ASVAB — All Questions

Which decimal equals the sum 1/4 + 3/5?

• a.0.74
• b.0.44
• c.0.85✓
• d.0.15

Turn each fraction into a decimal, then add: 1/4 = 0.25 and 3/5 = 0.6, so the total is 0.85. Adding numerators and denominators straight across to get 4/9 produces about 0.44, multiplying rather than adding produces 3/20 = 0.15, and misreading 1/4 as 0.14 produces 0.74.

If 3x + 2y = 16 and x = 2y, what is the value of y?

• a.8
• b.2✓
• c.3.2
• d.4

Substitute 2y everywhere x appears: 3(2y) + 2y = 16, which is 6y + 2y = 8y = 16, so y = 2. Then x = 2y = 4, which answers the other variable. Dropping the coefficient 3 during the substitution gives 5y = 16 and about 3.2, and dividing 16 by 2 alone gives 8.

If x^2 + 5x = 24, what is the negative solution for x?

• a.3
• b.-8✓
• c.-6
• d.-3

Move everything to one side before factoring: x^2 + 5x - 24 = 0. Now find two numbers whose product is -24 and whose sum is +5, namely +8 and -3, so (x + 8)(x - 3) = 0 and x = -8 or x = 3. The pair 6 and -4 also multiplies to -24 but sums to 2, and it points at -6 instead. Flipping one sign gives -3.

If 3/(x + 2) = 9/24, what is the value of x?

• a.6✓
• b.22
• c.8
• d.2.7

Cross multiply the proportion: 3 x 24 = 9(x + 2), so 72 = 9x + 18, then 9x = 54 and x = 6. Check it, since 3/8 really does equal 9/24. A shortcut reduces 9/24 to 3/8 and forces x + 2 = 8; stopping there and reporting 8 skips the last subtraction, and 24 / 9 = 2.7 divides the wrong pair.

If f(x) = 5x + 8 and f(a) = 43, what is the value of a?

• a.223
• b.35
• c.5.1
• d.7✓

When the OUTPUT is given, set the rule equal to it and solve backwards: 5a + 8 = 43. Subtracting 8 gives 5a = 35, and dividing by 5 gives a = 7. Stopping after the subtraction reports 35, adding the 8 instead gives 51 / 5, about 5.1, and substituting 43 for x gives 5 x 43 + 8 = 223.

If x = 1/2, what is the value of 8x^2 + 4x - 1?

• a.9
• b.5
• c.4
• d.3✓

Square the fraction BEFORE multiplying: (1/2)^2 = 1/4, so 8 x 1/4 = 2. Then 4 x 1/2 = 2, and 2 + 2 - 1 = 3. Multiplying 8 by 1/2 rather than by 1/4 gives 4 + 2 - 1 = 5, doubling x instead of squaring it gives 8 + 2 - 1 = 9, and forgetting to subtract the 1 leaves 2 + 2 = 4.

A trapezoid has parallel sides measuring 8 inches and 14 inches, and the distance between those two sides is 6 inches. What is its area?

• a.84 square inches
• b.66 square inches✓
• c.56 square inches
• d.48 square inches

The area of a trapezoid is the average of the two parallel sides times the height: (8 + 14) / 2 = 11, and 11 x 6 = 66 square inches. Using only the longer base, 14 x 6, gives 84, using only the shorter one gives 48, and halving the product of the two parallel sides gives 56.

A parallelogram has a base of 12 centimeters, a slanted side of 9 centimeters, and a perpendicular height of 7 centimeters. What is its area?

• a.54 square centimeters
• b.84 square centimeters✓
• c.63 square centimeters
• d.42 square centimeters

Area of a parallelogram is base times perpendicular height, never base times the slanted side: 12 x 7 = 84 square centimeters. Taking the 9-centimeter slanted side as the height and then halving as though the figure were a triangle gives 54, halving alone gives 42, and 9 x 7 gives 63.

A square and an equilateral triangle each have a perimeter of 48 inches. How much longer is one side of the triangle than one side of the square?

• a.16 inches
• b.28 inches
• c.12 inches
• d.4 inches✓

Each side of a regular figure is the perimeter divided by the number of sides. The triangle has 3 sides, so 48 / 3 = 16 inches; the square has 4, so 48 / 4 = 12 inches. The difference is 16 - 12 = 4 inches. The values 16 and 12 are the sides themselves, and 28 is their sum.

A triangle has an area of 54 square inches and a base of 12 inches. What is its height?

