What it tests, how it is scored, full lessons, and 10 practice questions with answers
What MK Tests
Mathematics Knowledge (MK) tests high-school math directly: algebra, equations and inequalities, exponents and roots, factoring, geometry (area, perimeter, volume, angles, triangles, circles), and basic probability. No calculator is allowed.
Scored questions (CAT-ASVAB)
15
Time limit
31 minutes (about 124 seconds per question)
Part of the AFQT?
Yes
MK is one of the four AFQT sections. It also feeds the Army CL, EL, FA, GM, and ST line scores. See ASVAB scores explained for how section scores become your AFQT and line scores.
How to Score Higher
Memorize the core formulas. The real test gives you none of them.
Plug the answer choices back in when an equation looks messy. One of the four has to work.
Check for extraneous answers. A value that makes a denominator zero is never a solution.
When a figure is scaled by k, areas scale by k squared and volumes by k cubed.
Mathematics Knowledge Lessons
Algebra and geometry concepts. Each lesson below is also in the interactive study guide, where you can take a quiz to check yourself.
Lesson 1 Solving Equations
Equations are like a balance scale. Whatever you do to one side, you MUST do to the other. Once you get this, algebra becomes straightforward.
The Balance Rule
An equation is balanced. To keep it balanced, any operation you do to the left side, you must do to the right side too. Add 5 to the left? Add 5 to the right.
Isolating the Variable
Your goal is to get x (or whatever variable) alone on one side. Do this by 'undoing' operations in reverse order: first add/subtract, then multiply/divide.
Variables on Both Sides
If x appears on both sides, first move all x terms to one side by adding or subtracting. Then solve as usual.
Worked example
Solve: 3x + 7 = 22
Subtract 7 from both sides: 3x = 15
Divide both sides by 3: x = 5
Check: 3(5) + 7 = 15 + 7 = 22 ✓
Tip: Always check your answer by plugging it back in!
Worked example
Solve: 5x - 8 = 2x + 7
Subtract 2x from both sides: 3x - 8 = 7
Add 8 to both sides: 3x = 15
Divide by 3: x = 5
Tip: Move all x terms to the side with the larger coefficient.
Key takeaway: To solve any equation: (1) Move variables to one side, (2) Move numbers to the other side, (3) Divide by the coefficient. Always check your answer.
Lesson 2 Exponents & Radicals
Exponents are just shorthand for repeated multiplication. x³ means x × x × x. Once you learn the rules, these problems become quick wins.
Multiplying Same Base = ADD Exponents
x³ × x⁴ = x⁷. Why? Because (x·x·x) × (x·x·x·x) = x⁷. When multiplying same bases, add the exponents.
Dividing Same Base = SUBTRACT Exponents
x⁸ ÷ x³ = x⁵. You're canceling out 3 of the x's from the 8, leaving 5.
Power of a Power = MULTIPLY Exponents
(x³)² = x⁶. You're squaring x³, which means x³ × x³ = x⁶.
Special Cases
x⁰ = 1 (anything to the zero power is 1). x⁻ⁿ = 1/xⁿ (negative exponent means reciprocal).
Worked example
Simplify: x⁵ × x³
Same base (x), so ADD exponents
x⁵⁺³ = x⁸
Tip: ADD for multiplication, SUBTRACT for division.
Worked example
Simplify: (x⁴)³
Power of a power: MULTIPLY exponents
x⁴ˣ³ = x¹²
Tip: MULTIPLY when you have an exponent raised to another exponent.
Key takeaway: Three rules: Multiply same base → ADD exponents. Divide same base → SUBTRACT. Power of power → MULTIPLY. x⁰ = 1.
Lesson 3 FOIL & Factoring
FOIL is a method for multiplying two binomials. Factoring is the reverse: breaking an expression back into two binomials. Master both!
FOIL: First, Outer, Inner, Last
To multiply (a + b)(c + d): First (a×c), Outer (a×d), Inner (b×c), Last (b×d). Then combine like terms.
Factoring x² + bx + c
Find two numbers that MULTIPLY to c and ADD to b. Those become your factors: (x + ?)(x + ?).
Difference of Squares
a² - b² = (a + b)(a - b). This is a special pattern. Example: x² - 9 = (x + 3)(x - 3).
Worked example
Expand: (x + 3)(x + 5)
First: x × x = x²
Outer: x × 5 = 5x
Inner: 3 × x = 3x
Last: 3 × 5 = 15
Combine: x² + 5x + 3x + 15 = x² + 8x + 15
Tip: Watch your signs! A negative in either binomial changes signs.
Worked example
Factor: x² + 7x + 12
Need two numbers that multiply to 12 and add to 7
3 × 4 = 12 ✓ and 3 + 4 = 7 ✓
Answer: (x + 3)(x + 4)
Tip: Always check by FOILing your answer back out!
Key takeaway: FOIL: First-Outer-Inner-Last, then combine like terms. Factoring: find two numbers that multiply to the constant and add to the middle coefficient.
Lesson 4 Pythagorean Theorem
The Pythagorean Theorem is THE most tested geometry concept. It only works for right triangles (with a 90° angle).
The Formula
a² + b² = c², where c is the hypotenuse (longest side, across from the right angle). a and b are the legs.
If you need to find a leg, rearrange: a² = c² - b². Then take the square root.
Worked example
A right triangle has legs 3 and 4. Find the hypotenuse.
a² + b² = c²
3² + 4² = c²
9 + 16 = 25 = c²
c = 5
Tip: 3-4-5 is the most common Pythagorean triple!
Worked example
A right triangle has hypotenuse 13 and one leg 5. Find the other leg.
a² + 5² = 13²
a² + 25 = 169
a² = 144
a = 12
Tip: Recognize 5-12-13 triple? It saves calculation time!
