AFQT Core Practice 9: Signed Numbers and Supporting Details

Answer: A

Correct answer: 3/7. Add the ratio parts to get the whole:

Diesel trucks are 3 of those 7 parts, so they make up $\frac{3}{7}$ of the trucks.

• A (correct): The ratio has 3 + 4 = 7 parts in all, and 3 of them are diesel.
• B: This compares diesel to gasoline trucks; a fraction of the total must use all 7 parts.
• C: This is the fraction of trucks that run on gasoline.
• D: This is the gasoline-to-diesel ratio written as a fraction; a part of a group cannot be more than the whole group.

Answer: C

Correct answer: 3.75 hours. Minutes are sixtieths of an hour:

So 3 hours 45 minutes is

• A: This treats the minutes as hundredths of an hour, but minutes are sixtieths of an hour.
• B: This divides 60 by 45 instead of 45 by 60.
• C (correct): Since 45 minutes is 45/60 = 0.75 hour, the shift is 3.75 hours.
• D: This finds 225 total minutes but then divides by 100 instead of 60.

Answer: A

Correct answer: 15 gallons per minute. A rate is gallons divided by minutes:

gal/min.

• A (correct): Dividing gallons by minutes gives 210 ÷ 14 = 15 gallons per minute.
• B: This subtracts 14 from 210 instead of dividing.
• C: This misplaces the decimal point; 1.5 × 14 is only 21 gallons.
• D: This adds 14 to 210 instead of dividing.

Answer: B

Correct answer: $300. Use

• A: This is the interest for just one year; it must be multiplied by 3 years.
• B (correct): Interest is 5,000 × 0.02 × 3 = $300.
• C: This uses 0.2 instead of 0.02 for 2%, making the interest ten times too large.
• D: This is the account balance after 3 years; the question asks only for the interest.

Answer: A

Correct answer: 6 hours. First state what one job is: one driveway. Combined rate:

driveway per hour.

For 3 driveways:

• A (correct): Together they seal 1/3 + 1/6 = 1/2 driveway per hour, so 3 driveways take 3 / (1/2) = 6 hours.
• B: This is the time for one driveway; the job here is three driveways.
• C: This averages the two workers' times (4.5 hours) and multiplies by 3 instead of adding rates.
• D: This multiplies 3 driveways by the rate of 1/2 driveway per hour instead of dividing.

Answer: C

Correct answer: 40 minutes. Label the units: minutes over miles on both sides gives

The pace is 8 minutes per mile, so

• A: This puts the proportion upside down, computing 24 × 3 ÷ 5.
• B: This adds the 2 extra miles as 2 extra minutes instead of using the pace.
• C (correct): The pace is 24 ÷ 3 = 8 minutes per mile, and 8 × 5 = 40 minutes.
• D: This is the time for only 4 miles, 24 + 8.

Answer: C

Correct answer: 54 mph. Find the total distance:

so 270 miles in all. Average speed =

More time was spent at 60 mph, so the average is above the simple mean of the two speeds.

• A: This is the simple mean of 45 and 60, which ignores that the bus spent longer at 60 mph.
• B: This swaps the times, giving 3 hours at 45 mph and 2 hours at 60 mph.
• C (correct): Total distance 270 miles divided by total time 5 hours gives 54 mph.
• D: This uses only the speed of the longer leg and ignores the slower 2 hours.

Answer: A

Correct answer: 46 workers. Convert 15 percent to 0.15. The increase is

workers.

New size:

• A (correct): The increase is 0.15 x 40 = 6 workers, so the crew has 40 + 6 = 46.
• B: This is the number of workers added, not the new crew size.
• C: This adds 15 workers instead of 15 percent of 40.
• D: This decreases the crew by 15 percent, but the crew grew.

Answer: A

Correct answer: $840. Only the crew size, hourly rate and hours matter; the stories and the year built are extra facts.

One worker:

Crew:

• A (correct): Each electrician earns 7.5 × $28 = $210, and 4 × $210 = $840.
• B: This is one electrician's pay, not the pay for the whole crew.
• C: This doubles the cost because the house has two stories, but the number of stories does not affect the pay.
• D: This rounds 7.5 hours up to 8 hours for each worker.

