Prealgebra

Prealgebra Practice Test: Practice Test 2

Practice Test 2 for Prealgebra: real questions and explanations from the Varsity Tutors practice-test pool.

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Question 1 of 25

A manufacturing process requires the width $w$ of a component to satisfy $|w - 12| \leq 0.5$ . Additionally, quality control rejects components where $w < 11.7$ or $w > 12.4$ . What range of widths passes both requirements?

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Question 1

$11.5 \leq w \leq 12.5$
$11.7 \leq w \leq 12.4$ (correct answer)
$11.7 \leq w \leq 12.5$
$11.5 \leq w \leq 12.4$

Explanation: First, solve $|w - 12| \leq 0.5$ : this gives $-0.5 \leq w - 12 \leq 0.5$ , so $11.5 \leq w \leq 12.5$ . Quality control accepts components where it's NOT true that $w < 11.7$ or $w > 12.4$ , meaning $11.7 \leq w \leq 12.4$ . The intersection of both requirements is $11.7 \leq w \leq 12.4$ . Choice A gives only the absolute value constraint. Choice C incorrectly takes the left bound from quality control and right bound from absolute value. Choice D incorrectly takes the left bound from absolute value and right bound from quality control.

Question 2

Marcus is planning a school fundraiser selling two types of items: keychains for $3 each and bookmarks for $2 each. He needs to raise at least $240 and wants to sell exactly 100 items total. If he sells $ $k$ $ keychains, which inequality represents the constraint for reaching his fundraising goal, and what is the minimum number of keychains he must sell?

$3k + 2(100-k) \geq 240$ ; he must sell at least 40 keychains (correct answer)
$3k + 2(100-k) \geq 240$ ; he must sell at least 50 keychains
$2k + 3(100-k) \geq 240$ ; he must sell at least 40 keychains
$3k + 200 \geq 240$ ; he must sell at least 60 keychains

Explanation: The inequality $3k + 2(100-k) \geq 240$ correctly represents the constraint where Marcus sells k keychains at $3 each and (100-k) bookmarks at $2 each. Solving: $ $3k + 200 - 2k \geq 240$ $, so$ $k \geq 40$ $. Choice B uses the correct inequality but miscalculates the solution. Choice C switches the prices. Choice D incorrectly assumes all non-keychains cost $0.

Question 3

A local theater sells adult tickets for $12 each and student tickets for $8 each. For a particular show, they need to sell exactly 100 tickets and collect exactly $1000 in revenue.

If this situation is modeled as a system of equations and graphed, what does the intersection point represent in this real-world context?

The theater sells 50 adult tickets and 50 student tickets for maximum profit
The theater sells 25 adult tickets and 75 student tickets to meet both constraints
The theater sells 40 adult tickets and 60 student tickets to satisfy the revenue requirement
The theater sells 50 adult tickets and 50 student tickets to meet both the ticket count and revenue requirements (correct answer)

Explanation: The system is: a + s = 100 (total tickets) and 12a + 8s = 1000 (total revenue), where a = adult tickets and s = student tickets. Solving: from the first equation, s = 100 - a. Substituting: 12a + 8(100 - a) = 1000, so 12a + 800 - 8a = 1000, giving 4a = 200, so a = 50 and s = 50. Verification: 50 + 50 = 100 tickets ✓ and 12(50) + 8(50) = 600 + 400 = 1000 ✓. The intersection represents the unique combination that satisfies both constraints simultaneously. Choice A mentions profit maximization which isn't specified. Choices B and C give incorrect ticket combinations.

Question 4

A machine can produce 450 widgets in 6 hours. Working at the same rate, how many widgets can 3 such machines produce in 8 hours?

900 widgets
1200 widgets
1800 widgets (correct answer)
2400 widgets

Explanation: First find the rate per machine: $\frac{450 \text{ widgets}}{6 \text{ hours}} = 75$ widgets per hour per machine. With 3 machines for 8 hours: $3 \times 75 \times 8 = 1800$ widgets. Choice A incorrectly uses only 2 machines or 6 hours instead of 8. Choice B uses 2 machines for 8 hours. Choice D incorrectly calculates $450 \times \frac{8}{6} \times 3$ with an arithmetic error.