• a.4.5 inches
• b.9 inches✓
• c.2.25 inches
• d.27 inches

Since area = 1/2 x base x height, rearranging gives height = 2 x area / base: 2 x 54 / 12 = 9 inches. Dividing the area by the base and forgetting to double it first gives 4.5 inches, and halving the area instead of doubling it gives 2.25 inches.

A rectangular floor measures 20 feet by 14 feet. A square rug 8 feet on a side lies flat on it. How many square feet of the floor are left uncovered?

• a.248 square feet
• b.272 square feet
• c.216 square feet✓
• d.280 square feet

Work out both areas and subtract. The floor is 20 x 14 = 280 square feet and the rug is 8 x 8 = 64 square feet, so 280 - 64 = 216 square feet stay uncovered. Subtracting the rug perimeter of 32 feet gives 248, and 280 is the floor before the rug is taken out.

A circular flower bed covers 154 square feet. Using 22/7 for pi, what is its radius?

• a.49 feet
• b.7 feet✓
• c.14 feet
• d.24.5 feet

Run the area formula backwards. Area = pi x radius squared, so divide the area by pi: 154 divided by 22/7 is the same as 154 x 7 / 22 = 49, and that 49 is the radius squared. The radius is the square root of 49, or 7 feet. Stopping at 49 skips the square root, and 14 feet is the diameter.

A circle has a radius of 10 inches. Using 3.14 for pi, how long is an arc that cuts off a central angle of 72 degrees?

• a.6.28 inches
• b.62.8 inches
• c.31.4 inches
• d.12.56 inches✓

An arc is the same fraction of the circumference as its central angle is of a full 360 degrees. The circumference is 2 x 3.14 x 10 = 62.8 inches, and 72/360 is one fifth of that, or 12.56 inches. The value 62.8 is the whole way round, 31.4 is half, and one fifth of 31.4 gives 6.28.

A lawn sprinkler waters a quarter of a circle with a radius of 12 feet. Using 3.14 for pi, how large an area does it water?

• a.452.16 square feet
• b.226.08 square feet
• c.18.84 square feet
• d.113.04 square feet✓

Find the whole circle first, then take the stated fraction of it: 3.14 x 12 x 12 = 452.16 square feet, and a quarter of that is 113.04 square feet. The value 452.16 is the full circle, 226.08 is a half, and 18.84 is a quarter of the circumference rather than the area.

The circumference of a circle is 37.68 inches. Using 3.14 for pi, what is its radius?

• a.3 inches
• b.18.84 inches
• c.12 inches
• d.6 inches✓

Circumference = 2 x pi x radius, so radius = circumference / (2 x pi): 37.68 / 6.28 = 6 inches. Dividing by pi alone gives 12 inches, which is the diameter and not the radius, and halving the circumference gives 18.84, which is not a length across the circle at all.

A circle is inscribed in a square whose sides measure 10 inches, touching all four sides. Using 3.14 for pi, how much of the square lies outside the circle?

• a.68.6 square inches
• b.214 square inches
• c.78.5 square inches
• d.21.5 square inches✓

The inscribed circle has a diameter equal to the side of the square, so its radius is 5 inches. The square is 10 x 10 = 100 square inches and the circle is 3.14 x 5 x 5 = 78.5, leaving 21.5 square inches. Subtracting the circumference 31.4 instead of the area gives 68.6.

The diagonal of a square measures 12 inches. What is the area of the square?

• a.144 square inches
• b.288 square inches
• c.72 square inches✓
• d.36 square inches

A diagonal splits a square into two right triangles, so side squared plus side squared equals diagonal squared, which makes the area equal to the diagonal squared divided by 2: 12 x 12 / 2 = 72 square inches. Treating the diagonal as a side gives 144, and doubling rather than halving gives 288.

In a 30-60-90 right triangle, the side opposite the 30 degree angle measures 7 inches. How long is the hypotenuse?

• a.12 inches
• b.10 inches
• c.14 inches✓
• d.21 inches

The sides of a 30-60-90 triangle are always in the ratio 1 to the square root of 3 to 2, with the shortest side opposite the 30 degree angle. The hypotenuse is twice that short side: 7 x 2 = 14 inches. About 12 inches is the longer leg, and 10 inches fits a 45-45-90 triangle instead.

A 25-foot ladder leans against a wall with its base 7 feet out from the wall. How high up the wall does the top of the ladder reach?

