Mathematics Knowledge asks you to use high-school math concepts directly. Begin by solving an equation without losing track of its structure, then review the rules behind exponents, geometry and number operations.
What you’ll learn
Learn the method
Solve 4(2x − 3) = 20 for x.
The entire expression in parentheses is multiplied by four. Divide both sides by four to get 2x − 3 = 5.
Add three to both sides: 2x = 8. Then divide both sides by two: x = 4. Each operation preserves the equality.
Substitute four for x: 4(2 × 4 − 3) = 4(8 − 3) = 4 × 5 = 20. Both sides agree.
x = 4. Undoing operations and substitution give a complete solve-and-check method.
If you choose to distribute first, four must multiply both terms: 4(2x − 3) = 8x − 12. Writing 8x − 3 changes the original expression and leads to a different equation.
Why can you divide the original equation by four without changing its solution?
You divide both equal sides by the same nonzero number. Dividing only the left side would break the equality. Here it simplifies the outside multiplication before you handle the terms inside.
The example teaches one method. Open a concept below when you need a rule or a different type of example during practice.
Perform the same operation on both sides. To solve 3x + 5 = 20, subtract five from both sides and then divide by three, giving x = 5. Substitute that value into the original equation to check it. With inequalities, multiplication or division by a negative number reverses the inequality sign.
Multiplication of powers with the same base adds exponents: x² × x³ = x⁵. Raising a power to a power multiplies exponents: (x²)³ = x⁶. Addition follows neither rule: x² + x³ cannot generally be combined into one power. Parentheses also matter: (−3)² is 9, while −3² means −(3²), or −9.
Perimeter measures the boundary, area measures a surface and volume measures space. For a rectangle of sides six and four, the perimeter is 20 units and the area is 24 square units. For a triangle, the height in one-half × base × height must be perpendicular to the chosen base, even when the diagram is tilted.
Label known lengths and angles before calculating. In a right triangle, the side opposite the right angle is the hypotenuse, and a² + b² = c² applies with that side as c. On a coordinate grid, slope is change in y divided by change in x; use the same point order in both differences.
Now apply what you know
Try a few questions without opening the solutions, then explain why each answer works. Use scratch paper for calculations and work without a calculator. These original, untimed questions sample the subject; the result is not an official score.
Work through one question at a time. Explain your approach, check the worked answer, then move on. Use scratch paper for calculations.
0 of 12 checked · 0 correct.
Progress is saved in this browser. These original study questions do not produce an official AFQT score.
Answer: B. 2n + 1
An even integer is twice an integer. Thus 2n, 2n + 2 = 2(n + 1), and 4n = 2(2n) are all even. Adding 1 to the even number 2n gives the odd integer 2n + 1. This reasoning also applies when n is zero or negative.
Answer: C. y = 2x − 5
Reflection across the x-axis keeps each x-coordinate and reverses the sign of its y-coordinate. Negate the entire original expression: y = −(−2x + 5) = 2x − 5. Changing only the intercept would give y = −2x − 5 and would not reflect the whole line.
Answer: C. 17
Set the expressions equal: 2x + 7 = 5x − 8. Then 15 = 3x, so x = 5. Substituting gives 2(5) + 7 = 17 and 5(5) − 8 = 17. The question asks for the expressions' common value, not the value 5 of the variable.
Answer: C. 25
Since x is positive, √(4x) = 2√x. The equation becomes 3√x = 15, so √x = 5 and x = 25. Five is the square root of the solution, not the solution itself. Substitution confirms √25 + √100 = 5 + 10 = 15.
Answer: C. 7
Multiplication gives x² + 5x + 2x + 10 = x² + 7x + 10, so b = 7. Ten is the constant term from 2 × 5; it is not the coefficient of x. The two cross terms must be added.
Answer: B. 68°
The angle above m and to the left of the transversal corresponds to the 112° angle, so it also measures 112°. That angle and x form a straight angle above m. Thus x = 180° − 112° = 68°. Choosing 112° identifies the corresponding angle on the wrong side of the transversal.
Answer: A. 2
Every prime exponent in a perfect square is even. The exponent of 3 is already even, but the exponent of 2 is 3. Multiplying by 2 makes 2⁴ × 3² = 144 = 12². Adding a factor of 3 would make its exponent odd instead.
Answer: C. 31/45
It is shorter to count the complement. The same-color probability is (5/10)(4/9) + (3/10)(2/9) + (2/10)(1/9) = 28/90 = 14/45. Thus the different-color probability is 1 − 14/45 = 31/45. The 14/45 choice is the opposite event. The denominator for the second draw is 9 because the first counter is not replaced.
Answer: D. 7
A constant term contains no variable. In 7 − 3x, that term is 7. The number −3 is the coefficient of x, not the constant term. Setting x = 0 also leaves 7, but the distinction is between the variable term −3x and the constant 7.
Answer: A. −3/2
Perpendicular nonvertical lines have slopes whose product is −1. The negative reciprocal of 2/3 is −3/2, since (2/3)(−3/2) = −1. Changing only the sign gives −2/3, which is not the required reciprocal.
Answer: B. −9
Write the integers as n, n + 1, and n + 2. Their sum is 3n + 3 = −24, so 3n = −27 and n = −9. The middle integer is −8, but the question asks for the least. The integers −9, −8, and −7 do sum to −24.
Answer: A. −1/(xy)
Negative exponents give x⁻¹ − y⁻¹ = 1/x − 1/y = (y − x)/(xy). Since y − x = −(x − y), dividing by x − y leaves −1/(xy). The restriction x ≠ y permits that cancellation, and the two nonzero-variable restrictions preserve the reciprocals. The positive 1/(xy) loses the minus sign when reversing the numerator difference.
Before moving on, choose one error you can explain. Rework that question with the solution closed, then revisit the skill in a later session to check what you remember.
For each missed skill, record the rule, a small example and the condition under which the rule applies. For instance, note that the Pythagorean theorem requires a right triangle. Practice recalling the method without looking at the sheet.
Substitute a solution back into an equation, multiply to check division, or estimate a geometric result. A second method can reveal an error that repeating the same calculation would preserve.
Choose a focused lesson in fractions, percentages, ratios, algebra or geometry →
Mathematics Knowledge contributes to the AFQT alongside Arithmetic Reasoning, Word Knowledge and Paragraph Comprehension. It is also part of several job-qualification composites. Your raw result on this practice set is not an official standard score.
The computer test includes unscored tryout questions in selected subtests. The longer time shown is the total for a section with tryouts, not extra time to add. These figures describe the proctored CAT-ASVAB, not untimed PiCAT or our practice sets. See the official What to Expect page's questions and timing table and the guide to test versions.
Sources and materials. Subtest descriptions and timing were checked on September 29, 2026 against the Department of Defense’s What to Expect guide, sample questions and score guide on officialasvab.com. Teaching examples, diagrams and practice questions are independently authored Test Ninjas materials. Test Ninjas is not affiliated with or endorsed by the Department of Defense.