Question 5

A researcher wants to determine if the relationship between hours of sleep and reaction time is linear. She collected data from 20 participants and now needs to decide between creating a scatter plot with trend line, calculating correlation coefficient, or making a frequency table of sleep hours. Her goal is to assess linearity and determine the strength of the relationship. What tool choice provides the most complete analysis?

Frequency table, because it shows the distribution of sleep hours across all participants
Correlation coefficient, because it quantifies both the strength and direction of linear relationship
Scatter plot with trend line, because it visually shows linearity and reveals any outlier patterns
Both scatter plot and correlation coefficient, because visualization shows linearity while calculation measures strength (correct answer)

Explanation: The researcher has two distinct goals: assess linearity (best shown visually with scatter plot) and measure relationship strength (quantified by correlation coefficient). Using both tools provides complete analysis - the scatter plot reveals whether the relationship appears linear and identifies outliers, while the correlation coefficient gives a numerical measure of strength. Single tools address only one goal (A, B, C).

Question 6

Consider the system $\begin{cases} x - 2y = 4 \\ y = \frac{1}{2}x + c \end{cases}$ where $c$ is a constant. If this system has no solution, what can you conclude about the relationship between the two equations?

The equations represent intersecting lines with different slopes, so the system cannot have no solution
The equations represent perpendicular lines because their slopes are negative reciprocals of each other
The equations represent the same line when $c = -2$ , but parallel lines for any other value of $c$
The equations represent parallel lines because they have the same slope when written in slope-intercept form (correct answer)

Explanation: When you encounter a system of equations and need to determine when it has no solution, focus on the relationship between the lines represented by each equation. A system has no solution when the lines are parallel (same slope, different y-intercepts). To analyze this system, convert both equations to slope-intercept form ( $y = mx + b$ ). The second equation is already in this form: $y = \frac{1}{2}x + c$ . For the first equation, solve for $y$ : $x - 2y = 4$ $-2y = -x + 4$ $y = \frac{1}{2}x - 2$ Both equations now have the same slope of $\frac{1}{2}$ . For the system to have no solution, the lines must be parallel, which occurs when they have identical slopes but different y-intercepts. This happens when $c \neq -2$ . Choice A is incorrect because lines with the same slope don't intersect (unless they're identical). Choice B is wrong because perpendicular lines would have slopes that are negative reciprocals; here both slopes equal $\frac{1}{2}$ . Choice C reverses the logic—when $c = -2$ , the equations represent the same line (infinitely many solutions), but when $c \neq -2$ , they're parallel (no solution). Choice D correctly identifies that the equations represent parallel lines because they have identical slopes when written in slope-intercept form. Strategy tip: When analyzing systems for solution types, always convert to slope-intercept form first. Same slope with different y-intercepts = no solution; same slope with same y-intercept = infinite solutions; different slopes = one solution.

Question 7

A student runs a simulation to estimate the probability of getting exactly 2 heads when flipping 4 coins. She uses a random number generator producing digits 0-9, where even digits represent heads and odd digits represent tails. After 500 trials, she gets 180 successes. Based on these results, what should she conclude about her simulation?

The simulation is working correctly; the theoretical probability is 0.375 and her result of 0.36 is very close. (correct answer)
The simulation has an error; she should have gotten exactly 187.5 successes since that's what the theory predicts.
The simulation is flawed because even and odd digits don't represent a fair coin; she needs to use a different method.
The simulation needs more trials; 500 is too few to get an accurate estimate of the probability.

Explanation: The theoretical probability of exactly 2 heads in 4 flips is C(4,2) × (0.5)^4 = 6/16 = 0.375. Her simulation result is 180/500 = 0.36, which is very close. Even and odd digits do represent a fair 50-50 split. Choice B is wrong because simulations give estimates, not exact theoretical values. Choice C is incorrect about the digit assignment. Choice D is wrong because 500 trials is sufficient for a reasonable estimate.

Question 8

In the expression $4x^2 - 6x + 9 - 2x(x + 3)$ , which of the following correctly identifies the terms after expanding but before any combining of like terms?

