AFQT Core Practice 4: Money Math, Algebra and Main Idea

Answer: A

Correct answer: $402.00. Multiply the hourly rate by the hours:

• A (correct): Gross pay is rate times hours: 16.75 × 24 = $402.
• B: This drops the 75 cents and multiplies 16 × 24.
• C: This misplaces the decimal point; 16.75 × 24 is about 17 × 24, roughly $400.
• D: This is a digit slip; 16.75 × 24 is 384 + 18 = $402.

Answer: D

Correct answer: 23 miles per gallon. The word per signals division: miles per gallon =

• A: The digits are right, but the decimal point is misplaced; 9 times 2.3 is only 20.7.
• B: This multiplies miles by gallons; a per-gallon rate comes from dividing.
• C: This divides gallons by miles, which gives gallons per mile, not miles per gallon.
• D (correct): Miles per gallon is miles divided by gallons: 207 divided by 9 is 23.

Answer: B

Correct answer: 8,000 miles. Round to friendly numbers: 412 is about 400 and 19 is about 20.

The choices are far apart, so the estimate is enough; the exact product is 7,828.

• A: This is ten times too small; two weeks of driving alone is more than 800 miles.
• B (correct): Rounding gives 400 × 20 = 8,000; the exact total, 7,828 miles, is closest to this value.
• C: This is ten times too large, from adding an extra zero while rounding.
• D: This is a hundred times too large for 19 weeks of driving.

Answer: A

Correct answer: 6.5%. To change a decimal to a percent, move the decimal point two places to the right: 0.065 becomes 6.5, so the defect rate is 6.5 percent.

• A (correct): Moving the decimal point in 0.065 two places to the right gives 6.5 percent.
• B: This moves the decimal point three places right; 65 percent would be 0.65.
• C: This moves the decimal point only one place right.
• D: This adds a percent sign without moving the decimal point at all.

Answer: B

Correct answer: 6 7/12 miles. Add the whole numbers:

Add the fractions with denominator 12:

= $1\frac{7}{12}$.

Total:

• A: The fraction sum 19/12 is more than 1, but the extra whole was not carried into the whole-number part.
• B (correct): Adding wholes gives 5 and adding fractions gives 9/12 + 10/12 = 19/12 = 1 7/12, for 6 7/12 miles in all.
• C: This adds the numerators and the denominators (3 + 5 over 4 + 6), which is not allowed with unlike denominators.
• D: Only one whole should be carried from 19/12; carrying two adds a mile that is not there.

Answer: A

Correct answer: 60. Total for 8:

Total for 7:

Removed score:

A low score was removed, which is why the average went up.

• A (correct): The 8 scores total 592 and the remaining 7 total 532, so the removed score is 60.
• B: This assumes the removed score is 2 points below 74, but removing one score must shift the whole total, not just the average.
• C: This moves 2 points in the wrong direction; removing a high score would lower the average, not raise it.
• D: This multiplies both averages by 8, forgetting that only 7 trainees remain.

Answer: D

Correct answer: 20,000. Round each factor to a friendly number: 52 is about 50 and 398 is about 400.

which separates the choices easily. The exact product, 20,696, confirms it.

• A: This is ten times too small; 50 crates of about 400 is far more than 2,000.
• B: This is ten times too large, usually from adding an extra zero while rounding.
• C: This is about the size of a single crate, not 52 crates.
• D (correct): Rounding gives 50 × 400 = 20,000; the exact total, 20,696, is closest to this choice.

Answer: C

Correct answer: $49.98. Sale price:

Add 5% tax:

• A: This nets the percents into one 10% discount (15% − 5%); each percent must be applied to the current amount.
• B: This is the sale price before tax; the 5% tax still has to be added.
• C (correct): The sale price is 0.85 × 56 = $47.60, and 47.60 × 1.05 = $49.98.
• D: This adds tax to the original price but never applies the 15% discount.

Answer: D

Correct answer: 40 pieces. Convert to one unit: 3 inches is

so each piece is 6.25 feet.

