AFQT Core Practice 2: Rates, Exponents and Connotation

Answer: D

Correct answer:

In a true proportion the cross products are equal. For 6/8 and 15/20:

so the ratios are equal (both reduce to 3/4).

• A: Cross products are 6 × 12 = 72 and 9 × 10 = 90, which are not equal.
• B: Cross products are 4 × 16 = 64 and 5 × 12 = 60, which are not equal.
• C: Cross products are 3 × 14 = 42 and 7 × 9 = 63; the numerator was tripled but the denominator only doubled.
• D (correct): Cross products are 6 × 20 = 120 and 8 × 15 = 120, which are equal.

Answer: A

Correct answer: 15 gallons. Add the amounts:

Divide by the 4 weeks:

• A (correct): The four amounts add to 60 gallons, and 60 divided by 4 is 15.
• B: 60 gallons is the total for all four weeks, not the mean.
• C: This divides the total of 60 by 5 instead of by the 4 weeks.
• D: 14.5 is halfway between the smallest and largest amounts, which ignores the other two weeks.

Answer: B

Correct answer: $295.20. Total hours:

Pay:

Estimate check: 18 × $16 is about $290.

• A: This is the pay for a single 6-hour shift, not all three shifts.
• B (correct): The worker puts in 3 × 6 = 18 hours, and 18 × $16.40 = $295.20.
• C: This misplaces the decimal point; 18 hours at about $16 is roughly $300, not about $3,000.
• D: This multiplies $16.40 by 3 only, treating each shift as one hour.

Answer: C

Correct answer: 3 hours. When two vehicles approach each other, the gap closes at the sum of their speeds:

Time =

• A: This uses the difference of the speeds, 10 mph, which applies only to a chase.
• B: This divides by the faster van's speed alone and ignores the other van.
• C (correct): The gap closes at 90 mph, and 270 ÷ 90 = 3 hours.
• D: This divides by the slower van's speed alone and ignores the other van.

Answer: D

Correct answer: $65. A 40 percent markup means the price is 1.40 times the cost.

Cost:

Check: 40 percent of 65 is 26, and 65 plus 26 is 91.

• A: This takes 40 percent off the selling price, but the markup was figured on the cost, so you must divide by 1.40.
• B: This marks up the selling price another 40 percent instead of undoing the markup.
• C: This subtracts 40 dollars, treating the percent as a dollar amount.
• D (correct): The selling price is 140 percent of the cost, and 91 divided by 1.40 is 65.

Answer: D

Correct answer: 737. To multiply a two-digit number by 11, put the sum of its digits between them. Here

so write 3 in the middle and carry 1 into the 6.

Check:

• A: This writes the 3 from 13 in the middle but forgets to carry the 1 into the left digit.
• B: This places all of 13 between the digits instead of carrying the 1.
• C: This puts the wrong middle digit; 67 × 11 = 670 + 67 = 737.
• D (correct): Place the digit sum 6 + 7 = 13 between the digits and carry 1 into the 6: 737.

Answer: D

Correct answer: 75 boxes. Since 3 fifths make 45 boxes, one fifth is

boxes. The whole shipment is

boxes, the same as $45÷\frac{3}{5}$.

• A: This finds 3/5 of 45, but 45 is already 3/5 of the unknown total.
• B: This is the 2/5 of the shipment still to unload, not the whole shipment.
• C: This is only 1/5 of the shipment; it must be multiplied by 5.
• D (correct): If 3/5 of the shipment is 45, then 1/5 is 15 and the whole (5/5) is 75.

Answer: C

Correct answer: 3.375. Multiply each grade by its credits:

The weights are credits, not percents, so divide by their sum:

• A: 3.0 is the simple average of 4.0, 3.0, and 2.0, which ignores that the courses carry different credits.
• B: This divides the 27 weighted points by the 3 courses instead of by the 8 credits.
• C (correct): The weighted points are 16 + 9 + 2 = 27, and 27 divided by 8 credits is 3.375.
• D: This pairs the 4 credits with the C and the 1 credit with the A, reversing the weights.

Answer: B

Correct answer: 3 hours 20 minutes. In 6 hours the mason finishes

leaving $\frac{5}{8}$.