Four terms: $4x^2$ , $-6x$ , $9$ , and $-2x(x + 3)$
Five terms: $4x^2$ , $-6x$ , $9$ , $-2x^2$ , and $6x$
Five terms: $4x^2$ , $-6x$ , $9$ , $-2x^2$ , and $-6x$ (correct answer)
Six terms: $4x^2$ , $-6x$ , $9$ , $-2x$ , $x$ , and $3$

Explanation: First, expand $-2x(x + 3) = -2x^2 - 6x$ . The complete expanded expression is $4x^2 - 6x + 9 - 2x^2 - 6x$ , which has 5 terms: $4x^2$ , $-6x$ , $9$ , $-2x^2$ , and $-6x$ . Choice A doesn't expand the last factor. Choice B incorrectly shows $+6x$ instead of $-6x$ from the expansion. Choice D incorrectly separates the factors within the parentheses.

Question 9

The distance from Earth to the nearest star (other than the Sun) is approximately $4.1 \times 10^{13}$ kilometers. The distance from Earth to the Moon is approximately $380,000$ kilometers. How many times farther is the nearest star than the Moon?

Approximately $1.08 \times 10^{8}$ times farther (correct answer)
Approximately $1.08 \times 10^{9}$ times farther
Approximately $9.27 \times 10^{7}$ times farther
Approximately $9.27 \times 10^{18}$ times farther

Explanation: First convert Moon distance to scientific notation: $380,000 = 3.8 \times 10^{5}$ . Then divide star distance by Moon distance: $\frac{4.1 \times 10^{13}}{3.8 \times 10^{5}} = \frac{4.1}{3.8} \times 10^{13-5} = 1.08 \times 10^{8}$ . Choice B uses incorrect exponent subtraction. Choice C inverts the coefficient division. Choice D multiplies the distances instead of dividing.

Question 10

A digital scale displays weights to the nearest gram. The scale shows three different readings throughout the day: 847 g in the morning, 851 g at noon, and 849 g in the evening.

If the actual weight at noon was $850.7$ grams, what is the range of possible actual weights (in grams) for the morning reading, assuming the scale's rounding behavior is consistent?

846.5 to 847.4
846.0 to 848.0
846.5 to 847.5 (correct answer)
847.0 to 847.9

Explanation: When you encounter rounding problems, you need to understand the range of values that round to a specific displayed number. Since this scale rounds to the nearest gram, any actual weight from 846.5 up to (but not including) 847.5 grams would display as 847 g. Let's verify this logic using the given information. The noon reading shows 851 g, and we know the actual weight was 850.7 g. Since 850.7 is closer to 851 than to 850, it correctly rounds to 851 g. This confirms the scale rounds to the nearest whole number. For any displayed value, the actual weight falls within a range of ±0.5 grams around whole numbers. Specifically, a reading of 847 g means the actual weight is anywhere from 846.5 g (inclusive) up to 847.5 g (exclusive). However, since we're dealing with continuous measurements, we typically express this as 846.5 to 847.5 grams. Looking at the wrong answers: Choice A (846.5 to 847.4) cuts off too early at 847.4, missing values like 847.45 that would still round to 847. Choice B (846.0 to 848.0) is far too wide—weights like 846.2 would round to 846, not 847, and 847.8 would round to 848. Choice D (847.0 to 847.9) misses the lower half of the range entirely, excluding values like 846.8 that clearly round up to 847. Remember: when a value rounds to the nearest whole number, the actual range extends 0.5 units below and 0.5 units above that displayed number.

Question 11

A elevator has a weight limit and currently contains 4 adults averaging 160 pounds each and some children averaging 80 pounds each. The elevator can safely hold a maximum of 2,000 pounds total. If $c$ represents the number of children currently in the elevator, which inequality represents the constraint on the number of additional adults $a$ that can enter?