Pieces:

• A: This treats 6 feet 3 inches as 6.3 feet instead of 6.25 feet.
• B: This ignores the 3 inches and divides by 6 feet.
• C: This adds 6 and 3 as if both were feet, dividing by 9.
• D (correct): 6 feet 3 inches is 6.25 feet, and 250 / 6.25 = 40 pieces.

Answer: B

Correct answer: 3 3/20 meters. Keep the whole number 3. The decimal part 0.15 ends in the hundredths place, so

The length is $3\frac{3}{20}$ meters.

• A: This reads the digits 1 and 5 as the fraction 1/15, which is about 0.067, not 0.15.
• B (correct): 0.15 = 15/100, and dividing top and bottom by 5 gives 3/20, so 3.15 = 3 3/20.
• C: 3/200 is 0.015; this uses thousandths as the place value instead of hundredths.
• D: 3 1/5 is 3.2; this rounds 0.15 to 0.2 before converting.

Answer: C

Correct answer: 450 pounds. Add the ratio parts:

One part is

Gravel is 3 parts:

• A: This is the sand (2 parts), not the gravel.
• B: This is the size of one part, which is the cement, not the gravel.
• C (correct): The ratio has 6 parts, each part is 150 pounds, and gravel is 3 parts, or 450 pounds.
• D: This uses only 5 parts (2 + 3) as the total, leaving the cement out of the part count.

Answer: D

Correct answer: 6 liters. Water adds volume but no cleaner, so the pure amount stays at

At 30 percent, the total volume must be

Water to add:

• A: 18 liters is the total volume the 30 percent solution must have, not the amount of water to add.
• B: This multiplies 12 by the 15-point drop in percent, which does not find the volume that holds 5.4 liters at 30 percent.
• C: Adding 15 liters gives 5.4 liters of cleaner in 27 liters, which is only 20 percent.
• D (correct): The 5.4 liters of pure cleaner must make up 30 percent of the new volume, so the total is 18 liters and 6 liters of water are added.

Answer: D

Correct answer: 24. The difference between jeeps and vans is

parts, and that difference is 30 vehicles, so one part is

Trucks are 4 parts:

• A: This is the number of jeeps (7 × 6), not trucks.
• B: This is the number of vans (2 × 6), not trucks.
• C: This is the total of all three vehicle types, 42 + 24 + 12, not the trucks alone.
• D (correct): Jeeps exceed vans by 7 - 2 = 5 parts, so one part is 30 ÷ 5 = 6 and trucks are 4 × 6 = 24.

Answer: A

Correct answer: 5 hours. Use a common denominator of 30. Inlets add, the drain subtracts:

The tower fills at 1/5 per hour, so it takes 5 hours.

• A (correct): Net rate is 1/6 + 1/10 - 1/15 = 1/5 of the tower per hour, so filling takes 5 hours.
• B: This adds the drain's rate as if it were a third inlet; a drain's rate must be subtracted.
• C: This is the time for the two inlets with the drain closed; it ignores the drain.
• D: This treats the net rate of 1/5 tower per hour as the time, forgetting to invert it.

Answer: A

Correct answer: 9:50 a.m. First turn the 50-minute head start into distance:

The jeep gains

each hour, so it needs

Two hours after 7:50 a.m. is 9:50 a.m.

• A (correct): The 30-mile lead closes at 15 mph in 2 hours, counted from the jeep's 7:50 a.m. start.
• B: This finds the 2-hour chase time but adds it to the truck's 7:00 a.m. start instead of the jeep's start.
• C: This treats 50 minutes as 0.5 hour, giving an 18-mile lead and a 72-minute chase.
• D: This divides the 30-mile lead by the jeep's full speed of 51 mph instead of by the 15 mph it gains.

Answer: A

Correct answer: bulky and clumsy. Cumbersome means bulky, heavy, or clumsy to handle, like a thick winter coat or a slow paperwork process. It means hard, not impossible, so bulky and clumsy fits best.

• A (correct): Cumbersome describes something heavy, large, or slow that is awkward to deal with, whether an object or a process.
• B: This is the dramatic trap: a cumbersome load or process is hard to manage, but it can still be handled.
• C: This is close to the opposite; cumbersome things are usually heavy or bulky.
• D: Fragile means easily broken, a different idea; a cumbersome object may be very sturdy.