Team rate:

per hour. Time:

which is 3 hours 20 minutes.

• A: This is the total elapsed time including the mason's first 6 hours; the question asks only for the time after the partner joins.
• B (correct): After 6 hours 5/8 remains; together they lay 3/16 per hour, and 5/8 divided by 3/16 is 10/3 hours, or 3 hours 20 minutes.
• C: This is the time for the pair to lay the entire patio, ignoring the 3/8 already finished.
• D: This lets the partner finish the remaining 5/8 alone and leaves out the mason's continued work.

Answer: B

Correct answer: 20 minutes. Find each rate:

bottles per minute and

bottles per minute.

Combined:

per minute. Time:

• A: This averages the two rates (95 per minute) instead of adding them.
• B (correct): A fills 120 per minute and B fills 70 per minute, so together 190 per minute, and 3,800 / 190 = 20 minutes.
• C: This uses machine A's rate of 120 per minute alone and leaves out machine B.
• D: This uses machine B's rate of 70 per minute alone and leaves out machine A.

Answer: C

Correct answer: $2,724. Simple interest: I = principal × rate × time =

Total paid back:

• A: This is the interest alone; the question asks for the loan plus interest.
• B: This includes only one year of interest ($108) instead of three.
• C (correct): Interest is $2,400 × 0.045 × 3 = $324, and $2,400 + $324 = $2,724.
• D: This uses 4.5 as 0.045 per year but multiplies by 4 years instead of 3.

Answer: D

Correct answer: $120. Work backward from the end. Before lunch she had

After spending one-third on the jacket, two-thirds remained, so

Check: one-third of 120 is 40,

• A: This undoes the lunch but stops before undoing the jacket purchase.
• B: This undoes the steps in the wrong order, scaling $56 up first and adding $24 last.
• C: This treats the $80 as one-third of her money instead of the two-thirds left after the jacket.
• D (correct): Before lunch she had $56 + $24 = $80, which is two-thirds of her money, so she started with $80 / (2/3) = $120.

Answer: C

Correct answer: 2 hours. Each pump's rate is 1 divided by its time. Add the rates:

At one-half job per hour, the whole job takes 2 hours.

• A: This stops at the combined rate of 1/2 job per hour and reads it as half an hour instead of inverting it.
• B: This averages the three times; working together must be faster than even the fastest pump alone.
• C (correct): The rates 1/4, 1/6 and 1/12 add to 1/2 job per hour, so the job takes 2 hours.
• D: This adds the times, but it is the rates that add when pumps work together.

Answer: B

Correct answer: $116.00. Work backward through the steps in reverse order. Undo the tax:

Undo the 25% discount (the customer paid 75%):

Check:

• A: This is the sale price before tax; the 25% discount still has to be undone.
• B (correct): Removing tax: 93.96 ÷ 1.08 = $87; undoing the discount: 87 ÷ 0.75 = $116.
• C: This adds 25% to the sale price; the discount was 25% of the original, so divide by 0.75.
• D: This combines the steps into one 17% net discount; each percent applies to a different base.

Answer: C

Correct answer: $1,223.85. Regular pay:

Overtime rate:

for

Total:

• A: This pays all 46.5 hours at the regular $24.60 rate and ignores the overtime rate.
• B: This counts only 6 overtime hours instead of 6.5, losing the half hour.
• C (correct): Regular pay is 984.00, overtime is 6.5 hours at 36.90, or 239.85, and the total is 1,223.85.
• D: This pays all 46.5 hours at the overtime rate of $36.90.

Answer: B

Correct answer: Cooked beforehand. Pre- means before or ahead of time. Precooked food was cooked beforehand and only needs warming.

• A: Cooked too long is the meaning of overcooked; pre- deals with time, not amount.
• B (correct): Pre- means before, so precooked food was cooked ahead of time.
• C: Cooked again is the meaning of recooked or reheated; it uses re-, not pre-.
• D: Not yet cooked reverses the meaning; precooked food has already been cooked.

Answer: A

Correct answer: cooking tools. Use the example clue after such as: ladles, tongs, and a spatula. What they have in common is that they are cooking tools.