Answer: A

Correct answer: 120 miles. If 65 percent is finished, then 100 minus 65, or 35 percent, remains.

Whole equals part divided by percent:

Check: 35 percent of 120 is 42.

• A (correct): The 42 remaining miles are 35 percent of the project, and 42 divided by 0.35 is 120.
• B: This treats 42 miles as 65 percent of the project, but 65 percent is the finished part, not the remaining part.
• C: This is the finished part of a 120-mile project, not the length of the whole project.
• D: This multiplies 42 by 0.35 instead of dividing; the whole must be larger than 42 miles.

Answer: A

Correct answer: 38. Exponent first: 5² = 25. Then multiplication and division:

Finally subtract:

• A (correct): Square first (25), then 2 × 25 = 50 and 24 ÷ 2 = 12, so 50 − 12 = 38.
• B: This subtracts 50 − 24 before dividing, but division comes before subtraction.
• C: This squares 2 × 5 = 10 to get 100; the exponent applies only to the 5.
• D: This treats 5² as 5 × 2 = 10 instead of 5 × 5 = 25.

Answer: D

Correct answer: 0.4287. Convert the fraction:

Pad each choice to compare: 0.4287 is greater than 0.42857 and less than 0.4300, so it lies between the two values. Carrying enough decimal places is the key, since 0.4285 falls just short.

• A: This is just below 3/7, because 0.42850 is less than 0.42857.
• B: Padded to 0.4280, this is less than 3/7, which is about 0.4286.
• C: Padded to 0.4310, this is greater than 0.4300, so it is outside the range.
• D (correct): 3/7 is 0.428571..., so 0.4287 is just above it and still below 0.4300.

Answer: A

Correct answer: 21 miles. Turn each average into a total: 5 days

first 3

last 3

The two 3-day totals together count day 3 twice:

Day 3:

• A (correct): The first 3 days total 45 and the last 3 total 66, which is 111; the 5 days total only 90, so day 3, counted twice, is 21.
• B: 18 is the overall average; nothing requires the middle day to equal it.
• C: 7 is the difference between the two 3-day averages, which does not use the totals.
• D: 24 is 90 minus the last-three total of 66, which is the sum of days 1 and 2, not day 3.

Answer: C

Correct answer: 3 hours. Let d be the one-way distance:

Using 600 as the common denominator,

so d = 600 miles. The flight out took

(and the return took 2 hours).

• A: This splits the 5 hours evenly, but the slower leg takes longer.
• B: This is the time for the faster return flight, not the flight out.
• C (correct): The airstrip is 600 miles away, and 600 ÷ 200 = 3 hours.
• D: This uses the simple mean of 250 mph for the round trip, which is not the true average speed.

Answer: B

Correct answer: 48 mph. Find each leg's time:

Average speed = total distance divided by total time:

• A: This averages the two speeds, but the driver spends more time at the slower speed, so that shortcut fails.
• B (correct): The trip is 240 miles in 2 + 3 = 5 hours, and 240 divided by 5 is 48.
• C: This adds the two speeds; a single driver's average speed cannot exceed the faster speed.
• D: This divides only the one-way distance of 120 miles by the total time of 5 hours.

Answer: D

Correct answer: weak. Frail means weak and easily hurt. Weak matches the core meaning; old and sick are associations, and tiny describes size.

• A: Frail people are often elderly, but age is an association, not the meaning.
• B: Illness can make someone frail, but sick names a cause, not the quality of being weak.
• C: Small size does not mean weakness.
• D (correct): Frail means physically weak or delicate, which matches weak.

Answer: C

Correct answer: desert water source. The phrase set off by the comma renames OASIS: a green desert patch where springs bring up water. The matching choice is desert water source.

• A: A sand ridge, or dune, is a desert feature, but it has no springs or green plants.
• B: A mirage only looks like water; the convoy actually refilled its tanks there.
• C (correct): The phrase after the comma renames the oasis as a green desert spot with springs, which is a water source in the desert.
• D: The convoy refilled tanks, but the renaming phrase says the spot provides water from springs, not fuel.

