ASVAB Mathematics Knowledge: Formulas, Traps and Timing

Mathematics Knowledge has the narrowest scope of any subtest on the ASVAB, which makes it the best return on study time anywhere on the test.

Every question is a procedure with a right way to do it, the procedures are all things a high school algebra class covers, and the list of them is finite. That is nothing like Arithmetic Reasoning, where the difficulty is in reading a paragraph correctly and no amount of formula memorizing helps. Here, learning the procedure is what produces the points.

It also happens to be one of two subtests that pull triple duty. Mathematics Knowledge counts toward your AFQT and feeds both the Administrative and Electrical line scores. Points here move the number that decides whether you can enlist and two of the four numbers that decide what you can do afterwards.

The short version:

• 15 questions, 31 minutes on the computer version - about 2 minutes each
• No calculator, on this or any other subtest
• Counts toward your AFQT, plus the Administrative and Electrical line scores
• Content stops at first-year algebra and basic geometry. No trig, no calculus
• You are not given a formula sheet. The formulas are on you
• Most lost points are sign errors, order of operations and forgotten formulas - not concepts

Mathematics Knowledge versus Arithmetic Reasoning

Learn this distinction before anything else, because it determines which subtest you should be studying.

To apply that table, work twenty practice questions from each subtest and look at what the wrong answers have in common.

• Wrong on Mathematics Knowledge, right on Arithmetic Reasoning. You can read a problem but you have forgotten procedures. Study rules and formulas. This is the easier hole to fill, and the fastest points on the whole ASVAB.
• Right on Mathematics Knowledge, wrong on Arithmetic Reasoning. Your math is fine and your translation is not. Formula drills will do nothing for you. Go work on Arithmetic Reasoning instead.
• Wrong on both. Start here anyway. Mathematics Knowledge is more teachable, and the arithmetic fluency you build carries over.

The first group is the largest of the three, and it is the one with the most points left on the table.

Where the score goes

Two composites plus the AFQT. Arithmetic Reasoning is the only other subtest that does this much, which is why if you only have time to study two subtests, these are the two. Check the minimums for the job you want in the career field library and you will usually find the Administrative or Electrical number is what stands between you and it.

The formulas worth memorizing

You are not handed a formula sheet, so this table is the one to commit to memory.

Nine lines. Use 3.14 for pi unless the answer choices are written in terms of pi, in which case leave pi in place and save yourself the multiplication.

Two things that are not formulas but belong on the same index card:

• The angles in a triangle add to 180 degrees, and in any four-sided figure to 360.
• A radius is half the diameter. A startling share of circle errors are just this, and the wrong answer you get from using the diameter as the radius will be among the choices.

What is actually on the subtest

Roughly in order of how often each turns up:

1. Order of operations and signed numbers. Negatives, absolute value, and the sequence of parentheses, exponents, multiplication and division, then addition and subtraction.
2. Solving linear equations. One variable, isolate it. The bread and butter of the subtest.
3. Geometry with a formula. Area, perimeter, volume, circles, angles, the Pythagorean theorem.
4. Exponents and roots. Multiplying and dividing powers, a negative exponent, a zero exponent, simplifying a square root.
5. Factoring and expanding. Distributing, pulling out a common factor, factoring a simple quadratic, and the difference of two squares.
6. Inequalities. Same as equations with one extra rule that catches almost everyone.
7. Systems of two equations. Two unknowns, solved by substitution or elimination.
8. Ratios, proportions and functions. Cross-multiplying, and evaluating something like f(x) at a given value.
9. Basic probability. Favorable outcomes over total outcomes, and simple independent events.

The list does not grow. There is no hidden tenth category waiting for you on test day, which is why this subtest rewards a finite study plan in a way General Science never will.

Where people lose points

Almost none of these are conceptual. They are execution errors, and execution errors are fixable.