• A (correct): The examples ladles, tongs, and a spatula are all tools used in cooking.
• B: The examples are tools, not ingredients or rations.
• C: None of the examples are containers; they are tools for handling food.
• D: Recipes are instructions, while the examples are objects used by hand.

Answer: A

Correct answer: bad-tempered. Cantankerous means grouchy and quick to argue. The sentence gives behavior clues (grumbled, argued with every deckhand) that point to a sour temper.

• A (correct): Grumbling at passengers and arguing with each deckhand shows a cranky, quarrelsome temper, which is what cantankerous means.
• B: A ferry captain may well be experienced, but nothing in his behavior points to skill; the clues are all about his mood.
• C: A cheerful person would not grumble and argue all morning; this is the opposite tone.
• D: Asking and answering questions does not suggest memory trouble; this invents a detail the sentence does not give.

Answer: D

Correct answer: friendly. Cordial means warm, courteous, and friendly. Substitute friendly into the sentence and it keeps the meaning; brief and firm describe other features of a handshake.

• A: A handshake may be brief, but cordial describes its warmth, not its length.
• B: Rivals might be tense, but cordial points the other way; this assumes a story the word rules out.
• C: A very firm grip describes strength; cordial describes friendliness of manner.
• D (correct): Cordial means warm and friendly in a polite way, the right word for a sporting handshake between rivals.

Answer: A

Correct answer: Dried out. De- means remove, and hydrate means to supply with water. An empty canteen after hours in the heat means water has been lost from the body, so the recruit was dried out.

• A (correct): De- means remove, and hydrate refers to water, so dehydrated means having lost needed water, or dried out.
• B: Waterlogged means soaked with water, which ignores the remove sense of de-.
• C: Overheated fits the desert heat but describes too much heat, not a loss of water.
• D: Underfed means too little food; it uses under- and the wrong base.

Answer: A

Correct answer: the same throughout. Homo means same and the -gene part refers to kind, so homogeneous means of the same kind throughout. Stirring blends the paint until every part is alike.

• A (correct): Homo means same, so a homogeneous mixture looks and behaves the same in every part, which is the goal of stirring.
• B: This is heterogeneous; swapping homo, same, for hetero, different, reverses the meaning.
• C: Stirring may change thickness, but the root homo is about sameness, not thinness.
• D: Temperature would need the root therm; nothing in this word refers to heat.

Answer: B

Correct answer: average. MEAN follows "the" and is linked to "of the five quiz scores," a math setting. The mean is the average: add the scores and divide by how many there are.

• A: Unkind is the adjective sense of mean, but here mean is a noun after "the."
• B (correct): In math, the mean of a set of scores is their average, which fits "of the five quiz scores."
• C: Intent connects to the verb mean (to intend), which does not fit a set of scores.
• D: The lowest score is the minimum, not a single number describing all five scores together.

Answer: A

Correct answer: distant. Aloof means remote and cool toward other people. The clues, polite nods but no conversation or participation, show distance, not anger or modesty.

• A (correct): Aloof means keeping apart from others and staying cool and uninvolved, matching someone who nods but never joins in.
• B: Modest is about not bragging about oneself; the sentence is about keeping away from others, not about boasting.
• C: Nothing in the sentence shows anger; the neighbor is even polite. Adding an emotion the text does not show is a trap.
• D: A cheerful person usually joins in; the neighbor's avoidance points the other way.

Answer: C

Correct answer: arrogant. The clue, speaking to the crew as if they were beneath him, shows proud scorn. Remove the opposite (humble) and the sound-alike (naughty), then compare: wealthy is merely related, so arrogant is the precise match.

• A: Humble is the opposite: modest rather than proud.
• B: Naughty only rhymes with the target; it means badly behaved, which is a sound-alike trap.
• C (correct): Arrogant captures the core sense of acting superior and looking down on others, which the sentence shows.
• D: A manager may earn more, but wealth is loosely related, not the meaning of the word.

Answer: D

Correct answer: easily noticed. The root spic is a variant of spect, look. Something conspicuous is easy to look at and notice, which is exactly where an evacuation sign should be placed.

• A: This is the opposite; a hidden sign would defeat its purpose.
• B: Guarding involves watching, but the word describes how easily the sign is seen, not protected.
• C: This sounds a little alike, but suspicious means causing distrust, not easy to see.
• D (correct): Spic means look, and a sign meant for an evacuation is placed where people will see it easily.

