Mathematical Principles - Armed Services Vocational Aptitude Battery ASVAB

What is the value of $5 + 3 \times (8 - 2)$ ?
Answer: 23 Following the order of operations (PEMDAS/BODMAS), first solve the expression in the parentheses: $8 - 2 = 6$ . The equation becomes $5 + 3 \times 6$ . Next, perform the multiplication: $3 \times 6 = 18$ . Finally, perform the addition: $5 + 18 = 23$ .
If $3x - 7 = 11$ , what is the value of $x$ ?
Answer: 6 To solve for $x$ , first add 7 to both sides of the equation: $3x - 7 + 7 = 11 + 7$ , which simplifies to $3x = 18$ . Then, divide both sides by 3: $3x \div 3 = 18 \div 3$ , which gives $x = 6$ .
What is the area of a rectangle with a length of 12 inches and a width of 5 inches?
Answer: 60 square inches The formula for the area of a rectangle is Area = length × width. In this case, Area = 12 inches × 5 inches = 60 square inches.
A square has a side length of 9 meters. What is its perimeter?
Answer: 36 meters The perimeter of a square is the sum of the lengths of all four equal sides, or $P = 4s$ . With a side length (s) of 9 meters, the perimeter is $4 \times 9 = 36$ meters.
A cylindrical storage tank has a volume of 96π cubic meters and a height of 6 meters. What is the diameter of the tank's base?
Answer: 8 meters When you encounter cylinder volume problems on the ASVAB, you're working with the formula $V = \pi r^2 h$ , where V is volume, r is radius, and h is height. Since the question asks for diameter, remember that diameter equals twice the radius. Starting with the given information: volume = 96π cubic meters and height = 6 meters. Substitute these values into the volume formula: $96\pi = \pi r^2 \cdot 6$ Divide both sides by π: $96 = 6r^2$ Divide by 6: $16 = r^2$ Take the square root: $r = 4$ meters Since diameter = 2 × radius, the diameter is 2 × 4 = 8 meters. Looking at the wrong answers: Choice A (4 meters) is the radius, not the diameter - this is a common trap where students forget to double their result. Choice C (12 meters) might result from incorrectly adding radius and height (4 + 6 = 10) or other calculation errors. Choice D (16 meters) equals r², which could happen if you solved for r² but forgot to take the square root, then doubled that incorrect value. The correct answer is B (8 meters). For ASVAB geometry problems, always double-check what the question is asking for. Many problems give you enough information to calculate radius, but then ask for diameter, circumference, or area. Write down the target measurement before you start calculating, and make sure your final answer matches what was requested.
What is $\frac{1}{2} + \frac{1}{3}$ ?
Answer: $\frac{5}{6}$ To add fractions, you must find a common denominator. The least common denominator for 2 and 3 is 6. Convert the fractions: $\frac{1}{2} = \frac{3}{6}$ and $\frac{1}{3} = \frac{2}{6}$ . Now add the new numerators: $\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$ .
What is the circumference of a circle with a diameter of 10 cm? (Use $\pi \approx 3.14$ )
Answer: 31.4 cm The formula for the circumference of a circle is $C = \pi \times d$ , where d is the diameter. Using the given values, $C \approx 3.14 \times 10\text{ cm} = 31.4\text{ cm}$ .
Express $\frac{3}{5}$ as a percentage.
Answer: 60% To convert a fraction to a percentage, first convert the fraction to a decimal by dividing the numerator by the denominator: $3 \div 5 = 0.6$ . Then, multiply the decimal by 100 to get the percentage: $0.6 \times 100\% = 60\%$ .
If two angles are supplementary and one angle measures 65 degrees, what is the measure of the other angle?
Answer: 115 degrees Supplementary angles are two angles that add up to 180 degrees. To find the measure of the other angle, subtract the known angle from 180: $180^{\circ} - 65^{\circ} = 115^{\circ}$ .
A rectangular box has a length of 6 feet, a width of 4 feet, and a height of 3 feet. What is its volume?
Answer: 72 cubic feet The formula for the volume of a rectangular box (prism) is Volume = length × width × height. In this case, Volume = 6 ft × 4 ft × 3 ft = 72 cubic feet.
Simplify the expression $y^5 \cdot y^3$ .
Answer: $y^8$ When multiplying exponential expressions with the same base, you add the exponents. Therefore, $y^5 \cdot y^3 = y^{5+3} = y^8$ .
A cylindrical tank has a radius of 3 feet and a height of 8 feet. What is the volume of the tank?
Answer: $72\pi$ cubic feet When you encounter cylinder volume problems on the ASVAB, you're working with one of the most common 3D geometry formulas: $V = \pi r^2 h$ , where r is the radius and h is the height. For this cylindrical tank, you have a radius of 3 feet and height of 8 feet. Substituting into the formula: $V = \pi \times 3^2 \times 8 = \pi \times 9 \times 8 = 72\pi$ cubic feet. Let's examine why the other answers represent common mistakes. Answer B ( $48\pi$ ) occurs when students forget to square the radius, calculating $\pi \times 3 \times 8 \times 2 = 48\pi$ . This treats the radius like it appears once in the formula rather than squared. Answer C ( $24\pi$ ) results from the same error but without the extra factor of 2: $\pi \times 3 \times 8 = 24\pi$ . Answer D ( $96\pi$ ) happens when students incorrectly use diameter instead of radius in their calculation, computing $\pi \times 6^2 \times 8 \times \frac{1}{3}$ or make similar algebraic errors that inflate the result. The correct answer is A: $72\pi$ cubic feet. ASVAB Strategy: Always write down the cylinder volume formula first: $V = \pi r^2 h$ . Double-check that you're squaring the radius (not just multiplying by 2) and using radius, not diameter. The arithmetic test often includes these geometry problems, so memorizing this formula and practicing careful substitution will serve you well.
