ASVAB — All Questions

A number increased by 25 percent equals 90. What is the number?

• a.65
• b.70
• c.75
• d.72✓

The result is 125 percent of the number, so the number is 90 / 1.25 = 72. Checking, 25 percent of 72 is 18, and 72 + 18 = 90.

What is the value of (3^2)^3?

• a.27
• b.729✓
• c.243
• d.81

Raising a power to a power multiplies the exponents, giving 3^6 = 729.

In a right triangle, one acute angle measures 35 degrees. What does the other acute angle measure?

• a.55 degrees✓
• b.65 degrees
• c.45 degrees
• d.145 degrees

The two acute angles of a right triangle add to 90 degrees, so the second is 90 - 35 = 55 degrees.

What is the value of 8 + 2 x (6 - 2)^2 / 4?

• a.16✓
• b.12
• c.40
• d.24

Work inside the parentheses first: 6 - 2 = 4, and 4 squared is 16. Then 2 x 16 = 32, 32 / 4 = 8, and 8 + 8 = 16.

What is the reciprocal of 2/7?

• a.7/2✓
• b.1/7
• c.2/7
• d.-2/7

A reciprocal flips the numerator and denominator, so 2/7 becomes 7/2. Their product is 1.

Two triangles are similar. The smaller has sides of 3, 4, and 5 inches, and the larger has a shortest side of 12 inches. How long is the larger triangle's longest side?

• a.20 inches✓
• b.25 inches
• c.15 inches
• d.16 inches

The scale factor is 12 / 3 = 4, so every side is multiplied by 4. The longest side becomes 5 x 4 = 20 inches.

Solve for x: 2(x - 3) = x + 5

• a.16
• b.8
• c.14
• d.11✓

Distributing gives 2x - 6 = x + 5. Subtracting x from both sides and adding 6 gives x = 11.

Using 3.14 for pi, what is the volume of a cylinder with a radius of 3 inches and a height of 10 inches?

• a.282.6 cubic inches✓
• b.314.0 cubic inches
• c.188.4 cubic inches
• d.94.2 cubic inches

Volume equals pi times radius squared times height: 3.14 x 9 x 10 = 282.6 cubic inches.

What is 5/4 expressed as a percent?

• a.125%✓
• b.145%
• c.45%
• d.80%

Dividing 5 by 4 gives 1.25, and moving the decimal two places right gives 125 percent. Any fraction greater than 1 exceeds 100 percent.

If f(x) = 2x^2 - 3x, what is f(4)?

• a.20✓
• b.26
• c.12
• d.44

Substituting 4 gives 2(16) - 3(4) = 32 - 12 = 20.

The sum of three consecutive even integers is 78. What is the largest of the three?

• a.24
• b.28✓
• c.26
• d.30

Calling the smallest x gives 3x + 6 = 78, so x = 24 and the integers are 24, 26, and 28. Their sum checks out at 78.

What is 3/4 divided by 2/3?

• a.8/9
• b.1/2
• c.3/2
• d.9/8✓

Dividing by a fraction means multiplying by its reciprocal: 3/4 x 3/2 = 9/8.

Two angles are supplementary, and one is three times the size of the other. How large is the smaller angle?

• a.45 degrees✓
• b.60 degrees
• c.135 degrees
• d.30 degrees

Supplementary angles add to 180 degrees. Splitting 180 into 4 equal parts gives 45 for the smaller angle and 135 for the larger.

For x not equal to 3, simplify (x^2 - x - 6) / (x - 3).

• a.x + 3
• b.x - 6
• c.x + 2✓
• d.x - 2

The numerator factors as (x - 3)(x + 2), and the (x - 3) terms cancel. Testing at x = 10 gives 84 / 7 = 12, which matches 10 + 2.

What is the least common multiple of 12 and 18?

• a.216
• b.54
• c.36✓
• d.6

Multiples of 18 are 18, 36, 54, and 36 is the first one that 12 also divides. The value 6 is the greatest common factor, not the least common multiple.

A cube has a volume of 216 cubic inches. How long is one edge?