• a.32 feet
• b.18 feet
• c.24 feet✓
• d.26 feet

The ladder is the hypotenuse, so square it and subtract the known leg: 625 - 49 = 576, and the square root of 576 is 24 feet. Adding the squares instead of subtracting gives about 26 feet, which is longer than the ladder, and simply working out 25 - 7 gives 18 feet.

A rectangular field measures 40 yards by 30 yards. Cutting across the diagonal instead of walking along two sides saves how many yards?

• a.10 yards
• b.30 yards
• c.50 yards
• d.20 yards✓

Walking two sides covers 40 + 30 = 70 yards. The diagonal is the hypotenuse of a 30-40 right triangle: 1600 + 900 = 2500, whose square root is 50 yards. The saving is 70 - 50 = 20 yards. The value 50 is the diagonal itself, and 10 is just 40 - 30.

A right triangle has a hypotenuse of 13 centimeters and one leg of 5 centimeters. What is its area?

• a.78 square centimeters
• b.60 square centimeters
• c.65 square centimeters
• d.30 square centimeters✓

Find the missing leg first: 169 - 25 = 144, so that leg is 12 centimeters. In a right triangle the two legs serve as base and height, so the area is 5 x 12 / 2 = 30 square centimeters. Forgetting to halve gives 60, and using the 13-centimeter hypotenuse as a side gives 78 or 65.

A closed rectangular box measures 6 inches long, 5 inches wide, and 4 inches high. What is its total surface area?

• a.148 square inches✓
• b.120 square inches
• c.296 square inches
• d.74 square inches

A box has three pairs of matching faces, so surface area = 2 x (length x width + length x height + width x height): 2 x (30 + 24 + 20) = 148 square inches. Counting only one face of each pair gives 74, and 120 is the volume, 6 x 5 x 4, which answers a different question.

A cylindrical tank has a diameter of 10 feet and a height of 6 feet. Using 3.14 for pi, what is its volume?

• a.188 cubic feet
• b.471 cubic feet✓
• c.1,884 cubic feet
• d.157 cubic feet

Volume of a cylinder is pi x radius squared x height, and the radius is half the given diameter: 3.14 x 5 x 5 x 6 = 471 cubic feet. Using the 10-foot diameter as the radius gives 1,884, four times too large; 188 is the curved side area and 157 is the cone with the same base and height.

A cone has a base radius of 3 inches and a height of 4 inches. Using 3.14 for pi, what is its volume?

• a.37.68 cubic inches✓
• b.56.52 cubic inches
• c.12.56 cubic inches
• d.50.24 cubic inches

A cone holds exactly one third of the cylinder with the same base and height: pi x radius squared x height / 3 = 3.14 x 3 x 3 x 4 / 3 = 37.68 cubic inches. Halving the cylinder instead of taking a third gives 56.52, and dividing by 3 a second time gives 12.56.

A sphere has a radius of 6 inches. Using 3.14 for pi, what is its volume?

• a.678.24 cubic inches
• b.452.16 cubic inches
• c.226.08 cubic inches
• d.904.32 cubic inches✓

Volume of a sphere is 4/3 x pi x radius cubed. Cube the radius first: 6 x 6 x 6 = 216, then 4/3 x 3.14 x 216 = 904.32 cubic inches. Dropping the 4/3 gives 678.24, and 452.16 is 4 x pi x radius squared, which is the sphere surface area rather than its volume.

A cube has 6 congruent square faces and a total surface area of 96 square inches. What is its volume?

• a.64 cubic inches✓
• b.96 cubic inches
• c.16 cubic inches
• d.24 cubic inches

Divide the surface area among the faces to get one face: 96 / 6 = 16 square inches. An edge is the square root of that face, 4 inches, so the volume is 4 x 4 x 4 = 64 cubic inches. Stopping at the face area gives 16, and 96 is the surface area the question started from.

Two parallel lines are cut by a transversal. One interior angle measures 118 degrees. How large is the interior angle on the same side of the transversal?

• a.28 degrees
• b.118 degrees
• c.72 degrees
• d.62 degrees✓

Interior angles on the same side of a transversal crossing parallel lines are supplementary, so they add to 180 degrees: 180 - 118 = 62 degrees. Repeating 118 is the answer for corresponding or alternate interior angles, which are equal, and 28 is the complement rather than the supplement.

A regular polygon has 8 equal sides and 8 equal interior angles. How large is each interior angle?

