AFOQT Math Knowledge practice and strategy

Math Knowledge format and timing

Source: AFOQT Information Pamphlet (Form T, updated 1 Aug 2015), Tables 1–2. Math Knowledge is subtest 4 of 12, in Part A. The test has been computer-only (the E-AFOQT) since 2023; whether that changed any timing isn't published, so these are the Form T figures our timed sets use. Our drills are written for scratch paper and no calculator.

What Math Knowledge tests

Math without a story: you get the expression, equation or figure directly. Our bank covers fractions and decimals, exponents and roots, scientific notation, factoring, linear and quadratic equations, systems, inequalities, absolute value, functions, ratios, primes and factors, angles and triangles, the Pythagorean theorem, circles, area and volume, and coordinate geometry. No topic list is published, so that's our coverage. Most of it is high-school algebra and geometry; the speed comes from knowing the rules cold.

The technique: rule, simplify, check

1. Name the rule before you compute: slope is rise over run, a² + b² = c², the exponent laws, the area formula.
2. Simplify first. Factor, cancel and combine; keep roots and exponents in their structure instead of converting to decimals.
3. One step per line, with every sign written. Most algebra errors are sign errors.
4. Use the options. Plug them in, estimate their size, compare their form.
5. Re-read the question. Is it asking for x or 2x? Area or perimeter? Which statement is NOT true?
6. Time-box it. No path after about 30 seconds? Enter your best guess and move on; revisit it only if the software lets you go back and time remains.

Three worked problems

Example 1. Simplify (x³)² · x⁴ ÷ x⁵.
A. x⁴ · B. x⁵ · C. x⁷ · D. x¹⁵ · E. x¹⁹

Rules: raising a power to a power multiplies the exponents (3 × 2 = 6), multiplying like bases adds them (6 + 4 = 10), and dividing subtracts (10 − 5 = 5). Answer B, x⁵. A comes from adding 3 + 2 instead of multiplying, D adds every exponent, and E multiplies 6 × 4 before dividing.

Example 2. √50 + √18 = ?
A. 8 · B. √68 · C. 8√2 · D. 15√2 · E. 16

Rule: simplify each root before adding by pulling out the largest perfect square. √50 = 5√2 and √18 = 3√2, so the sum is C, 8√2. B adds under the root, which is never allowed: √(a + b) ≠ √a + √b. D multiplies the coefficients instead of adding them, and A drops the √2.

Example 3. Which values of x satisfy 10 − 4x > 2x − 8?
A. x < 3 · B. x > 3 · C. x < −3 · D. x > −3 · E. x < 9

Collect x on the side where its coefficient stays positive: add 4x and 8 to both sides to get 18 > 6x, so x < 3 (A). With no division by a negative, there's no sign to flip. Collect on the left instead (−6x > −18) and you must flip when you divide by −6; B is what you get if you forget. Test a value: x = 0 gives 10 > −8, which is true.

Common traps

• Answering the wrong quantity. You find x, but the question wanted x² − 1 or 2x. The value of x is usually sitting among the options.
• Inequality signs. Multiplying or dividing by a negative flips the sign.
• Exponent and root laws. Powers multiply when raised and add when multiplied; √(a + b) is not √a + √b.
• Area against perimeter, and square units against linear units.
• Distributing a negative. −(x − 3) is −x + 3.
• Extraneous solutions. Check answers to radical and absolute-value equations in the original.

Pacing plan: 25 in 22 minutes

Twenty-two minutes for 25 questions is about 53 seconds each. Many items are 20-second rule checks; bank that time for multi-step algebra. Checkpoint: about 12 answered at 11:00. If you see no path within about 30 seconds, enter your best guess and move on (revisit it only if the software allows going back). Leave nothing blank.

Which composites it feeds

Math Knowledge counts toward Pilot, CSO, ABM, Academic Aptitude and Quantitative: five of the seven composites. It and Table Reading are the only subtests in all three rated composites. Quantitative carries a commissioning minimum of 10 for every line officer, and Pilot, CSO and ABM training each need 25 on the matching composite (DAFMAN 36-2664, Table A2.1). For pilot candidates the Pilot composite also feeds PCSM; see what Pilot score you need. Math skill builds slowly, so start it early.

Composites are percentiles from 1 to 99 against a reference group. The norms aren't published, so any composite we show is an estimate. More in AFOQT scores explained.

Practice it here

Work the five questions above, then a timed Math Knowledge set: 25 questions on the 22-minute clock. Each explanation names the rule behind the question, so keep a short list of the rules you had to look up and drill those first. Check your Quantitative and Pilot estimates with the free diagnostic or a full mock.

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