• a.36 inches
• b.6 inches✓
• c.12 inches
• d.8 inches

The edge is the cube root of the volume, and 6 x 6 x 6 = 216 inches.

What is the value of 40 - 6 x (9 - 5)^2 / 8?

• a.12
• b.28✓
• c.37
• d.68

Parentheses first, then the exponent, then multiply and divide left to right, and add or subtract last. 9 - 5 = 4 and 4^2 = 16; then 6 x 16 = 96 and 96 / 8 = 12, so 40 - 12 = 28. Skipping the exponent leaves 37, and doing 40 - 6 before multiplying gives 68.

A quantity increases from 40 to 50. What is the percent increase?

• a.25%✓
• b.10%
• c.80%
• d.20%

Percent change is the change divided by the ORIGINAL amount, times 100. The change is 50 - 40 = 10, and the original is 40, so 10 / 40 = 0.25, or 25 percent. Dividing by the new value 50 gives 20 percent, dividing the original by the new gives 80 percent, and 10 is the raw change, not a percent.

If x = 2, what is the value of (x^3)(x^4) / x^2?

• a.64
• b.2
• c.32✓
• d.8

Multiplying powers of the same base ADDS exponents; dividing SUBTRACTS them. So (x^3)(x^4) = x^7, then x^7 / x^2 = x^5, and 2^5 = 32. Dividing each factor by x^2 separately leaves x times x^2 = 8, subtracting the 4 instead of adding it leaves x^1 = 2, and multiplying 3 by 4 then halving gives x^6 = 64.

When (x + 6)(x - 2) is multiplied out and written as x^2 + bx + c, what is the value of b?

• a.6
• b.-12
• c.8
• d.4✓

In FOIL the Outer and Inner products combine into the middle term. Outer gives -2x and Inner gives 6x, so the middle term is 6x - 2x = 4x and b = 4. The Last product 6 x -2 = -12 is c, not b. Adding 6 + 2 without carrying the negative sign gives 8, and reading b straight off the 6 in the first factor gives 6.

What is the value of 3|4 - 9| - |2 - 6|?

• a.1
• b.19
• c.3
• d.11✓

Finish the subtraction INSIDE the bars first, then drop the sign. 4 - 9 = -5, so |4 - 9| = 5 and 3 x 5 = 15. Next 2 - 6 = -4, so |2 - 6| = 4, and 15 - 4 = 11. Adding that second term instead of subtracting gives 19, dropping the coefficient 3 gives 5 - 4 = 1, and multiplying both bars by 3 gives 15 - 12 = 3.

Solve for x: 7x - 4 = 3x + 20

• a.4
• b.6✓
• c.8
• d.2

Gather the variable terms on one side and the numbers on the other. Subtracting 3x leaves 4x - 4 = 20; adding 4 gives 4x = 24, so x = 6. Subtracting the 4 instead of adding it gives x = 4, adding the x terms into 10x while leaving the 4 behind gives 10x = 20 and x = 2, and dividing 24 by 3 rather than 4 gives 8.

Which of these is 0.000362 written in scientific notation?

• a.3.62 x 10^-4✓
• b.3.62 x 10^-5
• c.3.62 x 10^-3
• d.36.2 x 10^-5

Scientific notation keeps exactly one nonzero digit in front of the decimal point, and the exponent counts how many places the point moved. Going from 0.000362 to 3.62 moves it four places right, so the exponent is -4. A leading 36.2 breaks the one-digit rule, and miscounting the places by one gives -3 or -5.

What is the value of (the square root of 8) x (the square root of 18)?

• a.5.1
• b.6
• c.12✓
• d.26

The product of two square roots is the square root of the product: sqrt(8) x sqrt(18) = sqrt(8 x 18) = sqrt(144) = 12. Writing them as 2sqrt2 and 3sqrt2 and then multiplying only the whole numbers gives 6. Adding the numbers under the radicals instead of multiplying gives sqrt(26), about 5.1, and 26 drops the radical.