Answer: C

Correct answer: Live side by side. Co- means together or with, and exist means to be or to live. To coexist is to live together side by side.

• A: Fighting over territory describes conflict, while co- stresses existing together, and the sentence says they hunt different prey.
• B: Exist before would need pre-, not co-.
• C (correct): Co- means together, so to coexist is to exist together, or live side by side.
• D: Stop existing reverses the base and does not match the together idea of co-.

Answer: C

Correct answer: damaging. Here the -ive adjective does mean causing the base action: a destructive thing destroys. Termites ruin wood, so destructive means damaging.

• A: Destruction is the noun for the damage itself and cannot replace an adjective.
• B: Harmless is the opposite of destructive.
• C (correct): Destructive means causing destruction, so damaging is the synonym.
• D: Thorough is the meaning of exhaustive, another -ive word that does not simply cause its base action.

Answer: D

Correct answer: agree. Concur means to agree or hold the same opinion. 'Both pilots agree on the new flight plan' keeps the sense, while argue is an opposite and meet and compete name different actions.

• A: Competing means trying to beat each other, which does not fit two people sharing the same view of a plan.
• B: Arguing is the opposite of concurring; it means the pilots disagree.
• C: Meeting may come before an agreement, but getting together is not the same as sharing the same opinion.
• D (correct): Concur means to be of the same opinion, so agree is the closest match.

Answer: A

Correct answer: lack of interest. Apathy joins a-, without, and path, feeling. People without feeling or interest in an election stay home, which explains the empty polling stations.

• A (correct): A- means without and path means feeling, and empty polling stations show people did not care enough to vote.
• B: Anger is a strong feeling, the opposite of being without feeling.
• C: Sympathy shares the path root but adds sym, together; the a- prefix in apathy removes feeling instead.
• D: Confused voters might stay home, but the word is built from without plus feeling, which means not caring.

Answer: D

Correct answer: skillful. Deft means neat, quick, and skillful in movement. Hurried shares the speed but misses the skill, so it is only partly right.

• A: Deft is about skill and lightness of touch, not about power.
• B: Clumsy is the opposite of the smooth, quick work described.
• C: The work is fast, but hurried suggests rushing and possible mistakes; deft stresses skill.
• D (correct): Deft means quick and skillful with the hands, matching a technician who removes a cover in seconds.

Answer: B

Correct answer: enthusiasm. Cover -ness to find eager, meaning keen. The -ness ending forms a noun of quality, so eagerness is keenness. Only enthusiasm is both a noun and a match in meaning.

• A: This is the adjective base itself; it cannot replace a noun after "recruit's."
• B (correct): Base eager means keen, and -ness turns it into a noun naming that quality, so enthusiasm is an exact noun match.
• C: This noun names an unwillingness to act, the opposite of being keen.
• D: Anxiety is nervous worry, which is not the same as keen interest in learning.

Answer: C

Correct answer: direct. HEAD follows "to," so it is a verb, and its object is a program. Being chosen to head a program means being put in charge, so the word means direct.

• A: Top part is the noun sense, but "to head" uses the word as a verb with an object.
• B: Attending would make the colonel a participant, while "was chosen to" signals a leadership role.
• C (correct): To head a program is to be in charge of it, and "chosen to direct the program" fits.
• D: "Head toward" means move in a direction, but here there is no "toward" and the object is a program.

Answer: A

Correct answer: occurring at regular intervals. The clue every 200 hours shows a regular schedule. Periodic means recurring at regular intervals, which separates it from sporadic (scattered) and constant (nonstop).

• A (correct): Periodic means recurring on a regular schedule, like every 200 hours of operation.
• B: This describes sporadic or intermittent timing; the sentence gives a fixed interval, so the timing is regular.
• C: This is constant, which would mean the oil is changed nonstop, not on a schedule.
• D: Emergency work happens when something fails; the manual schedules these changes in advance.

Answer: D

Correct answer: basic supplies. "Such as" introduces examples: flour, rice, salt and beans. Their shared trait is that they are ordinary, everyday necessities, so STAPLES here means basic supplies, not the paper fasteners the word can also name.