Answer: B

Correct answer: required. Compulsory means required by a rule or law. Required matches exactly; recommended is too weak, voluntary is the opposite, and expensive is unrelated.

• A: This is the opposite; voluntary means left to a person's choice.
• B (correct): Compulsory means something must be done, which is what required means.
• C: A recommendation is advice, which is weaker than a rule that must be followed.
• D: Cost is unrelated to whether wearing a hard hat must be done.

Answer: A

Correct answer: shortage. Dearth means a scarcity or lack. The result 'slowed repairs' supports shortage; death is a look-alike word, surplus is the opposite, and training is only associated with mechanics.

• A (correct): Dearth means a lack or scarcity of something, which shortage expresses.
• B: This is a sound-alike trap; dearth has nothing to do with dying.
• C: This is the opposite; a surplus of mechanics would not slow repairs.
• D: Training is related to whether mechanics are qualified, but it does not mean a lack of them.

Answer: A

Correct answer: consider. ENTERTAIN takes the object "offers," and "before signing" shows the owners are still weighing other bids. Entertain here means consider.

• A (correct): To entertain an offer is to think it over, which fits owners not yet committed to a contract.
• B: Amuse is the familiar sense, but offers are not an audience to keep amused.
• C: Hosting fits guests at a party, not offers from buyers.
• D: Turning down offers is the opposite of agreeing to entertain them before signing.

Answer: D

Correct answer: accountable. The sentence says the supervisor must answer for later problems because she signed off. That is the accountable sense of responsible. Responsive looks similar but means quick to react, so treat the two as separate words.

• A: Quick to react is the meaning of responsive, a look-alike word that is a classic false friend.
• B: Sensible means practical; responsible can describe a sensible person, but here it means answerable for the problems.
• C: Blameless means free of fault, the reverse of being the one who must answer for problems.
• D (correct): Accountable means required to answer for something, which is what signing off on the check makes the supervisor.

Answer: A

Correct answer: unable to be undone. Read from the inside out: reverse means undo, -ible makes it able to be undone, and ir- (not) reverses that. So irreversible means unable to be undone. Skipping the prefix gives the exact opposite.

• A (correct): Reverse plus -ible means able to be undone, and ir- means not, so the word means unable to be undone.
• B: This reading uses -ible but skips the prefix ir-, which flips the meaning.
• C: This takes only the base reverse and ignores both the prefix and the suffix.
• D: This adds the idea of incomplete work, which is not part of the word.

Answer: B

Correct answer: The straw needed for the pens arrived late. The manager explains the delay: the straw truck was held up, and the barn could not open until straw was spread in every pen. Name the cause the passage states, not a cause that merely sounds reasonable.

• A: The goats are never described as late; judging simply started after the barn opened.
• B (correct): The manager says the straw truck was held up and the barn could not open until the straw was spread.
• C: The weigh station delayed the truck; it gave no order about the barn, so this assigns the action to the wrong actor.
• D: Crowd size is never mentioned, so this cause is not in the passage.

Answer: A

Correct answer: The torque wrench. To resolve a pronoun, find the earlier noun that fits what the sentence says about it. "It" was calibrated and has readings, which only makes sense for a measuring tool, so it refers to the torque wrench.

• A (correct): A tool is calibrated and gives readings, so "it" points back to the torque wrench.
• B: A checklist is not calibrated and does not give readings; it is mentioned separately in the last sentence.
• C: The inspection had not yet begun, and an inspection is not something that is calibrated.
• D: The landing gear is the part being worked on, but landing gear does not produce readings that need to be trusted.

Answer: C

Correct answer: In wet coastal pastures, treated pine held up better than untreated cedar. A comparison stated for one setting stays inside that setting. The study tested only wet coastal pastures, where treated pine showed little rot and cedar often rotted, so the supported conclusion is limited to that setting.

• A: The study covered only wet coastal pastures, so it says nothing about other kinds of fields.
• B: The one-third figure comes from the wet coastal pastures; dry upland fields were never tested.
• C (correct): This keeps the result inside the setting the study actually tested.
• D: This reverses the result; the cedar posts were the ones that rotted.