In the system 3x + 2y = 19 and x - 2y = -7, solve for x.

• a.6.5
• b.4
• c.3✓
• d.5

Adding the two equations eliminates y, because +2y and -2y cancel. The left sides give 3x + x = 4x and the right sides give 19 + (-7) = 12, so 4x = 12 and x = 3. Dividing that 12 by the 3 from the first equation rather than by the combined 4 gives 4, and substituting back gives 2y = 10 and y = 5, the other variable.

If f(x) = 3x^2 - 2x + 5, what is f(-2)?

• a.-3
• b.45
• c.13
• d.21✓

Replace every x with -2 and watch the signs. (-2)^2 = 4, so 3 x 4 = 12; then -2 times -2 = +4; then 12 + 4 + 5 = 21. Treating (-2)^2 as -4 gives -12 + 4 + 5 = -3, losing the sign on the middle term gives 13, and squaring 3x as a unit gives 36 + 4 + 5 = 45.

Evaluate: 6 + 18 / 3 x 2 - 4

• a.12
• b.20
• c.14✓
• d.-6

Multiplication and division have EQUAL rank and are worked left to right, ahead of the addition and subtraction. So 18 / 3 = 6, then 6 x 2 = 12, then 6 + 12 - 4 = 14. Adding before multiplying gives (6 + 6) x 2 - 4 = 20, reading strictly left to right from the first 6 gives 12, and doing 2 - 4 first gives 6 - 12 = -6.

Which expression is the complete factorization of 4x^2 + 6x?

• a.2x(2x + 3)✓
• b.4x(x + 6)
• c.x(4x + 3)
• d.2x(x + 3)

Pull out the greatest common factor, which means dividing EVERY term by it. The largest factor shared by 4x^2 and 6x is 2x: 4x^2 / 2x = 2x and 6x / 2x = 3. Always check by distributing back: 2x(x + 3) rebuilds only 2x^2 + 6x, 4x(x + 6) rebuilds 4x^2 + 24x, and x(4x + 3) rebuilds 4x^2 + 3x.

If (x - 4)(x + 7) = 0, what is the sum of the two solutions?

• a.4
• b.3
• c.11
• d.-3✓

The zero product rule says at least one factor must equal zero. From x - 4 = 0 comes x = 4, and from x + 7 = 0 comes x = -7, so the sum is 4 + (-7) = -3. Reading the roots straight off the parentheses as -4 and 7 flips both signs and gives 3, adding 4 + 7 ignores signs, and 4 is only one of the two roots.

What is 0.045 written as a percent?

• a.0.45%
• b.450%
• c.4.5%✓
• d.45%

To convert a decimal to a percent, multiply by 100, which slides the decimal point two places to the RIGHT. So 0.045 becomes 4.5 percent. Sliding it only one place gives 0.45 percent, sliding it three places gives 45 percent, and multiplying by 100 a second time gives 450 percent.

Simplify: 4(2x - 3) - 2(x - 5)

• a.6x + 2
• b.6x - 2✓
• c.6x - 22
• d.10x - 2

Distribute each factor, and carry the minus sign in front of the second group through BOTH of its terms. 4(2x - 3) = 8x - 12, and -2(x - 5) = -2x + 10. Combining, 8x - 2x = 6x and -12 + 10 = -2, giving 6x - 2. Distributing that minus as -2x - 10 produces 6x - 22 instead.

The variable y varies inversely with x. When x = 3, y = 24. What is y when x = 8?

• a.72
• b.9✓
• c.3
• d.64

Inverse variation means the PRODUCT xy stays constant, so find that constant first: 3 x 24 = 72. Then 8y = 72 and y = 9. The constant 72 is a step, not the answer. Treating the pair as direct variation instead gives 24 / 3 x 8 = 64, which makes y rise with x when it should fall.

What is the value of (5^0 + 4^-1) x 8?

