ASVAB — All Questions

Marcus deposits $1,500 in an account paying 8 percent simple interest a year and adds nothing more. After how many years does the balance first reach $2,100?

• a.4 years
• b.3 years
• c.5 years✓
• d.6 years

Find the interest needed, then divide by the interest earned each year. The balance must grow by $2,100 - $1,500 = $600, and simple interest pays 8 percent of the original $1,500, or $120 every year, so $600 / $120 = 5 years. Using 8 percent of the $2,100 target, $168 a year, gives about 4 years.

An account earned $1,050 in simple interest over 5 years at an annual rate of 3.5 percent. What was the principal?

• a.$183.75
• b.$6,000✓
• c.$30,000
• d.$5,250

Rearrange I = P x r x t into P = I / (r x t). Here r x t = 0.035 x 5 = 0.175, and $1,050 / 0.175 = $6,000. Dividing by the rate but not by the 5 years gives $30,000, multiplying by the rate and the years instead of dividing gives $183.75, and $1,050 x 5 gives $5,250.

An investor splits $4,800 between two accounts, putting $3,000 in one that pays 5 percent a year and the rest in one that pays 2 percent a year. What is the total interest after one year?

• a.$336
• b.$96
• c.$240
• d.$186✓

Interest is figured on each piece separately and then added. $3,000 x 0.05 = $150, and the rest, $4,800 - $3,000 = $1,800, earns $1,800 x 0.02 = $36, for $186 total. Putting the whole $4,800 at 5 percent gives $240, all of it at 2 percent gives $96, and adding the rates into 7 percent gives $336.

A roaster blends 12 pounds of coffee that costs $9 a pound with 8 pounds that costs $14 a pound. What is the cost per pound of the blend?

• a.$11.50
• b.$12.00
• c.$11.00✓
• d.$23.00

A blend's unit cost is total cost divided by total weight, weighted toward whichever lot is larger. 12 x $9 = $108 and 8 x $14 = $112, so $220 / 20 pounds = $11.00. Averaging $9 and $14 ignores the weights and gives $11.50, swapping the weights gives $12.00, and adding the prices gives $23.00.

A 6-quart container of coolant is 20 percent antifreeze. How many quarts of pure antifreeze must be added to bring the mixture to 50 percent antifreeze?

• a.1.8 quarts
• b.4.5 quarts
• c.3.6 quarts✓
• d.3.0 quarts

Pure antifreeze adds to the antifreeze and to the total, so set (1.2 + x) / (6 + x) = 0.50. Then 1.2 + x = 3 + 0.5x, so 0.5x = 1.8 and x = 3.6 quarts. Forgetting that the total also grows gives 1.8 quarts, and half of the original 6 quarts gives 3.0.

A 15-gallon drum holds a solution that is 30 percent cleaner. How many gallons of water must be added to weaken the solution to 18 percent cleaner?

• a.9 gallons
• b.10 gallons✓
• c.25 gallons
• d.5 gallons

Adding water changes the total but not the amount of cleaner, so work from that fixed amount. The drum holds 0.30 x 15 = 4.5 gallons of cleaner, and 4.5 / 0.18 = 25 gallons of finished solution, so 25 - 15 = 10 gallons of water go in. Answering 25 gallons reports the new total, and scaling the drum by 18/30 gives 9 gallons.

A grocer wants 30 pounds of a nut mix worth $5.20 a pound, using peanuts at $4.00 a pound and cashews at $7.00 a pound. How many pounds of cashews are needed?

• a.18 pounds
• b.12 pounds✓
• c.10 pounds
• d.15 pounds

Price the whole batch, then see how far the cheap nut falls short. The mix must be worth 30 x $5.20 = $156, while 30 pounds of peanuts is only $120; each pound of cashews swapped in adds $7.00 - $4.00 = $3.00, so $36 / $3.00 = 12 pounds. The remaining 18 pounds are the peanuts, and 15 assumes an even split.