• A: This is the office-supply sense of staples; nothing in a list of foods is a fastener.
• B: Plain foods used every day are the opposite of rare luxuries.
• C: Rice is only one item in the list; the word names the whole group, not one example.
• D (correct): Flour, rice, salt and beans are everyday basics, and the cooks reached for them at every meal.

Answer: D

Correct answer: spiteful. MALICIOUS means intending harm, and in order to damage confirms the intent. The answer has to sound negative, so do not avoid unpleasant choices; careless lacks intent and mischievous is too mild, leaving spiteful.

• A: Mischievous suggests playful trouble-making, which is far milder than an attempt to damage a reputation.
• B: Careless means without attention; the phrase in order to shows the harm was deliberate.
• C: Generous is a pleasant word a test taker might pick to avoid a negative answer, but it is close to the opposite.
• D (correct): Spiteful matches malicious: wanting to hurt others on purpose.

Answer: B

Correct answer: truthfulness. Ver means true, as in verify. To question the veracity of an account is to doubt its truthfulness.

• A: This confuses veracity with velocity, which sounds similar but means speed.
• B (correct): Ver means true, the same root as verify, so investigators doubted whether the account was true.
• C: Nothing in ver refers to sound; investigators care whether the story is true.
• D: This sound-alike means fierceness, which has no connection to ver, true.

Answer: B

Correct answer: motionless. Stationary is an adjective built with -ary, meaning staying in one place. The truck waits without moving until waved through, so motionless fits. Do not confuse it with the noun stationery, which names writing paper.

• A: This is the meaning of stationery, a lookalike spelled with -ery that names paper and envelopes.
• B (correct): The -ary ending forms an adjective here, and stationary means standing still, which is motionless.
• C: This takes the base too literally; a checkpoint is not a station, and the word means not moving.
• D: Temporary means lasting a short time, which describes how long, not whether it moved.

Answer: B

Correct answer: patronizing. Condescending means treating someone as inferior, often by over-explaining simple things. The clue is that the foreman ignores the real question and explains something any mechanic knows; slow speech here carries a negative connotation, so patronizing fits and patient does not.

• A: The foreman speaks slowly and simply, not with shouting or rage; this imports an emotion the sentence does not show.
• B (correct): Condescending means talking down to someone as if they were less capable, which is what patronizing means and what the foreman does.
• C: Speaking slowly can look like patience, but here it is used to belittle a mechanic who obviously knows what a wrench is; the tone is negative, not kind.
• D: The word contains descend, but condescending means acting superior, the opposite of humble.

Answer: A

Correct answer: Nothing; a pool is mentioned only for Iron Works. In a two-column grid, the pool cell for Iron Works is filled, but the cell for Summit Fitness is empty. An empty cell is an unknown, not a difference.

• A (correct): The passage lists a pool for Iron Works and is silent about one at Summit, so Summit's pool is an unknown.
• B: Silence is not evidence that a feature is missing; the passage never says Summit lacks a pool.
• C: No pool is described at Summit, so no size comparison is possible.
• D: The 24-hour detail applies to the gym's hours, and no pool is mentioned there.

Answer: B

Correct answer: give way. The sentence after the target word restates it: drivers must slow down or stop and let the bus merge before moving ahead. Putting give way in place of yield keeps that meaning, so yield here means give way.

• A: This is a familiar farming sense of yield, but nothing in a passage about buses and traffic involves crops.
• B (correct): The next sentence explains that drivers must slow or stop and let the bus merge first, which is what giving way means.
• C: The passage says drivers must slow down or stop, the opposite of speeding up.
• D: This is the sense of yielding in a contest or argument; drivers here are simply letting a bus go first, not conceding anything.

Answer: D

Correct answer: Spare batteries should be packed in carry-on bags. A consequence tells what happens as a result; a recommendation tells what someone should do. Words such as should, must and ought mark a recommendation, and only the statement about packing spare batteries in carry-on bags uses one. The word therefore also shows it is drawn from the dangers described before it.