• a.42
• b.10✓
• c.40
• d.2

Any nonzero base raised to the zero power equals 1, and a negative exponent means take the reciprocal, so 4^-1 = 1/4 = 0.25. Then (1 + 0.25) x 8 = 10. Treating 5^0 as 0 gives 0.25 x 8 = 2, reading 4^-1 as 4 gives 5 x 8 = 40, and treating 5^0 as 5 gives 5.25 x 8 = 42.

What is the smallest integer value of x that satisfies -3x + 5 < 14?

• a.2
• b.-4
• c.-2✓
• d.-3

Isolate the variable term first: subtracting 5 from both sides gives -3x < 9. Dividing both sides by a NEGATIVE number reverses the inequality, so x > -3. The smallest integer strictly greater than -3 is -2, because -3 itself makes the two sides equal rather than unequal. Skipping the flip points at -4 instead.

If |2x - 6| = 10, what is the greatest possible value of x?

• a.16
• b.-2
• c.8✓
• d.2

An absolute value equation splits into two cases, because the inside can be positive or negative. From 2x - 6 = 10 comes 2x = 16 and x = 8; from 2x - 6 = -10 comes 2x = -4 and x = -2. The greater is 8. Forgetting to divide by 2 leaves 16, and subtracting 6 from 10 rather than adding gives 2.

Given y = 2x - 3 and 3x + y = 17, solve for y.

• a.4
• b.8
• c.5✓
• d.1

Substitute the first equation into the second: 3x + (2x - 3) = 17, so 5x - 3 = 17, 5x = 20 and x = 4. But y is what was asked for, so finish the job: y = 2(4) - 3 = 5. Stopping at x reports 4, forgetting to subtract the 3 gives 8, and dropping the coefficient 2 gives 4 - 3 = 1.

If a = -3 and b = 4, what is the value of 2a^2 - ab + b?

• a.10
• b.-2
• c.52
• d.34✓

Square before multiplying, and remember a negative times a negative is positive. a^2 = 9, so 2a^2 = 18. Then ab = -3 x 4 = -12, so subtracting it adds 12. Finally 18 + 12 + 4 = 34. Squaring 2a as one unit gives 36 + 12 + 4 = 52, losing the double negative gives 10, and treating a^2 as -9 gives -2.

What is (4 x 10^6) / (8 x 10^2) written as a whole number?

• a.500
• b.5000✓
• c.0.5
• d.2

Divide the coefficients and SUBTRACT the exponents: 4 / 8 = 0.5 and 6 - 2 = 4, giving 0.5 x 10^4. Shifting the decimal one place left of 10,000 makes that 5000. Moving the point the wrong way instead gives 500, stopping at the coefficient gives 0.5, and inverting that division gives 2.

Which expression is equivalent to (2x - 5)(x + 3)?

• a.2x^2 + 6x
• b.2x^2 + x - 15✓
• c.2x^2 - 15
• d.x^2 + x - 15

FOIL produces four products: 2x times x is 2x^2, 2x times 3 is 6x, -5 times x is -5x, and -5 times 3 is -15. The two middle terms combine as 6x - 5x = x. Multiplying only First and Last leaves 2x^2 - 15, writing 2x times x as x^2 gives x^2 + x - 15, and stopping after First and Outer gives 2x^2 + 6x.

What is the value of the square root of 0.36?

• a.0.6✓
• b.0.06
• c.0.1296
• d.0.18

Look for the number that multiplies by itself to give 0.36. Since 6 x 6 = 36, and 0.6 x 0.6 = 0.36, the root is 0.6. A square root carries HALF the decimal places of the original, so two places in means one place out and 0.06 has too many. Squaring instead of rooting gives 0.1296, and halving 0.36 gives 0.18.

Solve for x: (2x + 1)/5 = 3

• a.7✓
• b.14
• c.1
• d.8

Clear the denominator first by multiplying both sides by 5, which gives 2x + 1 = 15. Subtract 1 to get 2x = 14, then divide by 2 for x = 7. Forgetting the final division leaves 14, adding the 1 instead of subtracting gives 16 / 2 = 8, and never multiplying by 5 gives 2x + 1 = 3 and x = 1.

28 is what percent of 80?