Answer: C

Correct answer: burst out. E- means out and rupt means break or burst. Cheers that erupt burst out all at once, the way a volcano erupts.

• A: This is the opposite of a sudden burst of sound.
• B: Rupt suggests a sudden break or burst, not slow spreading.
• C (correct): E- means out and rupt means burst, so the cheers burst out suddenly.
• D: Silencing stops the cheers, while erupting releases them.

Answer: D

Correct answer: wipe out. Eradicate means to eliminate entirely. Reduce is the tempting near-miss, but it lacks the complete degree the word carries.

• A: Crews may study a disease closely, but the goal described is getting rid of it, not learning about it.
• B: Reduce is partly right but too weak; eradicate means remove completely.
• C: Spread is the opposite of what health crews try to do.
• D (correct): Eradicate means to destroy completely, root and all, which is what wipe out means.

Answer: B

Correct answer: Germ-killing. Septic traces to a Greek word meaning rotting, and in medicine it describes infection caused by bacteria and other microbes; the prefix anti- means against. An antiseptic is put on skin or wounds to kill or slow germs, which is why a medic cleans a cut with it before bandaging. The tempting choice is healing, but the wipe does not repair tissue; it lowers the chance of infection so the body can heal on its own. Related words include sepsis and aseptic, meaning free of germs.

• A: Healing is a later result, not what the prefix anti- says the wipes do.
• B (correct): Anti- means against, and septic refers to infection, so antiseptic wipes act against germs.
• C: Pain-relieving fits a medic but names a different product, such as a numbing spray.
• D: Disposable is true of many wipes but has nothing to do with anti- or the base septic.

Answer: C

Correct answer: support. Cover -ment to find encourage, to give someone confidence. The ending -ment forms a noun naming that act, so encouragement is support. Courage and encourage are the wrong word class.

• A: Courage is the inner base of the word, not the act of urging someone on.
• B: This is the verb base; the target is a noun, so a verb cannot replace it.
• C (correct): The ending -ment turns encourage into a noun naming the act of giving confidence, which is support.
• D: Criticism points out faults, the opposite of cheering someone on.

Answer: C

Correct answer: Supporter. Pro- usually means forward or in favor of. Someone who speaks in favor of a plan at meetings puts it forward, so a proponent is a supporter.

• A: Opponent is the opposite; she spoke in favor of the bike lanes.
• B: Inventor names who created something, but she is arguing for the lanes, not designing them.
• C (correct): Pro- means for or in favor of, and she spoke in favor of the plan, so a proponent is a supporter.
• D: Early version reads pro- as before, a sense that only a few pro- words carry.

Answer: C

Correct answer: supported. As a verb, advocate means to argue for or recommend something. Supported fits; opposed is the opposite, invented is a different act, and explained lacks the sense of favoring the plan.

• A: Explaining a route describes it, but does not show the speaker was arguing in its favor.
• B: This is the opposite of speaking in favor of the route.
• C (correct): Advocate as a verb means to speak publicly in favor of something, so supported is the best match.
• D: Creating the route is a different act; a person can advocate a plan someone else designed.

Answer: A

Correct answer: lacking a definite shape. Amorphous joins a-, without, and morph, shape. A plan that still has no definite shape or structure leaves contractors unable to estimate a price.

• A (correct): A- means without and morph means shape, and a plan without a clear shape cannot be priced.
• B: This is the opposite; a detailed plan would be easy to price.
• C: Cost is the result the sentence mentions, not the meaning of the word describing the plan.
• D: Change of shape is metamorphosis; the a- prefix in amorphous means without shape, not changed shape.

Answer: C

Correct answer: view. Perception follows a possessive, so it is a noun. It names the act of noticing or a way of seeing something. Shoppers' way of seeing the neighborhood is their view.

• A: Observant is an adjective that matches perceptive; it cannot follow shoppers' as a noun.
• B: Noticeable matches perceptible, an adjective for something that can be noticed.
• C (correct): Perception can mean a way of seeing a situation, so shoppers' perception of the area is their view of it.
• D: Purchase is unrelated; the sentence is about how shoppers saw the area, not what they bought.