• a.45 degrees
• b.135 degrees✓
• c.1,080 degrees
• d.120 degrees

The interior angles of a polygon with n sides total (n - 2) x 180 degrees, so here (8 - 2) x 180 = 1,080 degrees. Because the polygon is regular, divide by the 8 angles: 1,080 / 8 = 135 degrees each. The 1,080 is the total, and 45 degrees is one exterior angle, 360 / 8.

The exterior angles of any polygon add up to 360 degrees. Each exterior angle of a certain regular polygon measures 24 degrees. How many sides does the polygon have?

• a.15 sides✓
• b.7 sides
• c.24 sides
• d.12 sides

In a regular polygon every exterior angle is the same size and together they total 360 degrees, so the number of sides is 360 / 24 = 15. Using 180 instead of 360 gives 7.5, which is not even a whole number of sides, and 12 comes from halving the correct count.

In a triangle, the exterior angle at one vertex measures 130 degrees, and the two interior angles not touching that vertex are equal to each other. How large is each of them?

• a.130 degrees
• b.25 degrees
• c.50 degrees
• d.65 degrees✓

An exterior angle equals the sum of the two non-adjacent interior angles. Those two are equal here, so each is 130 / 2 = 65 degrees. The value 50 degrees is the interior angle right beside the exterior one, 180 - 130, and 25 degrees comes from halving that 50 instead.

A clock face is a circle divided into 12 equal hour marks. What is the measure of the smaller angle between the hour hand and the minute hand at exactly 4:00?

• a.30 degrees
• b.90 degrees
• c.240 degrees
• d.120 degrees✓

One hour mark covers 360 / 12 = 30 degrees. At 4:00 the hands sit 4 marks apart, so the angle is 4 x 30 = 120 degrees. The value 30 degrees is a single mark, and 240 degrees is the reflex angle measured the long way round, which is the larger of the two angles.

A man 6 feet tall casts a 4-foot shadow at the same moment a flagpole casts a 22-foot shadow. How tall is the flagpole?

• a.33 feet✓
• b.24 feet
• c.44 feet
• d.15 feet

The sun creates similar triangles, so height divided by shadow is the same for both: 6 / 4 = 1.5, and 1.5 x 22 = 33 feet. Flipping the ratio and working out 4 / 6 x 22 gives about 15 feet, which would make the pole shorter than its own shadow, and 24 comes from 22 + 6 - 4.

Two similar rectangles have corresponding sides in the ratio 2 to 5. The smaller rectangle has an area of 12 square inches. What is the area of the larger one?

• a.2 square inches
• b.60 square inches
• c.75 square inches✓
• d.30 square inches

Areas of similar figures grow with the square of the scale factor, not the factor itself: 5/2 squared is 25/4, and 12 x 25/4 = 75 square inches. Applying the ratio only once gives 30, using 5 on its own gives 60, and turning the ratio upside down before squaring gives about 2.

Two similar cylinders have heights in the ratio 1 to 3. The smaller one holds 5 gallons. How much does the larger one hold?

• a.27 gallons
• b.15 gallons
• c.45 gallons
• d.135 gallons✓

Volume grows with the cube of the linear ratio: 3 x 3 x 3 = 27, so the larger holds 5 x 27 = 135 gallons. Scaling just once gives 15 gallons, which is the rule for a length, squaring gives 45 gallons, the rule for a surface, and 27 is the multiplier and not the volume.

In triangle ABC, D lies on side AB and E lies on side AC, with DE parallel to BC. AD is 4 units, DB is 6 units, and DE is 8 units. How long is BC?

• a.48 units
• b.13 units
• c.12 units
• d.20 units✓

DE parallel to BC makes triangle ADE similar to triangle ABC, and AD matches the whole side AB, not the piece DB. AB = 4 + 6 = 10, so the scale factor is 10/4, and BC = 8 x 10/4 = 20 units. Comparing 8 against the ratio 6/4 gives 12 units, and against 10/6 gives about 13.

Two similar triangles have corresponding sides in the ratio 3 to 7. The perimeter of the smaller triangle is 24 inches. What is the perimeter of the larger one?

• a.10 inches
• b.56 inches✓
• c.72 inches
• d.131 inches

A perimeter is just a total length, so it scales by the plain ratio rather than its square: 24 x 7/3 = 56 inches. Squaring the ratio, which is the rule for areas, gives about 131 inches, and turning the ratio upside down gives about 10 inches, smaller than the original.

What are the coordinates of the midpoint of the segment joining (-3, 5) and (7, 1)?

