so x = -5 or x = 3.
• The trap: stopping at one root. A quadratic equation has two solutions, and the other one is usually sitting among the choices.

14. Solving a system of two equations

Solve for x and y: x + y = 21 and x - y = 5

• Line the equations up and add them — the y terms cancel: (x + y) + (x - y) = 21 + 5, so 2x = 26.
• Divide: x = 13. Then substitute back: 13 + y = 21, so y = 8.
• Check it in the other equation: 13 - 8 = 5 ✓.
• When adding cancels nothing, substitute instead: y = 2x - 1 and x + y = 14 gives x + 2x - 1 = 14, so 3x = 15 and x = 5, y = 9.
• The trap: answering one variable. Read whether the question asks for x, for y, or for the pair — both numbers are true, and the choices will include the one you were not asked for.

15. Solving an inequality

Solve for x: -4x + 3 > 23

• Subtract 3 from both sides: -4x > 20.
• Divide both sides by -4 — and reverse the sign: x < -5.
• Check with a number: x = -6 gives -4(-6) + 3 = 27, and 27 > 23 ✓; x = 0 gives 3, which is not > 23 ✓.
• Dividing by a positive number changes nothing at all: 3x + 1 < 16 gives 3x < 15, so x < 5.
• The trap: keeping the direction. -4x > 20 is x < -5, not x > -5 — the flip is the whole procedure.

16. Absolute value, and the two-case equation

What is |-7|? And solve for x: |x - 4| = 6

• The bars mean distance from zero, so |-7| = 7 — and |7| is 7 as well.
• An absolute-value equation splits into two cases: x - 4 = 6, or x - 4 = -6.
• Solve each one: x = 10 or x = -2.
• Check both: |10 - 4| = 6 ✓ and |-2 - 4| = |-6| = 6 ✓.
• The trap: answering only 10. Two numbers sit six units from 4, one on each side of it, and both are in the choices.

17. Slope, midpoint and distance on the coordinate plane

Find the slope of the line through (2, 4) and (7, 14), the midpoint of the segment joining (2, 4) and (6, 8), and the distance from (0, 0) to (6, 8)

• Slope is the rise over the run: (14 - 4) / (7 - 2) = 10 / 5 = 2.
• The midpoint averages each pair of coordinates: ((2 + 6) / 2, (4 + 8) / 2) = (4, 6).
• Distance is the Pythagorean theorem on the differences — (0, 0) to (6, 8) are 6 and 8 apart, and the square root of 36 + 64 = 100 is 10.
• Check the midpoint against both ends: (4, 6) is one step from (2, 4) and one from (6, 8) ✓.
• The trap: subtracting where you should average. (4, 4) is the gap between those endpoints, not the point halfway along them.

18. Primes, the greatest common factor and the least common multiple

Is 51 prime? What is the greatest common factor of 36 and 60, and the least common multiple of 4 and 10?

• 51 = 3 × 17, so it has a factor besides 1 and itself: not prime.
• Prime factors: 36 = 2 × 2 × 3 × 3 and 60 = 2 × 2 × 3 × 5. The primes they share are 2, 2 and 3, so 2 × 2 × 3 = 12.
• The least common multiple of 4 and 10: multiples of 10 are 10, 20 — and 20 divides by 4, so the answer is 20.
• Check the pair against the numbers: the greatest common factor divides both (36 / 12 = 3 and 60 / 12 = 5), and the least common multiple is never smaller than the larger number.
• The trap: calling a common factor the greatest one. 3 divides both 36 and 60, and 12 is larger — the two answers are usually both sitting among the choices.

19. Scientific notation: writing it, and multiplying it

Write 45,000 in scientific notation, and multiply 4 × 10³ by 2 × 10⁵

• One digit before the point: 45,000 = 4.5 × 10⁴ — the point moves four places.
• Multiply the front numbers and add the exponents: 4 × 2 = 8 and 3 + 5 = 8, so the product is 8 × 10⁸.
• Small numbers work the same way with a negative exponent: 0.0006 = 6 × 10⁻⁴.
• Dividing subtracts the exponents: (8 × 10⁶) / (4 × 10²) = 2 × 10⁴.
• The trap: multiplying the exponents. They add in a product and subtract in a quotient — 10³ × 10⁵ is 10⁸, never 10¹⁵.

20. Parallel lines, and the angle sums of polygons

Two parallel lines are cut by a transversal, and one angle measures 70 degrees. What is the same-side interior angle, and what do the interior angles of a hexagon total?

• Angles on a straight line add to 180, so the same-side partner of the 70-degree angle is 180 - 70 = 110 degrees.
• The alternate interior angle — the Z shape — is 70 degrees, equal to the first.
• A polygon's interior angles total (n - 2) × 180: a hexagon has 6 sides, so 4 × 180 = 720 degrees.
• A regular hexagon divides that equally: 720 / 6 = 120 degrees per angle.
• The trap: treating the same-side pair as equal. The equal pairs form the Z; the same-side interior pair adds to 180.

21. Circumference, cylinders and surface area

A circle has a radius of 4. What is its circumference, and what is the volume of a cylinder with that base and a height of 15?

• Circumference is pi × diameter, and the diameter is 2 × 4 = 8: the circumference is 8 pi.
• The base area is pi × r² = pi × 16 = 16 pi square units.
• Volume is the base area × the height: 16 pi × 15 = 240 pi cubic units.
• A surface area counts every face — a cube with an edge of 2 has 6 faces of 4, so 24 square units.
• The trap: putting the radius into a diameter formula. pi × r is half the circumference, not the circumference.