• a.35%✓
• b.0.35%
• c.286%
• d.52%

Percent equals the part divided by the whole, times 100, and the whole is the number that follows the word of. So 28 / 80 = 0.35, which is 35 percent. Dividing 80 by 28 reverses the pair and gives about 286 percent, subtracting 80 - 28 = 52 treats a difference as a percent, and 0.35 skips the times 100.

What is the value of 5 + 2[9 - 3(4 - 2)]?

• a.17
• b.29
• c.11✓
• d.49

Work from the innermost grouping outward. First 4 - 2 = 2, then 3 x 2 = 6, then 9 - 6 = 3 inside the brackets. Now 2 x 3 = 6 and 5 + 6 = 11. Doing 9 - 3 first gives 6 x 2 = 12 and an answer of 29, and ignoring the brackets to compute 5 + 18 - 6 gives 17.

For the relationship y = kx, y = 20 when x = 8. What is y when x = 14?

• a.11.4
• b.35✓
• c.5.6
• d.26

In y = kx the constant k is the ratio y / x, and it never changes. Here k = 20 / 8 = 2.5, so y = 2.5 x 14 = 35. Adding the change in x rather than scaling gives 20 + 6 = 26. Treating the pair as an inverse relationship gives 8 x 20 / 14, about 11.4, and inverting the ratio gives 8 / 20 x 14 = 5.6.

Which expression is 3 times the square root of 12 with no perfect square left under the radical?

• a.3 times the square root of 6
• b.2 times the square root of 3
• c.6 times the square root of 3✓
• d.6 times the square root of 2

Split the number under the radical into a perfect square times a leftover: 12 = 4 x 3, so the square root of 12 equals 2 times the square root of 3. Multiplying by the 3 already outside gives 6 times the square root of 3. Stopping at 2 times the square root of 3 forgets that outside factor entirely.

If 3x^2 - 27 = 0, what is the positive value of x?

• a.9
• b.24
• c.81
• d.3✓

Isolate the squared term before taking any root. Adding 27 to both sides gives 3x^2 = 27, and dividing by 3 gives x^2 = 9, so x = 3 or x = -3, and the positive value is 3. Reporting 9 stops at x^2, squaring 9 instead of rooting it gives 81, and 27 - 3 = 24 mixes up the two operations.

If |x + 3| = 7, what is the sum of the two possible values of x?

• a.-6✓
• b.4
• c.14
• d.-10

Split the equation into two cases: x + 3 = 7 gives x = 4, and x + 3 = -7 gives x = -10. Their sum is 4 + (-10) = -6. A useful shortcut: the two roots sit the same distance either side of -3, so their sum is always twice that center. Adding 7 + 7 gives 14, and each single root answers only half the question.

If f(x) = 4x - 1 and g(x) = x + 3, what is f(g(2))?

• a.19✓
• b.7
• c.10
• d.20

Work the INNER function first, then feed its output into the outer one. g(2) = 2 + 3 = 5, and f(5) = 4 x 5 - 1 = 19. Order matters here: reversing it and computing g(f(2)) gives 7 + 3 = 10. Stopping at f(2) alone gives 7, and forgetting to subtract the 1 gives 20.

Solve for x: 0.25x + 1.5 = 4

• a.22
• b.2.5
• c.10✓
• d.6

Handle a decimal coefficient exactly as you would a whole one. Subtracting 1.5 from both sides gives 0.25x = 2.5, and dividing by 0.25 gives x = 10, since 0.25 goes into 2.5 ten times. Adding the 1.5 instead gives 5.5 / 0.25 = 22, and dividing 1.5 by 0.25 before subtracting gives 6.

What is the value of 47^2 - 43^2?

• a.90
• b.16
• c.8100
• d.360✓

A difference of two squares factors as (a - b)(a + b), which avoids squaring anything large. Here 47 - 43 = 4 and 47 + 43 = 90, so the value is 4 x 90 = 360. Squaring the difference instead, as (47 - 43)^2, gives 16, squaring the sum gives 8100, and stopping at one factor alone reports 90.