Answer: B

Correct answer: hollow. The partners "rainwater collected," "shallow," and "in the parking lot" describe a physical low spot. DEPRESSION here means a hollow.

• A: Depression can mean low mood, but water cannot collect in a feeling.
• B (correct): A shallow sunken spot where water pools is a hollow, which fits the parking lot.
• C: An economic depression is a downturn, a sense unrelated to rainwater.
• D: A ridge is a raised strip, the opposite shape; water runs off a ridge rather than collecting.

Answer: C

Correct answer: used up. The action, stopping to refill, tells you the canteens had little or no water left. DEPLETED means used up or emptied.

• A: This is the opposite; full canteens would not need refilling.
• B: A march can dent gear, but dented canteens would not explain stopping to refill them.
• C (correct): Stopping to refill the canteens shows the water had run out.
• D: Cleaning is never mentioned and does not explain the need for more water.

Answer: C

Correct answer: limited with conditions. Qualify has two common senses: to meet requirements, and to limit or soften a statement. The object here is her statement, and the added words only fairly sure narrow it, so the meaning is limited with conditions. Choosing met the requirements for is the primary-definition trap.

• A: This is the most familiar sense, as in qualifying for a job, but a statement cannot meet requirements.
• B: Saying she was only fairly sure weakens her claim rather than confirming it.
• C (correct): Adding that she was only fairly sure narrows and softens the statement, which is this sense of qualify.
• D: She kept the statement but added a limit; she did not take it back.

Answer: D

Correct answer: continued without letup. Sustain means to keep something going, and the -ed form describes something already kept up over time. The clue "eleven straight hours" confirms it. Sustainable (can be kept up) and the injury sense of sustain are both traps.

• A: This is the meaning of sustainable, which looks ahead to what can be kept up; sustained looks back at what was kept up.
• B: This is the injury sense of the verb sustain, as in sustain a wound, which does not fit an effort that people kept going.
• C: Brief contradicts the eleven straight hours in the sentence.
• D (correct): Sustained describes something that has been kept up over a period, like eleven straight hours of pumping.

Answer: D

Correct answer: do away with entirely. Watch the negation: the rule did not abolish overtime; it only capped it. Since capping is the lesser action, abolish must mean the stronger one, to end entirely. Choosing limit falls into the trap of matching the word to what actually happened.

• A: Limiting is what the rule actually did; the word not shows that abolish is something stronger.
• B: Approving would not be contrasted with capping, and it reverses the idea of ending something.
• C: Paying extra describes overtime itself, not what the rule did or did not do to it.
• D (correct): The sentence contrasts abolish with merely capping, so abolish must mean ending something completely.

Answer: B

Correct answer: scolded. Admonish means to warn or reprimand someone firmly in words. Scolded matches that action; punished adds a penalty the word does not include, praised is the opposite, and dismissed is an unrelated action.

• A: Punishing imposes a penalty; admonishing is a verbal warning or scolding, so this overstates the word.
• B (correct): Admonish means to warn or criticize someone firmly, which scolded captures.
• C: This is the opposite of a firm criticism.
• D: Sending the soldiers away is a different action from speaking sharply to them.

Answer: D

Correct answer: Packing a rucksack carefully makes the load easier to carry. The opening sentence states the main claim, and the following sentences show how to place heavy, bulky, and often-needed items. A purpose phrase such as "to explain" describes why an author writes, not what the passage says.

• A: This states a purpose, not a main idea, and the passage never mentions training.
• B: This is one detail used to illustrate where heavy items belong, not the point of the passage.
• C: The passage is about arranging the load, not reducing it; this changes its subject.
• D (correct): The first sentence states this claim, and the rest of the passage explains how placing each kind of item helps.

Answer: C

Correct answer: 3. Count the listed topics: reading channel markers, rules for passing, and, signaled by "also," what to do if someone falls overboard. That makes three. The "two" in the passage refers to days, not topics.

• A: Two is the length of the class in days; counting that way also misses the topic introduced by "also."
• B: The passage lists more than one topic in its very first sentence.
• C (correct): The class covers channel markers and passing rules, and "also" adds a third topic, falling overboard.
• D: Only three topics are named; the final test and the card are not topics the class covers.