Which of the following numbers is a prime number?
Answer: 17 A prime number is a whole number greater than 1 that has only two divisors: 1 and itself. 9 is divisible by 3. 15 is divisible by 3 and 5. 21 is divisible by 3 and 7. 17 is only divisible by 1 and 17.
Factor completely: $x^{3}-9x$ .
Answer: $x(x-3)(x+3)$ When you encounter a polynomial factoring problem, start by looking for common factors among all terms, then check for special patterns like difference of squares. For $x^3 - 9x$ , first factor out the greatest common factor. Both terms contain $x$ , so factor it out: $x^3 - 9x = x(x^2 - 9)$ . Now examine what's left inside the parentheses: $x^2 - 9$ . This is a difference of squares pattern: $a^2 - b^2 = (a-b)(a+b)$ . Since $x^2 - 9 = x^2 - 3^2$ , it factors as $(x-3)(x+3)$ . Therefore, the complete factorization is $x(x-3)(x+3)$ , which is answer D. Let's see why the other options are incorrect: A) $x(x^3-9)$ only factors out the $x$ but leaves $x^3-9$ unfactored. This expression can be factored further, so it's not completely factored. B) $(x-3)(x^2+3)$ doesn't include the factor of $x$ that should come from the original expression. If you expand this, you get $x^3 + 3x - 3x^2 - 9$ , which doesn't equal the original expression. C) $x(x-3)(x-3) = x(x-3)^2$ expands to $x^3 - 6x^2 + 9x$ , which is completely different from the original expression. Remember: always factor completely by first pulling out common factors, then looking for special patterns. Verify your answer by expanding it back to the original expression to catch any mistakes.
What is the y-intercept of the line given by the equation $y = 4x - 5$ ?
Answer: -5 The equation is in slope-intercept form, $y = mx + b$ , where 'm' is the slope and 'b' is the y-intercept. In the equation $y = 4x - 5$ , the value of 'b' is -5. Therefore, the y-intercept is -5.
Solve $2x^{2}-7x+3=0$ by factoring.
Answer: $x=3$ or $x=\dfrac12$ When you encounter a quadratic equation like $2x^2 - 7x + 3 = 0$ , factoring is often the most efficient solution method. You're looking for two binomials that multiply to give you the original quadratic. To factor $2x^2 - 7x + 3$ , you need two numbers that multiply to give $ac = (2)(3) = 6$ and add to give $b = -7$ . Those numbers are $-6$ and $-1$ since $(-6) \times (-1) = 6$ and $(-6) + (-1) = -7$ . Rewrite the middle term: $2x^2 - 6x - x + 3 = 0$ Factor by grouping: $2x(x - 3) - 1(x - 3) = 0$ This gives you: $(2x - 1)(x - 3) = 0$ Using the zero product property, set each factor equal to zero: - $2x - 1 = 0 \Rightarrow x = \frac{1}{2}$ - $x - 3 = 0 \Rightarrow x = 3$ Therefore, $x = 3$ or $x = \frac{1}{2}$ , which matches choice A. Choice B gives negative values for both solutions, which would result from factoring errors or sign mistakes. Choice C combines the correct positive solutions but with an incorrect negative sign on one. Choice D mixes one correct positive value with incorrect negative values. Study tip: Always check your factored form by expanding it back to the original equation, and verify your solutions by substituting them into the original quadratic. Both $x = 3$ and $x = \frac{1}{2}$ should make $2x^2 - 7x + 3 = 0$ true.
A quadrilateral has angles measuring 100°, 80°, and 90°. What is the measure of the fourth angle?
Answer: 90° The sum of the interior angles of any quadrilateral is 360°. First, sum the known angles: $100° + 80° + 90° = 270°$ . Then, subtract this sum from 360° to find the fourth angle: $360° - 270° = 90°$ .
If $2x + 5 > 15$ , which of the following must be true?
Answer: $x > 5$ To solve the inequality, first subtract 5 from both sides: $2x + 5 - 5 > 15 - 5$ , which simplifies to $2x > 10$ . Then, divide both sides by 2: $2x \div 2 > 10 \div 2$ , which results in $x > 5$ .
A triangle has two angles that measure 40° and 60°. What is the measure of the third angle?
Answer: 80° The sum of the interior angles of any triangle is always 180°. To find the third angle, add the two known angles and subtract the result from 180. $40° + 60° = 100°$ . Then, $180° - 100° = 80°$ .
What is the principal square root of 144?
Answer: 12 The principal square root is the non-negative number that, when multiplied by itself, equals the original number. Since $12 \times 12 = 144$ , the principal square root of 144 is 12.

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