Prealgebra Practice Test: Practice Test 5

Question 1

1. Wednesday's change was exactly $+12$ dollars (correct answer)
2. Wednesday's change was exactly $-12$ dollars
3. Wednesday's change could be either $+12$ or $-12$ dollars
4. Wednesday's change was exactly $0$ dollars

Explanation: Monday: +x, Tuesday: -x, Wednesday: ±|x|. Net change: x + (-x) + (±|x|) = ±|x| = +12. Since the net change is positive 12, Wednesday's change must be +12 (which equals +|x|). Choice B gives the wrong sign. Choice C suggests both are possible, but the net change constraint determines it uniquely. Choice D ignores the given net change.

Question 2

1. $2l + 2w = 60$ and $l = 2w + 8$ (correct answer)
2. $l + w = 60$ and $l = 2w + 8$
3. $2l + 2w = 60$ and $l = 8w + 2$
4. $l \times w = 60$ and $l = 2w + 8$

Explanation: The perimeter formula for a rectangle is $2l + 2w = 60$. The phrase 'length is 8 feet more than twice the width' translates to $l = 2w + 8$. Choice B uses the wrong perimeter formula (should be $2l + 2w$, not $l + w$). Choice C incorrectly models the length relationship as $l = 8w + 2$ instead of $l = 2w + 8$. Choice D confuses perimeter with area.

Question 3

1. The $x^2y$ term
2. The $xy$ term
3. No term has a coefficient that is a factor of all other coefficients
4. The constant term (correct answer)

Explanation: When you see an expression with multiple terms, your first step is always to combine like terms—terms that have identical variable parts. This question then asks you to find which coefficient is a factor of all the others. Let's combine the like terms in $6x^2y - 4xy + 8x^2y - 12xy + 2$. The $x^2y$ terms are $6x^2y + 8x^2y = 14x^2y$. The $xy$ terms are $-4xy + (-12xy) = -16xy$. The constant remains $2$. So our simplified expression is $14x^2y - 16xy + 2$. Now we need to check which coefficient divides evenly into all the others. The coefficients are 14, -16, and 2. Let's test each term: For choice A, we check if 14 divides into -16 and 2. Since $-16 ÷ 14$ doesn't give a whole number, 14 is not a factor of all coefficients. For choice B, we check if -16 divides into 14 and 2. Since $14 ÷ (-16)$ doesn't give a whole number, -16 is not a factor of all coefficients. For choice C, this would mean no coefficient works as a universal factor. For choice D, we check if 2 divides into all coefficients: $14 ÷ 2 = 7$, $-16 ÷ 2 = -8$, and $2 ÷ 2 = 1$. Since 2 divides evenly into all coefficients, the constant term has the coefficient that's a factor of all others. Study tip: When looking for common factors among coefficients, start with the smallest absolute value—it's the most likely candidate to divide into larger numbers.

Question 4

1. $\frac{1}{2}$ widget
2. $30$ widgets (correct answer)
3. $\frac{3}{2}$ widgets
4. $\frac{21}{8}$ widgets

Explanation: First, find the reduced rate: $\frac{4}{7} \times \frac{7}{12} = \frac{1}{3}$ widgets per minute. Convert time to minutes: $\frac{3}{2} \times 60 = 90$ minutes. Total widgets: $\frac{1}{3} \times 90 = 30$ widgets. Choice A comes from multiplying the fractions without converting time units. Choice C results from incorrect fraction multiplication. Choice D represents an intermediate calculation error.

Question 5

1. $49$ shapes total
2. $35$ shapes total
3. $42$ shapes total
4. $28$ shapes total (correct answer)

Explanation: When you see a pattern problem, your goal is to identify the rule for each type of object separately, then combine them to find the total. Let's track the squares and triangles in each row:

• Row 1: 1 square, 0 triangles
• Row 2: 2 squares, 1 triangle
• Row 3: 3 squares, 3 triangles
• Row 4: 4 squares, 6 triangles

Question 6

1. $a = 4, b = 4$
2. $a = 1, b = 7$
3. $a = 3, b = 5$ (correct answer)
4. $a = 5, b = 3$

Explanation: When you encounter an equation with unknown coefficients and a given solution, you need to use the fact that the solution must satisfy the equation to find relationships between those unknowns. Since $x = 2$ is the solution to $ax + 5 = 3x + b$, substituting $x = 2$ into the equation must make it true: $a(2) + 5 = 3(2) + b$ $2a + 5 = 6 + b$ $2a - b = 1$ Now you have two equations with two unknowns:

• $2a - b = 1$ (from substituting the solution)
• $a + b = 8$ (given condition)

Question 7

1. $\frac{2}{3} \cdot \frac{1}{3} \cdot \frac{s}{2}$
2. $\frac{2}{3} \cdot \frac{s}{6}$
3. $\frac{1}{3} \cdot \frac{s}{2} \cdot \frac{2}{3}$ (correct answer)
4. $\frac{s}{6} - \frac{2}{3}$

Explanation: For $s$ servings, the recipe requires $\frac{1}{3} \cdot \frac{s}{2} = \frac{s}{6}$ cups of flour. Sarah has $\frac{2}{3}$ of this amount, so she has $\frac{2}{3} \cdot \frac{s}{6} = \frac{s}{9}$ cups. This can be written as $\frac{1}{3} \cdot \frac{s}{2} \cdot \frac{2}{3}$ (order of multiplication doesn't matter). Choice A has an extra $\frac{1}{3}$ factor. Choice B is equivalent to the correct answer but not in the same form as choice C. Choice D subtracts $\frac{2}{3}$ instead of multiplying.

Question 8

1. $-15 < T \leq -10$
2. $-10 \leq T \leq 35$ (correct answer)
3. $-10 < T \leq 35$
4. $-15 < T < -10$ or $35 < T \leq 40$

Explanation: For safe operation without alarm, we need the intersection of safe operation ($-15 < T \leq 40$) and no alarm ($-10 \leq T \leq 35$). The alarm sounds when $T < -10$ or $T > 35$, so no alarm means $-10 \leq T \leq 35$. The intersection gives $-10 \leq T \leq 35$. Choice A gives only the unsafe region below -10. Choice C incorrectly excludes T = -10. Choice D gives the regions where the system is safe but the alarm IS sounding.

Question 9

1. 9.5% (correct answer)
2. 10.7%
3. 7.5%
4. 12.3%

Explanation: First, convert 0.275 to a percentage: 0.275 = 27.5%. Then find the difference: 27.5% - 18% = 9.5%. Choice B results from incorrectly adding the percentages (27.5% + 18% = 45.5%, then mistakenly dividing by some factor). Choice C comes from the error 0.275 - 0.18 = 0.095, then converting incorrectly as 0.095 × 100 = 9.5, but reporting 7.5%. Choice D results from calculation errors in the conversion process.

Question 10

1. $2,750$ widgets
2. $2,800$ widgets (correct answer)
3. $2,900$ widgets
4. $3,100$ widgets

Explanation: If $12.7\%$ are defective, then $87.3\%$ are good. Round to $87\%$ good and $2,450$ to $2,500$. We need $0.87x = 2,500$, so $x \approx 2,500 \div 0.87 \approx 2,500 \div 0.9 \approx 2,778$, closest to $2,800$. Choice A adds the defective amount incorrectly. Choice C uses $85\%$ good rate. Choice D significantly overestimates the defective rate impact.

Question 11

1. Any screen time greater than 8 hours should be considered unusually high
2. Any screen time greater than 11 hours should be considered unusually high
3. Any screen time greater than 14 hours should be considered unusually high (correct answer)
4. Any screen time greater than 17 hours should be considered unusually high

Explanation: Using the 1.5 × IQR rule: IQR = Q3 - Q1 = 8 - 4 = 4 hours. The upper fence = Q3 + 1.5(IQR) = 8 + 1.5(4) = 8 + 6 = 14 hours. Any value above 14 hours is considered unusually high (an outlier). Choice A uses Q3, choice B uses the maximum whisker value, and choice D incorrectly calculates the fence.

Question 12

1. $2$ (correct answer)
2. $7$
3. $14$
4. $21$

Explanation: When substituting $x = 3$ into $3x - 7$, we get $3(3) - 7 = 9 - 7 = 2$. Since Maria's answer gives 2 on the left side but the equation should equal 14, her solution is incorrect. The correct solution is $x = 7$.

Question 13

1. 84 square centimeters
2. 96 square centimeters
3. 48 square centimeters (correct answer)
4. 32 square centimeters

Explanation: Perimeter ratio is 7:4 (large:small), so the area ratio is (7/4)² = 49/16. If the large area is 147, then small area = 147 × (16/49) = 147 × 16 ÷ 49 = 2352 ÷ 49 = 48 square centimeters. Choice A incorrectly uses 4/7 instead of (4/7)². Choice B assumes incorrect linear relationship. Choice D uses wrong calculations.