Sign errors on negatives. The most expensive mistake on the subtest, by a distance. Subtracting a negative, distributing a negative across a parenthesis, squaring a negative. Negative 3 squared is 9; the negative of 3 squared is negative 9, and the notation decides which one you are looking at. When you distribute a minus sign, write every changed sign down instead of doing it in your head.

Order of operations. Given 2 plus 3 x 4, the answer is 14 and the answer 20 is sitting right there in the choices. Multiplication and division happen before addition and subtraction, and they happen left to right between themselves. Slow down for four seconds on any expression with mixed operations.

Forgetting a formula you knew last month. This is not a math failure, it is a storage failure, and it accounts for a lot of missed geometry. The fix is the index card above, reviewed for two minutes a day for a week. Nobody derives the volume of a cylinder under time pressure.

Flipping an inequality, or failing to. When you multiply or divide both sides of an inequality by a negative number, the direction reverses. Divide negative 2x is greater than 6 by negative 2 and you get x is less than negative 3, not greater than. This one rule is worth its own five minutes because the test knows people forget it and builds a choice out of the unreversed version.

Area versus perimeter, and area versus volume. Read the units in the answer choices. Square units mean area, plain units mean a length or a perimeter, cubic units mean volume. This is a free check and it catches the error before you commit to it.

Rushing the arithmetic once the algebra is done. A correctly built equation still loses the point when 14 times 6 comes out wrong. Two minutes per question is enough to do the last step carefully.

How to work one in two minutes

1. Identify the procedure before you write anything. Name it: factoring, solving for x, area of a circle. Naming it stops you from wandering into a longer method than you need.
2. Look at the answer choices. On this subtest they often tell you what form the answer takes, and sometimes they are close enough together that you can eliminate two by rough size alone.
3. Work it on paper, one step per line. Compressed work is where sign errors hide. One line per step costs you five seconds and saves the question.
4. Consider back-solving. With four numeric choices and a messy equation, plugging in a choice is often faster than solving forward. Start with the middle values.
5. Check the units and re-read what was asked. Perimeter or area. x or 2x. Four seconds.

Two minutes is comfortable with the procedures in hand and nowhere near enough without them. The binding constraint on this subtest is knowledge, not speed.

What to study, in order

1. Order of operations and signed-number arithmetic. Everything else sits on top of this. Get it automatic before you touch anything harder.
2. The formula table above. Memorize it outright. Cheapest points on the subtest.
3. Solving linear equations in one variable. Including fractions and parentheses.
4. Exponent rules. Multiplying, dividing, negative and zero exponents, and simplifying roots.
5. Factoring. Common factors, the difference of two squares, and simple quadratics.
6. Inequalities, with the negative-multiplication rule drilled until it is reflexive.
7. Systems of two equations. Learn substitution properly, then elimination.
8. Probability and functions. Small slices. Learn them last, and only if the rest is solid.

Do all of it without a calculator from the first practice question. Practicing with one builds a skill you will not have in the room and hides the arithmetic weaknesses you need to find.

Question counts and timing, both versions

Nearly everyone takes the computer version at a MEPS. It adapts, so a right answer pulls the next question harder, and a run of difficult questions indicates you have been answering correctly. You cannot skip or return to a question, so there is no point banking easy ones for later. Work each one in order and keep an eye on the clock only if you notice yourself over four minutes on anything.

What counts as a good score

Mathematics Knowledge is reported as a standard score with 50 as the average, and about two thirds of test takers between 40 and 60. Two things then happen to that number.

First, it goes into the AFQT along with Arithmetic Reasoning and Verbal Expression. The Air Force requires a higher AFQT than the other services for high school graduates, and higher still if you are enlisting with a GED, so this subtest carries real weight on the question of whether you get in at all.

Second, it goes into your Administrative and Electrical composites, which are what a career field actually checks. Those minimums run from the high 30s to the mid 60s depending on the job.

The career table below lists the enlisted fields with an Administrative or Electrical minimum and the number each one wants. Pick the job, read the number, then decide how much studying you owe this subtest. Chasing a generic good score is how applicants end up qualified to enlist but not qualified for the job they wanted.