Answer: D

Correct answer: Several problems contributed to the team's last-place finish. The captain lists three separate problems: lost build time from the flood, a late motor, and the loss of experienced members. When a passage names several causes, the best answer treats them as contributing factors rather than naming one as the sole cause.

• A: This picks one of three named factors and calls it the sole cause, overstating the evidence.
• B: Other teams' budgets are never mentioned; this brings in a cause from outside the passage.
• C: The passage reports causes for this year only and never mentions repairs or next season.
• D (correct): The captain names three factors, the flood, the late motor and the graduated members, so the poor result had more than one cause.

Answer: D

Correct answer: The crop failures in Ireland helped increase the populations of some North American port cities. Follow the chain of cause and effect in the order the passage gives it: crop failure, then families unable to eat, then emigration, then growth in port cities. Joining the first and last links gives the supported conclusion. Reversing the cause, making the claim total or inventing a number all go beyond the text.

• A: This flips the order of events; the passage says the crop failures led people to leave, not the other way around.
• B: The passage says more than a million people left and many went to North America, not that every family left or that all went there.
• C: The passage says the populations swelled but gives no figure, so "doubled" invents a number.
• D (correct): The passage links the failures to emigration, and the emigrants' arrival to growth in cities such as Boston and New York.

Answer: A

Correct answer: Earlier falls led to the mats, and the first winter brought no falls but higher power bills. Sort the details by kind. Cause: two workers slipped last winter. Purpose: prevent falls. Results: no falls at the entrances and an electric bill about $300 a month higher.

• A (correct): The falls caused the change, and the first winter's results were zero falls and about $300 more a month for power.
• B: The purpose was to prevent falls; the electric bill actually went up.
• C: The higher bills were a result of the mats, not the reason they were installed.
• D: The purpose is right, but the result is reversed; no one fell at either entrance during the first winter.

Answer: A

Correct answer: It admits a drawback before the writer answers it. Sentence 1 makes a claim, sentence 2 concedes a point on the other side (higher price, charging stations), and sentence 3 begins with Even so to answer that concession. A concession admits a drawback without abandoning the writer's position.

• A (correct): The phrase It is true that grants a point against the plan, and Even so in sentence 3 turns back to the writer's side.
• B: The recommendation to switch to electric vans is in sentence 1, not sentence 2.
• C: Sentence 2 lists added costs, which cut against the savings argument rather than support it.
• D: The switch has not happened yet; the passage is a proposal, so there is no outcome to report.

Answer: D

Correct answer: Foot traffic on the main street rose in the two years after the trees were planted. The passage says the trees came first and the rise in foot traffic came later, but it states that the report did not explain the cause. A sequence is not proof of causation, so the supported answer keeps the order of events, the main-street limit, and no claim about why.

• A: The passage gives only a sequence and says the report did not explain what drew the shoppers, so a cause is added.
• B: The count covered only the main street, so this widens the scope beyond the evidence.
• C: The shopping center was not measured, so no comparison of preferences is supported.
• D (correct): This keeps the passage's order of events and its scope without claiming a cause, so it is fully supported.

Answer: D

Correct answer: Careful preparation before pouring helps a concrete patio last. A process passage still has an overall point, usually the purpose of the steps. Here the final sentence explains why the crew does all of them: so the slab stays level and resists cracking. The main idea is that preparation makes the patio last, not any single step.

• A: This is one step in the process, a detail that leaves out the digging, compacting, and mesh.
• B: The passage makes no comparison with decks; this brings in an unsupported claim.
• C: The passage mentions the compactor once and never ranks tools; this inflates one detail.
• D (correct): This states the purpose behind all of the steps, which the last sentence spells out.

Answer: B

Correct answer: The move had some problems but was still completed on time. Read the double negative slowly: not without problems means with some problems. The following sentence lists three of them, and the final sentence limits their effect (stocked by Monday, no late orders). The paraphrase must keep both parts: some problems, yet finished on time.

• A: This drops the word not and reverses the phrase; the next sentence lists three specific problems.
• B (correct): The double negative means problems did occur; the passage lists three, yet it also says the move finished on schedule with no late orders.
• C: This overstates the problems; the passage says the move finished on schedule and no orders were late.
• D: This shrinks the scope; misrouted trucks and soaked paper were problems too.