Answer: A

Correct answer: The marina prepared its dock staff to contain fuel spills quickly. Hold each choice against every sentence. Only the statement about preparing staff to contain spills covers the training, the specific skills, and the July example; the others are either bigger than the passage or only one sentence of it.

• A (correct): Every sentence fits under this: the training, the skills practiced, and the July spill that shows the training working.
• B: This is too broad; the passage discusses one marina's training, not pollution everywhere.
• C: This is one skill from the training, a detail that leaves out the pump shutoff, the phone calls, and the July spill.
• D: This is the most dramatic event, but it serves as an example of the training working rather than as the passage's point.

Answer: B

Correct answer: empty. Replace the word with each choice and reread the sentence. The planner warns that waiting until a storm is announced fails because supplies are gone and lines are long, so "shelves are often empty" keeps the sentence logical.

• A: This is a common meaning of bare, as with bare skin, but shelves that are uncovered would not stop anyone from buying supplies.
• B (correct): The planner is saying that supplies have already been bought up, so the shelves have nothing left on them.
• C: Plain describes something simple or undecorated, which does not explain why waiting to shop is a problem.
• D: Nothing in the sentence suggests the shelves are broken; the point is about missing goods.

Answer: C

Correct answer: The walk may take too much of a short lunch period. "Even so" tells you the outcome is surprising, which hints at a reason. The details the writer adds right afterward are the five-minute walk and the thirty-minute lunch, so the modest inference is that the trip eats into a short break. The wrong answers contradict the passage, invent a rule, or turn "most" into "all."

• A: The passage describes padded chairs, coffee and a view, so this conflicts with the details given.
• B: No rule against leaving the floor appears; this invents a restriction the passage never mentions.
• C (correct): The contrast word "even so" signals a hidden reason, and the passage stresses the distance and the thirty-minute lunch, so the time cost is the best-supported explanation.
• D: The passage says "most" workers eat at their machines, which leaves room for some who use the break room.

Answer: D

Correct answer: It gives a case that shows the common belief is false. The passage opens with a common belief and then challenges it. A single building struck many times a year is a counterexample, a case that does not fit the belief, so its job is to show the belief is wrong.

• A: The skyscraper is struck again and again, which disproves the belief rather than backing it up.
• B: The sentence gives no definition; it reports how often one building is hit.
• C: The explanation of why lightning returns comes in the last sentence; this sentence provides the example that challenges the saying.
• D (correct): A building hit repeatedly each year directly contradicts the idea that lightning never strikes the same place twice.

Answer: C

Correct answer: To reassure residents that the discolored water is safe and temporary. Ask which direction the details push. The passage admits the water may look brown, then stresses that it is not a health risk, that it clears in hours and that a simple step fixes it. Lessening a worry in this way is reassurance; the laundry tip and the flushing explanation support that goal.

• A: This reverses the passage, which states the water still meets drinking standards; reading a risk into it ignores the calming wording.
• B: The passage only briefly mentions how flushing works; it never describes water treatment, so this names a broader topic rather than the passage's job.
• C (correct): The passage names a worry (brown water), says it is not a health risk and will pass, and gives simple steps, which is the direction of reassurance.
• D: No judgment words blame the city; the passage says sediment settles over time, which is a neutral fact.

Answer: A

Correct answer: good enough, but not impressive. Use both sides of the evidence. Hale succeeded (top of the rope within the limit), but the but-clause lists flaws to fix. A word that holds both ideas means good enough to pass without being impressive. Choices that are all praise or all failure miss half of the clues.

• A (correct): Hale met the standard by reaching the top in time, yet the instructor lists several flaws, so the word means adequate but unremarkable.
• B: This overstates the word; the instructor lists wasted energy, weak footwork and a long rest, which rule out top praise.
• C: Service suggests repair work, but that is an associated idea, not the meaning here; the word judges the quality of a climb.
• D: Hale reached the top within the time limit, so the climb met the standard; this reads the criticisms too strongly.

Answer: A

Correct answer: Every scale the inspector tested was either accurate or read too high. Keep every quantity word tied to its group. The "none read too low" statement covers only the tested scales, and those scales were either accurate (most) or too high (three). The passage says nothing about the 15 untested stands, and it says the three owners must fix, not replace, their scales.