Question 14

1. $\frac{3}{8}$ of the allocation (correct answer)
2. $\frac{1}{3}$ of the allocation
3. $\frac{5}{12}$ of the allocation
4. $\frac{1}{4}$ of the allocation

Explanation: Originally, Zone C gets: $1 - \frac{3}{8} - \frac{1}{6}$. Converting to common denominator 24: $1 - \frac{9}{24} - \frac{4}{24} = \frac{24-9-4}{24} = \frac{11}{24}$. After malfunction, Zone A uses $\frac{3}{8} + \frac{1}{12} = \frac{9}{24} + \frac{2}{24} = \frac{11}{24}$. Zone B still uses $\frac{1}{6} = \frac{4}{24}$. Zone C now gets: $1 - \frac{11}{24} - \frac{4}{24} = \frac{9}{24} = \frac{3}{8}$. Choice B ($\frac{1}{3} = \frac{8}{24}$) might result from minor calculation errors. Choice C ($\frac{5}{12} = \frac{10}{24}$) could result from not accounting for the malfunction properly. Choice D ($\frac{1}{4} = \frac{6}{24}$) might result from various computational mistakes.

Question 15

1. $2$ (correct answer)
2. $4$
3. $6$
4. $12$

Explanation: To pack all widgets into identical boxes with none left over, the number of widgets per box must be a divisor of each batch size. This means we need the GCF of the three batch sizes. Batch 1: $2^2 \cdot 3^1 = 12$ widgets. Batch 2: $2^1 \cdot 3^3 = 54$ widgets. Batch 3: $2^3 \cdot 5^2 = 200$ widgets. To find GCF(12, 54, 200), we use the minimum exponent for each prime: For prime 2: $\min(2, 1, 3) = 1$. For prime 3: $\min(1, 3, 0) = 0$ (since 200 has no factor of 3). For prime 5: $\min(0, 0, 2) = 0$ (since 12 and 54 have no factor of 5). Therefore, GCF = $2^1 \cdot 3^0 \cdot 5^0 = 2$. Choice B (4) would not divide 54 evenly since 54 = 2¹ × 3³, and 4 = 2². Choice C (6) would not divide 200 since 200 has no factor of 3. Choice D (12) would not divide 200 evenly since 200 has no factor of 3.

Question 16

1. Slope represents climbing rate; using $\frac{2050}{3}$ instead of $\frac{1200}{2.5}$
2. Slope represents total elevation gained; using $\frac{2050-1200}{1.25}$ instead of $\frac{2050}{2.5}$
3. Slope represents climbing rate; using $\frac{2050}{3}$ instead of $\frac{2050-1200}{2.5}$ (correct answer)
4. Slope represents average elevation; using $\frac{2050-1200}{1.25}$ instead of $\frac{2050+1200}{2.5}$

Explanation: When you encounter linear relationships involving motion or change over time, always remember that slope represents the rate of change — in this case, how fast the climber gains elevation per hour. To find the correct slope (climbing rate), you need the change in elevation divided by the change in time. The climber starts at 1200 feet and reaches 2050 feet, so she climbs $2050 - 1200 = 850$ feet in 2.5 hours. Her climbing rate is $\frac{850}{2.5} = 340$ feet per hour. The mistake that gives 680 feet per hour comes from using $\frac{2050}{3}$. Someone might incorrectly think the denominator should be 3 (perhaps confusing 2.5 hours with something else) and use the final elevation 2050 instead of the elevation change. This gives $\frac{2050}{3} ≈ 683$ feet per hour, which rounds to about 680. Looking at the choices: Choice A incorrectly describes slope calculation methods. Choice B wrongly claims slope represents "total elevation gained" rather than rate, and describes an impossible calculation with 1.25 hours. Choice D incorrectly states slope represents "average elevation" and suggests adding elevations rather than finding their difference. Only choice C correctly identifies that slope represents climbing rate and pinpoints the exact mistake: using $\frac{2050}{3}$ instead of the proper formula $\frac{2050-1200}{2.5}$. Study tip: For any linear relationship involving change over time, always use $\text{slope} = \frac{\text{change in output}}{\text{change in input}}$. Don't forget to subtract initial values when finding the change.

Question 17

1. 150 messages
2. 200 messages (correct answer)
3. 250 messages
4. 300 messages

Explanation: Set up equation: 25 + 0.05t = 35, where t is number of texts. Solving: 0.05t = 10, so t = 200 messages. Choice A results from solving 25 + 0.05t = 32.50. Choice C comes from using $0.04 per text instead of $0.05. Choice D results from setting up the equation incorrectly as 25 + 0.05t = 40.