• A: This describes a possible consequence of damage; it does not tell anyone what to do.
• B: This is another possible consequence, a further effect of a battery fire, not advice.
• C: This explains a danger that follows from where the bag is stored; it is a consequence, not advice.
• D (correct): The word should signals advice about what travelers ought to do, which makes this a recommendation.

Answer: D

Correct answer: They are meant to serve sailors returning from late training. "Meant to" marks a purpose, which is an intention, not a measured result. The passage reports no sales figures yet and says the hours will be reviewed, so it supports only the stated goal of serving sailors back from late training.

• A: The sales review will not happen until the end of the quarter, so no result has been reported yet.
• B: No petition or request appears in the passage; this invents a cause.
• C: The manager will decide whether to keep them after reviewing sales, so their future is still open.
• D (correct): The manager says the later hours are "meant to" give those sailors a chance to shop, which states the purpose.

Answer: B

Correct answer: The ladders are alike in many ways, but only fiberglass is safe near electricity. In a comparison that stresses one difference, the similarities only set the scene. After "however," the passage focuses on the fact that fiberglass does not conduct electricity, and it ends by explaining why that matters, so the main idea centers on that difference.

• A: This is a real minor difference, but the passage uses "however" to shift attention to the electrical difference, which is the point.
• B (correct): The similarities set the scene, and the passage stresses one key difference: fiberglass does not conduct electricity, which matters for electrical work.
• C: This reverses the passage, which warns that an aluminum ladder could carry a shock near live wiring.
• D: These are the similarities that set up the comparison, not the difference the passage emphasizes.

Answer: B

Correct answer: To keep workers from slipping on the steps. Separate cause, purpose and outcome. The cause was ice on the steps, the purpose was stated with "hoping to keep workers from slipping," and the outcomes were no reported falls and no more salting. The question asks for the purpose.

• A: This is the cause of the problem, not the goal the manager was trying to reach.
• B (correct): The phrase "hoping to keep workers from slipping" states what the manager intended.
• C: This is the outcome that followed the change, not the intention behind it.
• D: The passage calls this an added benefit that came afterward, not the reason the mats were installed.

Answer: A

Correct answer: To accommodate personnel whose long shifts fall on weekdays. Separate the reason from the example. The first sentence gives the reason (serving people who work long weekday shifts), and the next sentence begins "For example" to show one group helped. The question asks why, so the reason is the answer.

• A (correct): The first sentence states the reason with the phrase "to serve service members who work long weekday shifts."
• B: The mechanics are introduced with "For example"; they illustrate the reason but are not the full reason itself.
• C: The passage never mentions a decline in weekday visits; this is an invented reason.
• D: The study hall is an additional change the library made, not the stated reason for the longer hours.

Answer: A

Correct answer: The author does not take a side and reports views from both supporters and opponents. Use the author-versus-source test: ask who says each opinion. Both opinions are attributed to named people, and the only other sentences are plain facts about parking and the vote. With no unattributed judgment, the passage is a neutral report of the range of opinion.

• A (correct): The author quotes a supporter and an opponent, adds a neutral fact, and never adds an evaluation of either, so no stance is revealed.
• B: That reason belongs to Maria Santos; quoting her does not show the author agrees.
• C: That concern is Phil Grady's; the author reports it without endorsing it.
• D: This assigns a hidden motive the passage never states; the author only reports Grady's stated reasons.

Answer: D

Correct answer: About 12 rooms will still be unfinished when the shutdown begins. Rooms left now:

In the 4 weeks before the shutdown, crews finish 4 x 6 = 24 more rooms, leaving

The prediction stays within the stated limits: the pace, the contract's work stoppage and nothing beyond them.

• A: This divides the 36 remaining rooms by 6 per week but forgets that no work happens during the two-week shutdown.
• B: This is the number of rooms left today; it ignores the 24 rooms crews will finish in the next 4 weeks.
• C: The manager only hopes for a spring reopening, and the passage never says when spring begins relative to the work.
• D (correct): With 36 rooms left and 24 more finished in the next 4 weeks, 12 rooms remain when work must stop.