• A (correct): Most tested scales were accurate, three read too high, and none of the tested scales read too low, so each tested scale falls into one of the first two groups.
• B: Only 20 stands were tested, and three of those scales read too high, so this is both overstated and contradicted.
• C: The passage makes the "none" claim only about the 20 tested scales; the 15 untested stands have not been checked.
• D: The owners were told to fix their scales, not replace them; the choice borrows passage words but changes the action.

Answer: D

Correct answer: 24 square feet. Area of a rectangle is length times width: 6 times 4 = 24 square feet. Covering a surface is an area question, so the units are square feet.

• A: This is the perimeter value, 2 times (6 + 4) = 20, which measures the edge, not the surface.
• B: The number is right, but area must be given in square units, not plain feet.
• C: This adds the length and width instead of multiplying them.
• D (correct): Area is length times width, 6 times 4, in square units.

Answer: A

Correct answer: Perimeter, in feet. Ask what the fence does: it goes around the edge. A boundary is a perimeter, measured in plain length units such as feet.

• A (correct): Fencing runs along the edge of the run, so it is a length around the boundary.
• B: Area measures the ground inside the run, which would be needed for sod, not fencing.
• C: Volume measures how much fits inside a solid, not the length of the boundary.
• D: Surface area covers the faces of a solid; a fence follows only the edge.

Answer: A

Correct answer: 15x⁶. To multiply monomials, multiply the coefficients and add the exponents of the same variable:

The product is $15x^6$.

• A (correct): Multiply the coefficients, 3 x 5 = 15, and add the exponents, 2 + 4 = 6.
• B: This multiplies the exponents instead of adding them.
• C: This adds the coefficients instead of multiplying them.
• D: This adds the coefficients and multiplies the exponents, reversing both rules.

Answer: A

Correct answer: (-6, 2). Across the y-axis, (x, y) becomes (-x, y). So (6, 2) becomes (-6, 2), the same height but on the other side of the y-axis.

• A (correct): Reflecting across the y-axis changes the sign of x and keeps y.
• B: This changes the sign of y, which is a reflection across the x-axis.
• C: This changes both signs, which is the match through the origin, not a single reflection.
• D: This swaps the coordinates, which is a reflection across the line y = x.

Answer: D

Correct answer: 2¹⁰. Evaluate each power as repeated multiplication:

The largest is $2^{10}$. A large exponent on a small base can beat a large base with a small exponent.

• A: 10^3 = 1,000, slightly less than 2^10 = 1,024.
• B: 3^6 = 729, smaller than both 1,000 and 1,024.
• C: 5^4 = 625, the smallest of the four values.
• D (correct): 2^10 = 1,024, which beats 1,000, 729 and 625.

Answer: D

Correct answer: 324π square inches. Convert the radius first: 1.5 feet = 18 inches. Then A = π(18)^2 = 324π square inches.

• A: This is the area in square feet, π times 1.5 squared, with the units never converted.
• B: This converts 2.25 square feet by multiplying by 12; a square foot is 144 square inches.
• C: This is 2π times 18, the circumference in inches, not the area.
• D (correct): Convert 1.5 feet to 18 inches before squaring: π times 324.

Answer: C

Correct answer: 18. Factor into primes: 54 = 2 × $3^3$ and 90 = 2 × $3^2$ × 5. Multiply the shared primes at their lower powers: 2 × $3^2$ = 18.

• A: 9 divides both numbers, but it misses the shared factor of 2.
• B: 6 is a common factor, but it uses only one 3 when both numbers share two.
• C (correct): 54 = 2 × $3^3$ and 90 = 2 × $3^2$ × 5 share 2 × $3^2$ = 18.
• D: 270 is the least common multiple, not the greatest common factor.

Answer: D

Correct answer:

The inputs do not rise by 1, so divide each output change by its input change: from x = 2 to x = 5 the output falls 6 over 3 steps, a rate of −2, and from 5 to 11 it falls 12 over 6, again −2. Then

gives b = 10, so

• A: This uses the output change of −6 as the slope without dividing by the input change of 3; it fits only the first point.
• B: This loses the sign of the slope; the outputs fall as x rises, so the slope is negative.
• C: This treats the first output, 6, as the starting value, but 6 is the output at x = 2, not at x = 0.
• D (correct): It gives 6, 0 and −12 for x = 2, 5 and 11.