Answer: C

Correct answer: The route is still only a proposal, and its cost has not been made public. Ask "Who says this?" for each opinion: the optimistic claim is the steward's and the cautious example is the planner's, and the author endorses neither. Then check the verbs: is weighing, has asked for an estimate and expects to vote all describe a proposal still under consideration with an unreleased cost.

• A: A spring vote is planned, not an approval; this treats a proposal as a final decision.
• B: That prediction is quoted from the union steward; the author never adopts or judges it.
• C (correct): The board is weighing the idea, has requested a cost estimate that is not yet released, and will vote in spring, so nothing is final and the cost is unknown.
• D: The author reports the planner's comment as one side of the discussion and adds no evaluation, so the author takes no stance.

Answer: D

Correct answer: 25π square feet. Halve the diameter first: r = 5 feet. Then A = π(5)^2 = 25π square feet.

• A: This squares the diameter instead of the radius, making the area four times too big.
• B: This is π times the diameter, which is the circumference, not the area.
• C: This is 2π times 10, a circumference formula with the diameter used as the radius.
• D (correct): The radius is 5, and π times 5 squared is 25π square feet.

Answer: D

Correct answer:

Divide both sides by 4. Since 4 is positive, the sign stays the same:

A negative number on the right side does not cause a flip.

• A: This flips the sign because the right side is negative, but only dividing by a negative number reverses it.
• B: This drops the negative sign of −28 when dividing.
• C: This subtracts 4 instead of dividing by 4; 4x means 4 times x.
• D (correct): Dividing by positive 4 keeps the direction, so x < −7.

Answer: B

Correct answer: −19. When both numbers have the same sign, add their absolute values and keep the common sign:

• A: The sizes are added correctly, but two losses make a larger loss, so the sign must stay negative.
• B (correct): Both numbers are negative, so add 7 and 12 to get 19 and keep the negative sign.
• C: This subtracts the sizes, which is the rule for unlike signs, not like signs.
• D: This subtracts the sizes and also drops the negative sign.

Answer: A

Correct answer: Add 9 to both sides. Use the inverse operation: addition undoes subtraction. Adding 9 to both sides gives

so x = 23. Whatever you do to one side must also be done to the other.

• A (correct): Adding 9 undoes the subtraction of 9 and is applied to both sides, giving x = 14 + 9 = 23.
• B: Subtracting 9 gives x − 18 = 5; it repeats the operation instead of undoing it.
• C: Changing only one side breaks the balance, so the new equation no longer has the same solution.
• D: Multiplying gives 9x − 81 = 126; multiplication does not undo subtraction.

Answer: C

Correct answer: 5/(x - 2). Factor the top with the GCF:

Factor the bottom as a difference of squares:

Cancel the common factor $x+2$ to get $\frac{5}{x-2}$.

• A: This cancels the x - 2 factor, but the shared factor is x + 2.
• B: This cancels a 2 out of single terms across plus and minus signs, which is not allowed.
• C (correct): Top is 5(x + 2) and bottom is (x - 2)(x + 2); canceling x + 2 leaves 5 over x - 2.
• D: This places x - 2 in the top instead of the bottom after canceling.

Answer: C

Correct answer: 30 inches. The width, height and diagonal form a right triangle with hypotenuse 50. Subtract squares to find the missing leg:

The screen is 30 inches tall (a 3-4-5 triangle scaled by 10).

• A: This subtracts the lengths, 50 - 40, instead of their squares.
• B: This adds the squares, which finds a hypotenuse, but the diagonal is already the hypotenuse.
• C (correct): $2500-1600=900$, and the square root of 900 is 30.
• D: 900 is $h^2$; the square root still has to be taken.

Answer: A

Correct answer: 9. Replace x with 3/4:

• A (correct): Eight times 3/4 is 6, and adding 3 gives 9.
• B: This multiplies 8 by 3/4 but forgets to add the 3.
• C: This substitutes only the numerator 3 for x, giving 8(3) + 3.
• D: This substitutes only the denominator 4 for x, giving 8(4) + 3.