$4 + c + a ≤ 2000$
$160 + 80c + 160a ≤ 2000$
$640 + 80c + 160a ≤ 2000$ (correct answer)
$640 + 80c + 160a ≥ 2000$

Explanation: When you encounter inequality word problems, the key is systematically translating each piece of information into mathematical terms, then combining them to represent the constraint. Let's build this inequality step by step. The elevator currently has 4 adults at 160 pounds each, which gives us $4 \times 160 = 640$ pounds from adults already inside. We also have $c$ children at 80 pounds each, contributing $80c$ pounds. Finally, $a$ additional adults will add $160a$ pounds. The total weight is $640 + 80c + 160a$ , and this must be less than or equal to the 2,000-pound limit. Answer choice C correctly captures this: $640 + 80c + 160a ≤ 2000$ . Answer choice A treats each person as weighing 1 pound instead of using their actual weights—this completely ignores the weight information given in the problem. Answer choice B makes a critical error by using 160 for the current adults instead of $4 \times 160 = 640$ , essentially treating the current load as just one adult rather than four. Answer choice D uses the correct weight calculation but has the inequality flipped—it suggests the total weight should be greater than or equal to 2,000 pounds, which would mean the elevator is overloaded. When solving inequality word problems, always identify what's already present, what's being added, and what the constraint is. Calculate existing quantities completely before setting up your inequality, and double-check that your inequality symbol points in the logical direction for the constraint.

Question 12

What is the value of $\frac{(2^3)^2 - 3^4}{2^2 \times 3^2 - 2^3}$ ?

$\frac{-65}{28}$
$\frac{17}{28}$
$\frac{-17}{28}$ (correct answer)
$\frac{65}{28}$

Explanation: When you encounter complex expressions with exponents, the key is to work systematically using the order of operations and exponent rules. This problem tests your ability to evaluate powers and simplify fractions carefully. Let's evaluate the numerator first: $(2^3)^2 - 3^4$ . Using the power rule $(a^m)^n = a^{mn}$ , we get $(2^3)^2 = 2^6 = 64$ . Then $3^4 = 81$ , so the numerator equals $64 - 81 = -17$ . For the denominator: $2^2 \times 3^2 - 2^3$ . We have $2^2 = 4$ , $3^2 = 9$ , and $2^3 = 8$ , giving us $4 \times 9 - 8 = 36 - 8 = 28$ . Therefore, our expression equals $\frac{-17}{28}$ , which is answer choice C. Now for the wrong answers: Choice A ( $\frac{-65}{28}$ ) likely comes from incorrectly calculating the numerator as $64 - 81 = -17$ but then making an arithmetic error. Choice B ( $\frac{17}{28}$ ) results from getting the correct absolute value but missing the negative sign in the numerator. Choice D ( $\frac{65}{28}$ ) combines both errors—wrong sign and possibly miscalculating $64 + 81$ instead of $64 - 81$ . Remember to always follow PEMDAS strictly with exponent problems: calculate all powers first, then handle multiplication and subtraction from left to right. Double-check your arithmetic, especially with negative numbers, as sign errors are common traps in these problems.

Question 13

Two hikers start at the same point. One hiker walks 5 miles north, then 12 miles east. The other hiker walks directly to the first hiker's final position. How much farther did the first hiker walk compared to the second hiker?

3.5 miles
4 miles (correct answer)
4.8 miles
5.2 miles

Explanation: When you see a problem involving perpendicular paths and direct distances, think about the Pythagorean theorem. The first hiker's path forms two legs of a right triangle, while the second hiker travels along the hypotenuse. Let's trace the first hiker's journey: 5 miles north plus 12 miles east equals a total distance of $5 + 12 = 17$ miles. The second hiker walks directly to the same endpoint, creating the hypotenuse of a right triangle with legs of 5 miles and 12 miles. Using the Pythagorean theorem: $c^2 = a^2 + b^2$ , so $c^2 = 5^2 + 12^2 = 25 + 144 = 169$ . Therefore, $c = \sqrt{169} = 13$ miles. The difference between their distances is $17 - 13 = 4$ miles. Looking at the wrong answers: A) 3.5 miles likely comes from incorrectly calculating the hypotenuse or making an arithmetic error. C) 4.8 miles might result from confusion about which distance to subtract from which, or from using the wrong formula entirely. D) 5.2 miles could stem from incorrectly using one of the leg lengths (5 miles) in the final calculation instead of properly applying the Pythagorean theorem. The correct answer is B) 4 miles. Study tip: When you see right-angle movement problems, immediately identify the right triangle. The two perpendicular movements form the legs, and the direct path is always the hypotenuse. Remember that 5-12-13 is a common Pythagorean triple worth memorizing for quick recognition.