Answer: D

Correct answer: Most of the interviewed farmers had added large fans within the previous five years. Check who, where, and when the evidence covers: 150 family-run farms with fewer than 200 cows, in three counties, interviewed in 2024. Only the statement limited to the interviewed farmers matches that scope; the others widen the group, add causation, or describe farms outside the sample.

• A: The interviews covered 150 farmers in three counties, so extending the result to the whole country widens the group and the place.
• B: This turns what some farmers reported into a proven cause, which the interviews do not establish.
• C: No farm with 200 or more cows was in the sample, so the passage gives no information about larger farms.
• D (correct): About 80 percent of the 150 interviewed farmers reported installing fans in the past five years, and this answer keeps that exact group and time frame.

Answer: B

Correct answer: Dividing both sides by −5. Only multiplying or dividing both sides by a negative number reverses the sign. For example, 2 < 5, but dividing by −5 gives

• A: Adding or subtracting any number, positive or negative, never changes the direction.
• B (correct): Dividing by a negative number reverses the order of numbers, so the sign must flip.
• C: Adding a negative number is still addition, which keeps the direction.
• D: Dividing by a positive number keeps the direction of the sign.

Answer: C

Correct answer: 14. Substitute in parentheses: $6-2(-4)$. Multiply first:

• A: This drops the sign of -4 and computes 6 - 8.
• B: This subtracts first, 6 - 2 = 4, and then multiplies by -4, breaking the order of operations.
• C (correct): 6 - 2(-4) = 6 + 8 = 14.
• D: This computes only -2(-4) and forgets to include the 6.

Answer: C

Correct answer: A flat fee charged before any hours are worked. The y-intercept is the output when the input is zero. With h = 0,

so $90 is the flat fee, and $65 is the added cost per hour.

• A: The cost per hour is the slope, 65, the number multiplied by h.
• B: One hour costs 65 plus 90, or 155, not 90.
• C (correct): When h is 0, C is 90, so 90 is the charge just for showing up.
• D: Hours are the input h, not a fixed number in the equation.

Answer: A

Correct answer: 8, 15, 17. Test

with c the longest side. For 8, 15, 17:

= $17^2$, so it is right. It is one of the common Pythagorean triples.

• A (correct): 64 + 225 = 289, which equals $17^2$.
• B: 36 + 64 = 100, which is less than 144, so this triangle is obtuse.
• C: 25 + 144 = 169, which is less than 196; the right triangle would be 5-12-13.
• D: 49 + 576 = 625, which is less than 676; the right triangle would be 7-24-25.

Answer: C

Correct answer:

Isolate y. Subtract 6x from both sides:

Divide every term by 3:

so the slope is −2 and the y-intercept is 4.

• A: This loses the negative sign created when 6x moves to the other side.
• B: This divides the x-term by 3 but forgets to divide the constant 12.
• C (correct): Subtracting 6x and dividing every term by 3 gives y = −2x + 4.
• D: This moves 6x across but never divides by the coefficient 3 on y.

Answer: B

Correct answer: x = 6. Because 0.15 has two decimal places, multiply every term by 100:

Subtract 15x and add 30: 270 = 45x, so x = 6.

Check:

• A: This comes from multiplying the x terms by 100 but the constants only by 10, which unbalances the equation.
• B (correct): Multiplying every term by 100 gives 15x + 240 = 60x − 30, so 270 = 45x and x = 6.
• C: This is a decimal slip: 270 was divided by 4.5 instead of 45.
• D: This comes from a sign slip when moving the constants, such as writing 240 + 30 as −270.

Answer: C

Correct answer: 56. Use every prime at its highest power: 8 = $2^3$ and

so the LCM is $2^3$ × 7 = 56. Both 8 × 7 and 14 × 4 equal 56.

• A: 112 = 8 × 14 is a common multiple, but not the smallest one.
• B: 2 is the greatest common factor, not the least common multiple.
• C (correct): 8 = $2^3$ and 14 = 2 × 7, so the LCM is $2^3$ × 7 = 56.
• D: 28 is a multiple of 14 but not of 8 (28 ÷ 8 = 3.5).