Answer: C

Correct answer: The lines are parallel, so the system has no solution. Rewrite both in slope-intercept form. The first becomes

and the second becomes

The slopes are equal but the y-intercepts differ, so the lines are parallel and never meet: the system has no solution.

• A: The variable terms of the first equation are double those of the second, but 8 is not double 6, so the lines are not identical.
• B: The lines would cross once only if their slopes differed; here both slopes are 0.5.
• C (correct): Both lines have slope 0.5 but intercepts −2 and −3, so they never cross.
• D: Perpendicular lines need slopes whose product is −1; these slopes are equal, which makes the lines parallel.

Answer: B

Correct answer: 4y⁶. Treat the numbers and the variable separately. Divide the coefficients:

For the same base, subtract exponents:

The result is $4y^6$.

• A: This subtracts the coefficients, 24 - 6, instead of dividing them.
• B (correct): Divide the coefficients, 24 / 6 = 4, and subtract the exponents, 10 - 4 = 6.
• C: This adds the exponents, which is the rule for multiplying, not dividing.
• D: This multiplies the exponents, which is the rule for a power of a power.

Answer: A

Correct answer: x > 5. Add 4x to both sides to keep a positive coefficient:

Add 6: 15 < 3x. Divide by 3: 5 < x, which is x > 5. Check x = 6:

which is less than −12.

• A (correct): Adding 4x gives 9 < 3x − 6, so 15 < 3x and x > 5.
• B: This reverses the sign without dividing by a negative; the coefficient 3 is positive.
• C: This comes from a sign slip that produces −15 < 3x instead of 15 < 3x.
• D: This subtracts 6 from 9 instead of adding 6 to both sides, getting 3 < 3x.

Answer: B

Correct answer: -2⁴ × (-1)³. Find each sign before any arithmetic. In $-2^4\times (-1{)}^3$,

two negative factors give a positive product, 16. Each of the other expressions ends with an odd count of negatives.

• A: (−2)³ = −8 and (−1)⁴ = 1, so the product is −8.
• B (correct): −2⁴ = −16 and (−1)³ = −1, and two negatives multiply to +16.
• C: (−3)² = 9, and the minus sign in front makes it −9.
• D: Three negative factors give a negative product, −6.

Answer: B

Correct answer: 25 square inches. In a 45-45-90 triangle each leg is the hypotenuse divided by $\sqrt{2}$:

The legs are the base and height, so the area is

• A: This forgets the one-half in the area formula.
• B (correct): Each leg is $5\sqrt{2}$, and half of $5\sqrt{2}\times 5\sqrt{2}=50$ is 25.
• C: This takes each leg as 5 (half the hypotenuse); the legs are $5\sqrt{2}$.
• D: This leaves a stray $\sqrt{2}$; $5\sqrt{2}\times 5\sqrt{2}$ is 50, a whole number.

Answer: A

Correct answer: x = 2. Distribute on both sides, flipping every sign after the minus:

Combine like terms:

Subtract 2x and add 5 to get x = 2.

Check: the left side is

and the right side is

• A (correct): Distributing gives 6x − 10 − 4x + 7 = 3x + 6 − 11, which simplifies to 2x − 3 = 3x − 5, so x = 2.
• B: This comes from writing −(4x − 7) as −4x − 7, flipping only the first sign inside the parentheses.
• C: This comes from writing 3(x + 2) as 3x + 2, multiplying only the first term by 3.
• D: This comes from writing 2(3x − 5) as 6x − 5, multiplying only the first term by 2.

Answer: A

Correct answer: 129°. Because OD bisects angle COB, angle COB = 2(x + 16). Angles AOC and COB form a linear pair:

so 4x = 140 and x = 35.

Then angle AOC = 78° and angle COD = 51°, so angle AOD =

°.

• A (correct): Angle AOD is angle AOC plus angle COD: 78 + 51 = 129°.
• B: 51° is angle COD, only half of angle COB.
• C: 102° is angle COB, the whole bisected angle, not angle AOD.
• D: 35 is x, not an angle measure.