A feed dealer blends 200 pounds of feed that is 12 percent protein with 300 pounds of feed that is 22 percent protein. How many pounds of protein are in the blend?

• a.90 pounds✓
• b.170 pounds
• c.34 pounds
• d.85 pounds

Take the protein out of each lot and add the two amounts: 200 x 0.12 = 24 pounds and 300 x 0.22 = 66 pounds, for 90 pounds. Averaging the percents to 17 and applying that to all 500 pounds gives 85 pounds, while adding the percents into 34 and applying that to 500 pounds gives 170 pounds.

A radiator holds 16 quarts of a mix that is 40 percent antifreeze. A mechanic drains 4 quarts of the mix and replaces them with pure antifreeze. What percent antifreeze is the radiator now?

• a.65 percent
• b.55 percent✓
• c.50 percent
• d.40 percent

Track quarts of antifreeze, not percents. Draining 4 of 16 quarts leaves 12 quarts still 40 percent, which is 4.8 quarts of antifreeze, and the 4 quarts poured back are pure, so 8.8 quarts sit in 16, or 55 percent. Adding 25 percent of the volume straight onto 40 gives 65 percent.

A map is drawn so that 1 inch represents 40 miles. Two bases are 260 miles apart on the ground. How far apart are they on the map?

• a.6.5 inches✓
• b.7.5 inches
• c.8.7 inches
• d.5.2 inches

Going from ground distance to map distance divides by the scale, and only the reverse trip multiplies: 260 / 40 = 6.5 inches. Check the size of the answer, since a bigger scale number means fewer inches. Using 30 miles per inch gives about 8.7 inches and using 50 miles per inch gives 5.2 inches.

A floor plan uses a scale of 1/4 inch to 1 foot. A wall measures 7 inches on the plan. How long is the actual wall?

• a.28 feet✓
• b.14 feet
• c.1.75 feet
• d.21 feet

Each 1/4 inch on the plan stands for 1 foot, so count the quarter inches in 7 inches: 7 / (1/4) = 7 x 4 = 28 feet. Dividing by a fraction smaller than 1 must make the answer larger, so multiplying by 1/4 instead and getting 1.75 feet is the wrong direction; 14 feet doubles rather than quadruples.

A model ship is built to a scale of 1 to 48. The model's mast stands 9 inches tall. How tall is the real mast in feet?

• a.18 feet
• b.36 feet✓
• c.432 feet
• d.3 feet

Scale up first, then convert the units the question asks for. The real mast is 9 x 48 = 432 inches, and 432 / 12 = 36 feet. Answering 432 feet skips the inches-to-feet step, dividing by 12 a second time gives 3 feet, and dividing by 24 instead gives 18 feet.

On a map scaled at 1 inch to 15 miles, a supply route runs 3.2 inches from the depot to a camp and then 4.8 inches from the camp to the field site. How long is the whole route on the ground?

• a.120 miles✓
• b.72 miles
• c.8 miles
• d.48 miles

Add the legs in map inches first, then convert once: 3.2 + 4.8 = 8 inches, and 8 x 15 = 120 miles. Converting only the longer leg gives 72 miles, only the shorter leg gives 48 miles, and 8 miles is the map measurement reported without ever applying the scale.

A landscape drawing uses a scale of 1 inch to 6 feet. On the drawing a garden plot measures 4 inches by 3 inches. What is the actual area of the plot?

• a.72 square feet
• b.12 square feet
• c.216 square feet
• d.432 square feet✓

Convert each side before finding area: 4 inches becomes 24 feet and 3 inches becomes 18 feet, so 24 x 18 = 432 square feet. Area scales by the square of the scale, 36 times over, so multiplying the drawing's 12 square inches by 6 gives only 72 square feet and by 18 gives 216. Reading that 12 as feet converts nothing at all.

A recruit owns 4 shirts, 3 pairs of trousers, and 2 pairs of boots, and any of them go together. How many different complete outfits can he put together?