Question 14

Consider the expression $3x^2 + 7xy - 2x^2 + 5y - 4xy + 8$ . After combining like terms, how many terms in the simplified expression have a coefficient with an absolute value greater than 4?

0 terms
1 term
2 terms (correct answer)
3 terms

Explanation: First, combine like terms: $3x^2 - 2x^2 = x^2$ , $7xy - 4xy = 3xy$ , $5y$ stays the same, and $8$ is the constant. The simplified expression is $x^2 + 3xy + 5y + 8$ . The coefficients are 1, 3, 5, and 8. Only 5 and 8 have absolute values greater than 4, so there are 2 terms. Choice A would result from not identifying any coefficients correctly. Choice B might come from only counting the constant term 8. Choice D would result from incorrectly counting the coefficient 1 as having absolute value greater than 4.

Question 15

A delivery truck can carry a maximum load of $2400$ pounds. It already has packages weighing $1650$ pounds loaded. If each additional package weighs $25$ pounds, what is the solution to the inequality representing how many more packages $p$ can be loaded?

$p \leq 30$ and the truck can carry exactly 30 more packages
$p \leq 30$ and the truck can carry at most 30 more packages (correct answer)
$p < 30$ and the truck can carry at most 29 more packages
$p \geq 30$ and the truck can carry at least 30 more packages

Explanation: The inequality is $1650 + 25p \leq 2400$ . Subtracting 1650: $25p \leq 750$ . Dividing by 25: $p ≤ 30$ . This means at most 30 packages. Choice A incorrectly states 'exactly' instead of 'at most'. Choice C uses strict inequality when equality is allowed. Choice D reverses the inequality direction completely.

Question 16

A machine produces $3\frac{3}{5}$ widgets in $\frac{4}{5}$ of a minute. What is the production rate in widgets per minute?

$4\frac{1}{2}$ widgets per minute (correct answer)
$\frac{36}{25}$ widgets per minute
$2\frac{4}{5}$ widgets per minute
$\frac{144}{100}$ widgets per minute

Explanation: Convert $3\frac{3}{5}$ to $\frac{18}{5}$ widgets. Find the unit rate: $\frac{\frac{18}{5}}{\frac{4}{5}} = \frac{18}{5} \times \frac{5}{4} = \frac{18}{4} = \frac{9}{2} = 4\frac{1}{2}$ widgets per minute. Choice B results from incorrectly computing $\frac{18}{5} \times \frac{4}{5}$ . Choice C comes from subtracting $\frac{4}{5}$ from $3\frac{3}{5}$ . Choice D represents an incorrect decimal conversion.

Question 17

A recipe calls for ingredients in the ratio $2:3:5$ by weight. If the total weight of these three ingredients is $4.8$ kg, and one ingredient makes up $30\%$ of the mixture, what is the weight of the heaviest ingredient?

$2.4$ kg (correct answer)
$2.0$ kg
$1.8$ kg
$1.44$ kg

Explanation: The ratio $2:3:5$ means the parts are $\frac{2}{10} = 20\%$ , $\frac{3}{10} = 30\%$ , and $\frac{5}{10} = 50\%$ of the total. The ingredient that makes up $30\%$ corresponds to the middle ratio part (3). The heaviest ingredient corresponds to ratio part 5, which is $50\%$ of $4.8$ kg = $2.4$ kg. Choice B assumes the $30\%$ ingredient is the heaviest. Choice C miscalculates $30\%$ of $4.8$ . Choice D calculates $30\%$ of $4.8$ , identifying the wrong ingredient as heaviest.

Question 18

A rectangular field is 80 meters long and 60 meters wide. A diagonal path cuts across the field from one corner to the opposite corner. If a person walks along the perimeter from the same starting corner to the opposite corner, how many meters longer is the perimeter route compared to the diagonal path?