Key takeaway: a² + b² = c² (c is hypotenuse). Memorize triples: 3-4-5, 5-12-13, 8-15-17. The hypotenuse is ALWAYS the longest side.
Lesson 5 Area & Perimeter
These formulas come up constantly. Memorize them, there's no formula sheet on the ASVAB!
Rectangle
Area = length × width. Perimeter = 2(length) + 2(width) or 2(l + w).
Triangle
Area = ½ × base × height. The height must be perpendicular to the base!
Circle
Area = πr². Circumference = 2πr = πd. Always use RADIUS, not diameter!
Worked example
Find the area of a rectangle with length 8 and width 5.
Area = length × width
Area = 8 × 5 = 40 square units
Tip: Area is always in square units (sq ft, sq m, etc.)
Worked example
Find the area of a circle with diameter 10.
Diameter = 10, so radius = 5
Area = πr² = π(5)² = 25π
Tip: ALWAYS use radius, not diameter. If given diameter, divide by 2 first!
Key takeaway: Rectangle: A = lw, P = 2l + 2w. Triangle: A = ½bh. Circle: A = πr², C = 2πr. Use RADIUS for circles!
Lesson 6 Inequalities
Inequalities are solved just like equations, with ONE critical rule: flip the sign when multiplying or dividing by a negative!
Solving Inequalities
Solve exactly like equations: add, subtract, multiply, divide. But there's one exception...
THE FLIP RULE
When you multiply or divide both sides by a NEGATIVE number, FLIP the inequality sign. > becomes <. ≥ becomes ≤.
Reading Inequality Signs
< means less than. > means greater than. ≤ means less than or equal. ≥ means greater than or equal.
Worked example
Solve: 3x + 5 > 14
Subtract 5: 3x > 9
Divide by 3: x > 3
No flip needed (divided by positive)
Tip: No negative division = no flip needed.
Worked example
Solve: -2x + 4 ≤ 10
Subtract 4: -2x ≤ 6
Divide by -2 AND FLIP: x ≥ -3
Sign changed from ≤ to ≥ because we divided by negative!
Tip: Divided by negative = MUST flip the sign!
Key takeaway: Solve like equations. FLIP the inequality sign when multiplying or dividing by a negative number.
MK Practice Questions
Try each question before opening the answer. On the real CAT-ASVAB the questions adapt to you, so a timed, adaptive practice test is the best way to check your level.
Question 1
What is the area of a triangle with base 8 and height 6?
48 square units
24 square units
14 square units
28 square units
Show answer
Answer: B. 24 square units
Area of a triangle = 1/2 × base × height. Here that is 1/2 × 8 × 6 = 1/2 × 48 = 24 square units. Remember the one-half factor that separates a triangle from a rectangle.
Question 2
In a right triangle, if one leg is 3 and the other is 4, what is the hypotenuse?
5
6
7
12
Show answer
Answer: A. 5
Use the Pythagorean theorem: leg² + leg² = hypotenuse². Here 3² + 4² = 9 + 16 = 25, and the square root of 25 is 5. This is the classic 3-4-5 triangle.
Question 3
Solve for x: 3(x + 1) = 5(x - 2) + 7
-2
2
1/2
3
Show answer
Answer: D. 3
Expand both sides: 3x + 3 = 5x − 10 + 7 = 5x − 3. Subtract 3x and add 3: 6 = 2x, so x = 3. Distribute first, then collect variables on one side.
Question 4
Simplify: 3x + 2 - (4x - 6)
-x + 8
-x - 8
x + 8
x - 8
Show answer
Answer: A. -x + 8
Distribute the negative: 3x + 2 − 4x + 6 = −x + 8. Subtracting a group flips every sign inside it.
Question 5
A rectangle is cut in half to create two squares, each with area 25. What is the perimeter of the original rectangle?
20
25
30
50
Show answer
Answer: C. 30
Each square has area 25, so each side is 5. Two squares side by side form a rectangle 10 by 5, giving perimeter 2(10 + 5) = 30.
Question 6
Simplify: (3⁴) ÷ (3²)
3²
3⁶
3⁸
9
Show answer
Answer: A. 3²
Dividing like bases subtracts exponents: 3⁴ ÷ 3² = 3^(4−2) = 3². Keep the same base and subtract.
Question 7
What is the area of a circle with radius 6? (Use π = 3.14)
18.84 sq units
37.68 sq units
113.04 sq units
56.52 sq units
Show answer
Answer: C. 113.04 sq units
Area of a circle = πr² = 3.14 × 6² = 3.14 × 36 = 113.04 square units. Square the radius, then multiply by π.
Question 8
Solve: x + 5 > 12. What values satisfy this inequality?
x > 7
x < 7
x > 17
x < 17
Show answer
Answer: A. x > 7
Subtract 5 from both sides of x + 5 greater than 12 to get x greater than 7. Solve an inequality like an equation, keeping the direction.
Question 9
Factor: x² - 25
(x + 5)(x + 5)
(x - 5)(x - 5)
(x + 5)(x - 5)
(x + 25)(x - 1)
Show answer
Answer: C. (x + 5)(x - 5)
This is a difference of squares, a² − b² = (a + b)(a − b). With a = x and b = 5, x² − 25 = (x + 5)(x − 5).
Question 10
If the area of a circle is 100π square units, what is the radius?
5 units
10 units
25 units
50 units
Show answer
Answer: B. 10 units
Area of a circle = πr². Set πr² = 100π, so r² = 100 and r = √100 = 10 units. Divide out π, then take the square root.
Test Yourself Under Real Conditions
Take a timed, adaptive Mathematics Knowledge section, the 20-question AFQT Predictor, or a full AFQT practice test. Free, no account needed.