• a.(4, 6)
• b.(2, 6)
• c.(5, 2)
• d.(2, 3)✓

A midpoint is the average of the endpoints, handled one coordinate at a time: x = (-3 + 7) / 2 = 2 and y = (5 + 1) / 2 = 3, giving (2, 3). Adding without halving gives (4, 6), and halving the difference of the coordinates rather than their sum gives (5, 2).

At what value of x does the line y = -2x + 9 cross the x-axis?

• a.x = 4.5✓
• b.x = 9
• c.x = 2
• d.x = -9

A line meets the x-axis where y is 0, so set -2x + 9 = 0, move the term across to get 2x = 9, and divide: x = 4.5. The number 9 is the y-intercept, where the line crosses the other axis, and -9 comes from moving the 9 across but never dividing by the coefficient.

What is the slope of a line perpendicular to the line that passes through the points (0, 5) and (2, 0)?

• a.0.4✓
• b.2.5
• c.-2.5
• d.-0.4

First get the slope of the given line, rise over run: (0 - 5) / (2 - 0) = -2.5. A perpendicular line has the negative reciprocal, so flip it to 2/5 and switch the sign, giving 0.4. Only flipping leaves -0.4, and only switching the sign leaves 2.5.

A rectangle has vertices at (1, 2), (9, 2), (9, 8), and (1, 8). What is its area?

• a.48 square units✓
• b.28 square units
• c.14 square units
• d.24 square units

Read the side lengths straight off the coordinates. The horizontal side runs from x = 1 to x = 9, a length of 8, and the vertical side runs from y = 2 to y = 8, a height of 6. Area is 8 x 6 = 48 square units. The value 28 is the perimeter, 2 x (8 + 6), and 14 is 8 + 6.

The endpoints of a diameter of a circle are (2, 3) and (10, 9). What is the radius of that circle?

• a.7 units
• b.5 units✓
• c.10 units
• d.8 units

The distance between the two endpoints is the diameter. The legs are 10 - 2 = 8 and 9 - 3 = 6, so the distance is the square root of 64 + 36, which is 10. The radius is half of that, 5 units. Stopping at 10 units reports the diameter, 8 units is only one leg, and 7 is the average of the legs.

What is the greatest common factor of 48 and 72?

• a.48
• b.24✓
• c.12
• d.16

Break both numbers into primes: 48 = 2 x 2 x 2 x 2 x 3 and 72 = 2 x 2 x 2 x 3 x 3. Take the primes they share, three 2s and one 3, and multiply: 8 x 3 = 24. The value 12 is a common factor but not the greatest, 16 divides 48 but not 72, and 48 is merely the smaller number.

How many prime numbers are there between 20 and 40?

• a.5
• b.3
• c.4✓
• d.6

A prime has no factors besides 1 and itself, so test the odd numbers only. Ruling out 21, 27, 33 and 39 as multiples of 3, and 25 and 35 as multiples of 5, leaves 23, 29, 31 and 37, so the count is 4. Counting 27 or 33 by mistake is how people arrive at 5 or 6.

Which of the numbers 4,518, 3,724, 5,231 and 6,142 is divisible by 9?

• a.3,724
• b.6,142
• c.4,518✓
• d.5,231

A number divides evenly by 9 exactly when the sum of its digits is a multiple of 9. For 4,518 the digits give 4 + 5 + 1 + 8 = 18, which is 9 x 2, so it works. The other digit sums come to 16, 11 and 13, none of them multiples of 9, so no long division is needed.

In how many different orders can 4 runners cross the finish line if there are no ties?

• a.10 orders
• b.16 orders
• c.12 orders
• d.24 orders✓

Arrangements of n distinct items come to n factorial. There are 4 choices for first place, 3 left for second, 2 for third and 1 for last, so 4 x 3 x 2 x 1 = 24 orders. Working out 4 x 4 gives 16, which would count orders only if a runner could repeat, and adding gives 10.

If n is an odd integer, which of the expressions n + 2, 3 x n, n x n and n + 5 must be an even integer?

• a.3 x n
• b.n + 5✓
• c.n x n
• d.n + 2

Odd plus odd is even, odd plus even is odd, and odd times odd is odd. Adding 5, itself odd, to an odd n gives an even result every time. Adding 2 keeps it odd, and both 3 x n and n x n multiply two odd numbers, so they stay odd. Try n = 7: the values are 9, 21, 49 and 12.