35 meters
45 meters
40 meters (correct answer)
50 meters

Explanation: When you encounter a problem involving rectangular shapes and diagonal paths, you're working with the Pythagorean theorem. This question asks you to compare two different routes between opposite corners of a rectangle. First, let's find the diagonal path length. Using the Pythagorean theorem with legs of 80 meters and 60 meters: $\sqrt{80^2 + 60^2} = \sqrt{6400 + 3600} = \sqrt{10000} = 100$ meters. Next, calculate the perimeter route. Walking along the edges from one corner to the opposite corner means traveling the length plus the width: $80 + 60 = 140$ meters. The difference between these routes is $140 - 100 = 40$ meters, confirming answer C. Looking at the wrong answers: A) 35 meters might result from calculation errors when finding the square root or from rounding mistakes. B) 45 meters could come from incorrectly calculating the diagonal or making arithmetic errors in the subtraction. D) 50 meters might occur if you mistakenly calculated half the full perimeter (280 ÷ 2 = 140, then confused this in your final calculation). Remember this key strategy: rectangle diagonal problems almost always involve the Pythagorean theorem, and when you see "nice" numbers like 60 and 80, check if they form a Pythagorean triple. Here, 60-80-100 is a multiple of the 3-4-5 triangle (multiplied by 20), which can speed up your calculations and help you avoid computational errors.

Question 19

The equation $\frac{x + 5}{3} = 8$ has the same solution as which of the following equations?

$x + 5 = 11$
$x + 5 = 24$ (correct answer)
$\frac{x}{3} + 5 = 8$
$3x + 15 = 8$

Explanation: Solving $\frac{x + 5}{3} = 8$ : multiply both sides by 3 to get $x + 5 = 24$ , then $x = 19$ . Choice B gives the same equation after the first step. Choice A would give $x = 6$ , Choice C would give $x = 9$ , and Choice D would give $x = -\frac{7}{3}$ .

Question 20

A water tank loses $\frac{1}{6}$ of its water each day due to evaporation. If the tank starts with $\frac{4}{5}$ of its capacity filled, what fraction of the tank's total capacity remains after 2 days?

$\frac{2}{3}$ of capacity
$\frac{20}{45}$ of capacity
$\frac{5}{9}$ of capacity (correct answer)
$\frac{1}{2}$ of capacity

Explanation: Each day, $\frac{5}{6}$ of the water remains (since $\frac{1}{6}$ evaporates). After day 1: $\frac{4}{5} \times \frac{5}{6} = \frac{20}{30} = \frac{2}{3}$ of capacity. After day 2: $\frac{2}{3} \times \frac{5}{6} = \frac{10}{18} = \frac{5}{9}$ of capacity. Choice A represents the amount after only 1 day. Choice B is the unsimplified form of $\frac{2}{3}$ . Choice D comes from incorrectly subtracting $\frac{1}{6}$ twice from $\frac{4}{5}$ .

Question 21

A student simplified $5a^2b - 3ab^2 + 2a^2b + 4ab^2 - ab^2$ and got $7a^2b + 0ab^2$ . The student then wrote the final answer as $7a^2b$ . What error, if any, did the student make?

The student incorrectly combined $a^2b$ terms; the coefficient should be $6$ instead of $7$
The student incorrectly combined $ab^2$ terms; the result should be $ab^2$ instead of $0ab^2$
The student made no error; $7a^2b$ is the correct simplified form of the expression (correct answer)
The student incorrectly assumed that $a^2b$ and $ab^2$ are like terms when they are not

Explanation: Let's check the student's work: For $a^2b$ terms: $5a^2b + 2a^2b = 7a^2b$ ✓. For $ab^2$ terms: $-3ab^2 + 4ab^2 - ab^2 = 0ab^2 = 0$ ✓. The student correctly identified that $a^2b$ and $ab^2$ are not like terms, combined each group separately, and properly wrote $7a^2b + 0 = 7a^2b$ . Choice A gives wrong coefficient calculation. Choice B miscalculates the $ab^2$ terms. Choice D incorrectly suggests the student confused unlike terms.

Question 22

A water tank is being filled at a rate of 15 gallons per minute and drained at a rate of 8 gallons per minute simultaneously. The tank starts with 120 gallons and must maintain at least 50 gallons at all times for safety reasons. Which inequality represents the time constraint $t$ (in minutes) for how long this process can continue?