Answer: D

Correct answer: 112°. First find the angle:

°. Then find its supplement:

°. (Shortcut: the supplement is always 90° more than the complement.)

• A: This subtracts 22 from 180, treating 22° as the angle itself instead of its complement.
• B: 68° is the angle itself; the question asks for its supplement.
• C: This subtracts 68 from 360, which finds the rest of a full turn, not a supplement.
• D (correct): The angle is 90 - 22 = 68°, and its supplement is 180 - 68 = 112°.

Answer: B

Correct answer: 6. Set the rule equal to the output and solve:

Subtract 7:

Divide by −3: x = 6. Check:

• A: This loses the sign when dividing −18 by −3; a negative divided by a negative is positive.
• B (correct): Solving 7 − 3x = −11 gives −3x = −18, so x = 6.
• C: This evaluates f(−11) = 7 + 33 instead of finding the input that produces −11.
• D: This adds 7 to −11 instead of subtracting it, giving −3x = −4.

Answer: D

Correct answer: a²b. Settle each sign. Squaring a negative gives a positive, so $a^2$ is positive, and multiplying by positive b keeps it positive. The other expressions all combine one negative with one positive and come out negative.

• A: A negative times a positive is negative.
• B: Subtracting a positive from a negative moves farther left, so the result is negative.
• C: A negative divided by a positive is negative.
• D (correct): a² is positive because a negative times a negative is positive, and a positive times positive b stays positive.

Answer: B

Correct answer: 5. Change subtracting a negative into adding a positive:

Then

• A: This treats subtracting −15 as subtracting 15, so every term ends up negative.
• B (correct): Rewrite as −4 + 15 + (−6); then −4 + 15 = 11 and 11 − 6 = 5.
• C: This changes the sign of the −6 as well as the −15; only the subtracted number gets its sign changed.
• D: This subtracts 15 instead of adding it and turns the −6 into +6.

Answer: D

Correct answer: 960 - 72π cubic inches. Find the whole block, then subtract the hollowed cylinder. Block: 12 times 10 times 8 is 960. Hole: pi times 3 squared times 8 is 72 pi.

• A: This subtracts 2 pi r h, the inside wall area of the hole, instead of its volume.
• B: This uses pi times r times h, forgetting to square the radius.
• C: This uses the diameter 6 in place of the radius when squaring.
• D (correct): The block is 960 and the hole is a cylinder of 9 pi times 8, or 72 pi, which is subtracted.

Answer: B

Correct answer: 135 degrees. A hexagon's angles total (6 - 2) times 180 = 720 degrees. So

6x = 630 and x = 105. The largest angles are

• A: This is x, the measure of the smaller angles; the question asks for x + 30.
• B (correct): With a 720-degree sum, 6x + 90 = 720 gives x = 105, so the larger angles are 135 degrees.
• C: This uses a 900-degree sum, which belongs to a 7-sided polygon, giving x = 135 and x + 30 = 165.
• D: This is one angle of a regular hexagon, but these angles are not all equal.

Answer: D

Correct answer: 45π square feet. The outer radius is

The ring is the big disk minus the fountain.

The walkway covers 45π square feet.

• A: This squares the radius difference, π times 3 squared; the ring needs R squared minus r squared.
• B: This is the whole outer disk; the fountain's 36π still has to be removed.
• C: This adds the two disk areas instead of subtracting the fountain from the outer disk.
• D (correct): Outer radius 9, inner radius 6: π times (81 - 36) = 45π square feet.

Answer: A

Correct answer: ab⁵. Square every factor in the numerator:

Divide by the denominator:

The result is $a{b}^5$.

• A (correct): The top is 4a^-4 b^6; dividing gives 4/4 = 1, a^(-4 - (-5)) = a^1 and b^(6 - 1) = b^5.
• B: This subtracts -5 as if it were +5, taking -4 - 5 = -9 for the exponent of a.
• C: This forgets to square the coefficient 2, so it divides 2 by 4.
• D: This adds 3 and 2 for the power of b instead of multiplying, giving b^5 on top and b^4 after dividing.