The traps, collected

1. One step short — answering 2x, or 2x − 1, instead of x. The classic error on this subtest.
2. Adding before multiplying — 6 + 4 × 3 answered as 30, and a parenthesized sum multiplied inside-out.
3. Dropping the parentheses around a negative — a − b with b = −3 answered as 2 − 3.
4. Perimeter for area, area for perimeter — the unit settles it: 26 feet is a length.
5. The wrong circle formula — pi r² is the area; 2 pi r is the circumference.
6. Exponents multiplied instead of added — x³ × x⁴ is x⁷, not x¹².
7. Rounding past the answer — 0.375 is exactly 37.5%; 38% is not.
8. The wrong count in an average — the total is the average × the number of items.
9. Square units where cubic ones belong — the area of one face offered as a volume.
10. One root instead of two — a quadratic equation has two solutions: x(x - 5) = 0 gives x = 5 and x = 0.
11. Answering the wrong variable — a system's question may ask for x, for y, or for the pair; both numbers are true, so read which one is wanted.
12. The un-flipped inequality — dividing by a negative number reverses the direction: -3x > 12 is x < -4.
13. One case of an absolute value — |x - 5| = 4 means x = 9 or x = 1; both numbers sit four units from 5.
14. Subtracting where you should average — the midpoint of (1, 2) and (7, 10) is (4, 6), not the difference (6, 8).
15. A common factor called the greatest — 4 divides both 24 and 36, and 12 is the greatest common factor. The least common multiple is the other answer in the choices.
16. Multiplying the exponents — in scientific notation they add in a product (10³ × 10⁵ = 10⁸) and subtract in a quotient (10⁶ / 10² = 10⁴).
17. The wrong pair of angles — the equal pairs form the Z (alternate angles); the same-side interior pair adds to 180 degrees.
18. One face instead of every face — a surface area counts all of them; 4³ is a volume, in cubic units.

Look at how many of these are one step short or the other formula. On this subtest the arithmetic is rarely the problem; the ending is.

Every formula, on one table

The twenty-one procedures above, reduced to the rules themselves. Copy this onto the top of your scratch paper before you answer anything: finding the formula is not what this subtest is testing, and a rule you have written down cannot be misremembered mid-question.

Two rules of thumb that save more points than any formula here: the unit names the procedure (a length is not an area, a square unit is not a cubic one), and the answer goes back into the question — an equation's solution substituted back, or a geometry answer checked to be bigger than the side it came from.

Answer-choice strategies: three moves for when the setup will not come

The procedures and the method above are what earn points, and these three moves are not a substitute for them. They are what to do when the algebra will not simplify or the figure will not resolve — because a question you never start is a question you cannot get right, and a question you grind for four minutes is three questions you did not reach.

Use them in this order. Each one narrows the choices rather than guessing at them.

1. Estimate the size before you compute it

The choices on a computation question are almost never close together. Bound the answer first — a square root between two you know, a circle area against 3 × r² — and let the choices do the rest.

What is the square root of 80?

• 9² = 81, so the answer is a shade under 9 — about 8.9.
• A choice of 40 is not a square root of anything in that range, and 6,400 is 80 squared, not the square root of 80.

Two cautions. An estimate tells you where the answer is, never what it is — finish the arithmetic. And when two choices sit close together (0.375 and 0.38), the estimate has done its job and the exact work is the only thing that decides between them.

2. Backsolve — put the choices back into the equation

"Which value of x satisfies ..." is a question you do not have to solve. Substitute each choice into the equation and keep the one that makes both sides equal.

Which value of x satisfies 3x − 5 = 16?

• Take the middle choice first. If x = 6 gives 3(6) − 5 = 13, the answer has to be bigger, so every choice below 6 is gone in one step.
• Try x = 7: 3(7) − 5 = 16. ✓

Starting in the middle is the whole trick — a wrong choice still tells you which way to move, so one substitution usually halves the field.

3. Pick a number for the variable

When the choices are expressions rather than numbers, replace the variable with a convenient value, work it out, and match the shape of your answer to the choices. Let x = 3 unless the question says otherwise.

If x is an odd integer, which expression represents the next odd integer after x?

• Let x = 3. The next odd integer is 5.
• Test the choices with 3: x + 1 gives 4, x + 2 gives 5, 2x gives 6.
• Only one choice matches, so that is the expression.

Three cautions, and these are the ones that cost points. Never pick 0 or 1 — they hide exactly the errors you are trying to catch (multiplying by 1 changes nothing, squaring 0 and 1 changes nothing). Respect every condition in the stem: if the question says x is negative, picking a positive number tests a different question. And avoid a value that makes two choices come out equal — if that happens, pick another number and try again.

How to practice this

• Work the free Mathematics Knowledge practice set with a pencil, writing every step down — the habit is the point.
• Drill the procedures you get wrong, not the ones you already know. Sorting your errors by procedure is more useful than counting them.
• Check every equation by substitution and every geometry answer by its unit. If you adopt one habit from this page, adopt that one.
• Read the explanation on every question, including the ones you got right.

How to study for the ASVAB has the rest of the method, and what the pass adds.

Where to go next

• Next topic — Electronics Information
• The exam itself — how the ASVAB is scored, how many questions, and what makes the ASVAB hard.

Mathematics Knowledge carries 15 of the 135 scored questions on the computer test — the published subtest table's count — and it is one of the four subtests the AFQT score is computed from.

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