$120 + 7t ≥ 50$ (correct answer)
$120 - 7t ≥ 50$
$120 + 15t - 8t ≥ 50$
$120 - 15t + 8t ≥ 50$

Explanation: Net rate = 15 - 8 = 7 gallons per minute (filling faster than draining). After t minutes: amount = 120 + 7t. The constraint is this amount ≥ 50, so 120 + 7t ≥ 50. Choice B incorrectly assumes the tank is losing water overall. Choices C and D are mathematically equivalent to A and B respectively, but don't simplify the net rate, making them less clear representations of the constraint.

Question 23

Point $A$ is located at $-12$ on a number line. Point $B$ is the same distance from zero as point $A$ , but in the opposite direction. Point $C$ is located so that the distance from $C$ to zero equals the distance from $A$ to $B$ . How many possible locations are there for point $C$ ?

Exactly one location for point $C$
Exactly four locations for point $C$
Exactly three locations for point $C$
Exactly two locations for point $C$ (correct answer)

Explanation: When you encounter problems about points on a number line and distances, focus on understanding absolute value and how distance works in both directions from any reference point. Let's work through this step by step. Point A is at $-12$ . Since point B is the same distance from zero as A but in the opposite direction, B must be at $+12$ . The distance from A to zero is $|-12| = 12$ , so B is 12 units on the positive side. Now find the distance from A to B: from $-12$ to $+12$ is $12 - (-12) = 24$ units total. Point C must be located so that its distance from zero equals this distance from A to B, which is 24 units. Since distance is always positive, C could be either 24 units to the right of zero (at $+24$ ) or 24 units to the left of zero (at $-24$ ). That gives us exactly two possible locations for point C. Answer choice A is wrong because there isn't just one location—distance from zero creates two possibilities. Answer choice B incorrectly suggests four locations, which would happen if you mistakenly counted distances in multiple directions or made calculation errors. Answer choice C suggests three locations, which has no mathematical basis for this type of distance problem. Remember: whenever a problem asks for points at a specific distance from zero (or any reference point), there are typically two solutions—one in each direction along the number line, unless the point is exactly at the reference point itself.

Question 24

Two competing cell phone plans charge data overage fees of 0.025 dollars per MB and 2.8% of a dollar per MB, respectively. What is the difference between these rates expressed as a percentage of a dollar?

2.5%
0.53%
0.3% (correct answer)
0.28%

Explanation: When comparing rates that are expressed in different formats, you need to convert them to the same units before finding the difference. Here you have one rate as a decimal (0.025 dollars per MB) and another as a percentage (2.8% of a dollar per MB). First, convert both rates to the same format. Since the question asks for the answer as a percentage, let's convert the decimal to a percentage. To convert 0.025 dollars to a percentage of a dollar, multiply by 100: $0.025 \times 100 = 2.5\%$ Now you can compare the two rates directly: 2.5% per MB versus 2.8% per MB. The difference is $2.8\% - 2.5\% = 0.3\%$ Looking at the wrong answers: Choice A (2.5%) gives you just the first rate converted to a percentage, but doesn't calculate the difference. Choice B (0.53%) likely comes from incorrectly converting 2.8% to a decimal (0.028) and then subtracting 0.025, giving 0.003, which someone might mistakenly read as 0.53%. Choice D (0.28%) appears to come from confusing 2.8% with 0.28% during the calculation. The correct answer is C. Study tip: When comparing rates in different formats, always convert to the same units first. Remember that to convert a decimal to a percentage, you multiply by 100, and to convert a percentage to a decimal, you divide by 100. Double-check your conversions before doing any arithmetic.

Question 25

A store marks up items by $37.5\%$ above cost. If an item costs the store $24, what fraction of the selling price represents the store's profit?

$\frac{3}{8}$
$\frac{3}{11}$ (correct answer)
$\frac{8}{11}$
$\frac{5}{13}$

Explanation: First convert the markup percentage: $37.5\% = \frac{3}{8} = 0.375$ . The selling price is $\$24 + \$24 \times 0.375 = \$24 + \$9 = \$33$ . The profit is $9, so the fraction of selling price that is profit is $ $\frac{9}{33} = \frac{3}{11}$ $. Choice A represents the markup rate as a fraction of cost, not selling price. Choice C represents cost as a fraction of selling price. Choice D results from incorrectly calculating$ $37.5\%$ $as$ $\frac{5}{13}$ $.