Answer: B

Correct answer: 240°. The minute hand makes a full 360° turn every 60 minutes, which is

° per minute. In 40 minutes it turns

°.

• A: This treats 40 minutes as 40% of an hour; it is 40/60, or two-thirds, of an hour.
• B (correct): At 6° per minute, 40 minutes gives 40 × 6 = 240°.
• C: This uses 5° per minute, mixing up the 5 minutes between numbers with the degrees per minute.
• D: This uses 3° per minute, which is half the actual rate.

Answer: B

Correct answer: (3, 5). Isolate x in the first equation:

Substitute into the second:

giving y = 5. Then

and (3, 5) checks in both equations.

• A: This lists the values in the order (y, x); with x = 5 the first equation gives 5 + 6 = 11, not 13.
• B (correct): x = 3 and y = 5 make x + 2y = 13 and 3x − y = 4 both true.
• C: This pair fits x + 2y = 13 but gives 3 − 6 = −3 in the second equation, so it fails the check.
• D: This pair fits 3x − y = 4 but gives 2 + 4 = 6 in the first equation instead of 13.

Answer: D

Correct answer: 3⁵. Product rule: with the same base, keep the base and add the exponents,

Check:

• A: This multiplies the exponents, which is the rule for a power of a power.
• B: This multiplies the bases as well as adding the exponents; the base stays 3.
• C: This multiplies both the bases and the exponents.
• D (correct): Same base, so keep 3 and add the exponents: 2 + 3 = 5.

Answer: B

Correct answer: 3. Simplify inside each set of bars first:

Take absolute values to get 9 and 6, then subtract:

• A: This adds 9 and 6 instead of subtracting the second absolute value.
• B (correct): |3 − 12| = |−9| = 9 and |−2 − 4| = |−6| = 6, so 9 − 6 = 3.
• C: This drops the bars on the first part, using −9 − (−6) = −3, but the absolute value of −9 is 9.
• D: This drops the signs before simplifying, using 3 + 12 = 15 and 2 + 4 = 6.

Answer: A

Correct answer: -3y - 5. Work from the inside out. Inner part:

Distribute -3: $7-3y-12$. Combine constants: $-3y-5$.

• A (correct): Inside: 2y - y + 4 = y + 4. Then 7 - 3y - 12 = -3y - 5.
• B: This forgets to flip -4 in the inner parentheses, getting y - 4 and then 7 - 3y + 12.
• C: This subtracts 7 - 3 first, but the 3 must multiply the bracket before any subtraction.
• D: This makes a sign slip at the end: 7 - 12 is -5, not 5.

Answer: A

Correct answer: 1. Perpendicular slopes multiply to -1, so the new slope is -1/3. Set up rise over run from (1, 2) to (4, k).

So

and k = 1.

• A (correct): The needed slope is -1/3, and (k - 2)/3 = -1/3 gives k - 2 = -1, so k = 1.
• B: This uses slope 3, which would make the lines parallel, not perpendicular.
• C: This uses slope 1/3, the reciprocal without the negative sign.
• D: This uses slope -3, changing the sign but not taking the reciprocal.

Answer: A

Correct answer: 35. An interior angle and its exterior angle add to 180 degrees, so

giving e = 36 degrees. The exterior angles total 360, so

sides.

• A (correct): Find the exterior angle (36 degrees), the number of sides (10), then apply n(n - 3)/2 = 10 times 7 over 2.
• B: This is the number of sides, not the number of diagonals; the question needs one more step.
• C: This is n(n - 3) without dividing by 2, so every diagonal is counted twice.
• D: This is n(n - 1)/2, which counts every pair of vertices, including the 10 pairs joined by sides.

Answer: B

Correct answer: 204 square centimeters. Block:

Cube: 6 times 9 = 54. The cube's bottom and the block patch it covers are each 9 square centimeters and are hidden.

• A: This adds both surface areas but does not remove the two hidden faces where they touch.
• B (correct): The block has 168, the cube 54; remove the cube's bottom and the covered patch, 9 each, to get 204.
• C: This removes only one hidden 9 square centimeter face; the cube's bottom and the patch it covers are both hidden.
• D: This also removes the cube's top face, but the top is still exposed to the air.