• a.9 outfits
• b.18 outfits
• c.24 outfits✓
• d.12 outfits

When one item is chosen from each group, the counts multiply: 4 x 3 x 2 = 24 outfits. Each of the 12 shirt-and-trouser pairings can go with either pair of boots, which is where the doubling comes from. Adding the groups gives 9, and 12 stops after the shirts and trousers and forgets the boots.

In the dark, a person reaches into a drawer holding 8 black socks and 6 white socks and pulls them out one at a time. How many socks must be pulled out to be certain of having a matching pair?

• a.3 socks✓
• b.7 socks
• c.8 socks
• d.2 socks

Certainty means planning for the worst possible draw, not the likely one. Only two colors are in the drawer, so the first two socks could be one of each, but a third sock must match one of them: 3 socks guarantee a pair. Answering 2 relies on luck, 7 guarantees one black sock, and 8 guarantees a black pair.

A bin holds 20 batteries, 5 of which are dead. One battery is drawn at random, tested, and put back; then a second is drawn. What is the probability that both draws are dead batteries?

• a.50 percent
• b.25 percent
• c.12.5 percent
• d.6.25 percent✓

Because the first battery goes back in, the draws are independent and the probabilities multiply. Each draw is 5/20 = 25 percent, so both dead is 0.25 x 0.25 = 6.25 percent. Reporting 25 percent answers only one draw, and adding the two chances gives 50 percent, which is far too likely for a double event.

A machine has a 10 percent chance of jamming during any given shift, and the shifts are independent of one another. What is the probability that it jams at least once in 2 shifts?

• a.19 percent✓
• b.10 percent
• c.20 percent
• d.81 percent

For at-least-once, find the chance the event never happens and subtract from 100 percent. No jam on one shift is 90 percent, so no jam on both is 0.90 x 0.90 = 81 percent, leaving 100 - 81 = 19 percent. Adding 10 and 10 gives 20 percent and double counts the shift where it jams twice.

Each of the 6 members of a squad shakes hands exactly once with every other member of that squad. How many handshakes take place in all?

• a.30 handshakes
• b.12 handshakes
• c.36 handshakes
• d.15 handshakes✓

Each of the 6 people shakes with the other 5, which counts 6 x 5 = 30, but that counts every handshake once from each side, so divide by 2 to get 15. Forgetting to halve leaves 30, and 6 x 6 = 36 also lets a person shake his own hand, which the problem does not allow.

A locker code is 3 digits long, each digit chosen from 1 through 5, and a digit may be used more than once. How many different codes are possible?

• a.125 codes✓
• b.15 codes
• c.243 codes
• d.60 codes

With repeats allowed, every position is filled independently from the same 5 digits, so multiply 5 by itself once per position: 5 x 5 x 5 = 125 codes. Multiplying 5 x 3 gives 15 and counts digits rather than codes, barring repeats gives 5 x 4 x 3 = 60, and 3 x 3 x 3 x 3 x 3 reverses the roles for 243.

A cafeteria meal is one main dish, one side, and one drink, chosen from 3 main dishes, 4 sides, and 2 drinks. If the cafeteria adds one more side, how many additional meals become possible?

• a.9 additional meals
• b.30 additional meals
• c.6 additional meals✓
• d.24 additional meals

Count the meals both ways and subtract: 3 x 4 x 2 = 24 now and 3 x 5 x 2 = 30 after, so 6 more. The shortcut is that the new side pairs with each of the 3 mains and 2 drinks, which is the same 6. Reporting 30 gives the new total rather than the increase, and 24 gives the old total.

Crates are stacked so that the bottom row holds 48 crates and each row above holds 5 fewer crates than the row just below it. How many crates are in the 7th row from the bottom?

• a.33 crates
• b.18 crates✓
• c.23 crates
• d.13 crates

Count the gaps between rows, not the rows themselves: the 7th row is only 6 steps above the first, so 48 - 6 x 5 = 18 crates. Subtracting 7 x 5 is the off-by-one error and gives 13, stopping at the 6th row gives 23, and stopping at the 4th gives 33.

A bacteria culture doubles every 3 hours. If it starts with 250 cells, how many cells are present after 12 hours?

• a.2,000 cells
• b.4,000 cells✓
• c.8,000 cells
• d.1,000 cells

Count the doubling periods before doubling anything: 12 / 3 = 4 periods. Then 250 x 2 x 2 x 2 x 2 = 250 x 16 = 4,000 cells. Two doublings gives 1,000, three gives 2,000, and five gives 8,000, so the error is almost always in counting periods rather than in the multiplying.

A sequence begins 3, 7, 15, 31, where each term after the first is found by doubling the term before it and adding 1. What is the next term?

• a.62
• b.63✓
• c.39
• d.47

Apply the stated rule to the last term shown: 31 x 2 + 1 = 63. Test the rule on the earlier terms first, since 3 x 2 + 1 = 7 and 7 x 2 + 1 = 15, which confirms it. Doubling but forgetting the 1 gives 62, and treating the gaps of 4, 8, 16 as if 16 repeated gives 47.

A theater has 14 seats in the first row, and each row behind it has 3 more seats than the row in front of it. How many seats are in the first 8 rows together?

• a.196 seats✓
• b.112 seats
• c.168 seats
• d.280 seats

For an evenly growing list, total = number of terms x (first + last) / 2. The 8th row holds 14 + 7 x 3 = 35 seats, so the total is 8 x (14 + 35) / 2 = 196 seats. Using 14 for every row gives 112, using 35 for every row gives 280, and the true answer always lands between those two.

A sequence begins 2, 5, 7, 12, 19, where each term after the second is the sum of the two terms immediately before it. What is the 8th term?

• a.50
• b.81✓
• c.62
• d.31

There is no shortcut formula here, so extend the rule one term at a time and label each position: 12 + 19 = 31 is the 6th, 19 + 31 = 50 is the 7th, and 31 + 50 = 81 is the 8th. Stopping one term early gives 50, 31 is the 6th term, and doubling 31 gives 62.

A supply sergeant must move 338 sandbags using a cart that carries 24 sandbags at a time. How many trips does it take to move all of them?

• a.14 trips
• b.13 trips
• c.16 trips
• d.15 trips✓

Divide, then decide what the remainder needs: 338 / 24 = 14 full cart loads with 2 sandbags left over. Those 2 still have to be hauled, so the count rounds up to 15 trips. Dropping the remainder gives 14 and leaves 2 bags on the ground, while adding a trip for a remainder that has already been rounded up gives 16.

A baker bakes 5 sheets holding 32 rolls each, then packs the rolls 12 to a bag. How many rolls are left over after every full bag has been packed?

• a.8 rolls
• b.4 rolls✓
• c.13 rolls
• d.16 rolls

Total first, then divide and keep the remainder rather than the quotient: 5 x 32 = 160 rolls, and 160 / 12 = 13 bags with 4 rolls left over. The 13 answers how many bags were filled, not how many rolls remain, and 16 is what is still loose after only 12 bags, one bag short of done.

A shelf is 92 inches long. Storage boxes are 7 inches wide and must sit side by side on the shelf without hanging over the end. How many boxes fit?

• a.13 boxes✓
• b.14 boxes
• c.12 boxes
• d.15 boxes

Only whole boxes fit, so divide and throw the fraction away: 92 / 7 = 13.1, meaning 13 boxes with 1 inch of shelf left unused. Rounding 13.1 up to 14 assumes a partial box can hang over the end, which the problem forbids, and 12 leaves room for another full box.

A crew of 4 people must move 92 boxes, and each person can carry 3 boxes on a trip. How many trips must the crew make?

• a.9 trips
• b.23 trips
• c.8 trips✓
• d.7 trips

Find what the whole crew moves per trip, then round up for the leftovers: 4 x 3 = 12 boxes a trip, and 92 / 12 = 7 trips with 8 boxes still waiting, so an 8th trip is needed. Stopping at 7 strands those 8 boxes, and 23 divides the boxes among the 4 people while ignoring that each carries 3.

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.