Question 18

1. 18 rotations
2. 55 rotations
3. 68 rotations (correct answer)
4. 214 rotations

Explanation: First convert units: 500 feet = 6000 inches. The wheel's circumference is $C = 2\pi(14) = 28\pi \approx 87.96$ inches. Number of rotations = $\frac{6000}{87.96} \approx 68.2$, so 68 complete rotations. Choice A forgets to convert feet to inches. Choice B uses diameter instead of circumference. Choice D uses the radius as if it were the circumference.

Question 19

1. $\frac{1}{2}[(2n + 1)(3n - 2) + (n + 5)(3n - 2)]$ (correct answer)
2. $\frac{1}{2}(3n + 6)(3n - 2)$
3. $\frac{1}{2}[(2n + 1) + (n + 5)] \cdot (3n - 2)$
4. $\frac{1}{2}(6n^2 - 4n + 3n - 2 + 3n^2 - 2n + 15n - 10)$

Explanation: The area formula $A = \frac{1}{2}(b_1 + b_2)h$ becomes $A = \frac{1}{2}[(2n + 1) + (n + 5)](3n - 2)$. Using the distributive property, this equals $\frac{1}{2}[(2n + 1)(3n - 2) + (n + 5)(3n - 2)]$. Choice A correctly shows this distribution step. Choice B incorrectly simplifies the bases first. Choice C doesn't apply the distributive property yet. Choice D shows the result after full expansion but doesn't clearly demonstrate the distributive property application.

Question 20

1. Round to $w = 9$ feet and accept that she'll use slightly more than 60 feet of fencing
2. Round to $w = 8$ feet and use the leftover fencing for a separate small garden bed (correct answer)
3. Keep the exact decimal value since precision is important for maximizing garden area
4. Recalculate assuming she can buy additional fencing to make the width exactly 9 feet

Explanation: With $w = 8$ feet, the length becomes $2(8) + 4 = 20$ feet, and the perimeter is $2(8) + 2(20) = 56$ feet, leaving 4 feet of fencing unused - a practical solution. Choice A would require 62 feet of fencing (more than available). Choice C ignores the practical constraint that fractional feet are difficult to measure and work with in construction. Choice D unnecessarily complicates the problem by changing the constraint rather than working within it.

Question 21

1. The total perimeter is 34 units (correct answer)
2. The total perimeter is 38 units
3. The total perimeter is 42 units
4. The total perimeter is 46 units

Explanation: Analyzing the pattern: Each hexagon normally has perimeter 6, but when hexagons share a side, both lose one unit of perimeter per shared side. Pattern 1: 1 hexagon, 0 shared sides, perimeter = 6(1) = 6. Pattern 2: 2 hexagons, 1 shared side, perimeter = 6(2) - 2(1) = 10. Pattern 3: 3 hexagons, 2 shared sides, perimeter = 6(3) - 2(2) = 14. General formula: P_n = 6n - 2(n-1) = 4n + 2. For Pattern 8: P_8 = 4(8) + 2 = 34.

Question 22

1. 85%
2. 95% (correct answer)
3. 105%
4. 125%

Explanation: Jenny has 65% of the required flour initially. She borrows 30% more of the required amount. Total: 65% + 30% = 95% of the required flour. Choice A subtracts instead of adding the borrowed amount. Choice C incorrectly calculates 30% of what Jenny has rather than 30% of what's required. Choice D adds 65% and 30% then multiplies by some factor.

Question 23

1. The customer can park for a maximum of 5 complete hours plus part of the 6th hour (correct answer)
2. The customer can park for a maximum of 6 complete hours plus part of the 7th hour
3. The customer must park for exactly 5.8 hours to use their entire budget most efficiently
4. The customer can park for any amount of time since 5.8 hours exceeds typical parking needs

Explanation: Since parking is charged by complete hours, $h \leq 5.8$ means the customer can park for at most 5 complete hours (paying $4 + 4(2.50) = 14$) plus part of the 6th hour. If they stayed a complete 6th hour, they'd pay $16.50$, exceeding their budget. Choice B incorrectly suggests 6 complete hours work. Choice C treats it as an exact requirement. Choice D ignores the budget constraint.

Question 24

1. 275 feet above sea level (correct answer)
2. 275 feet below sea level
3. 225 feet above sea level
4. 150 feet above sea level

Explanation: The submarine's path: starts at 0, goes to -150, then to -75, then to -275 feet. The final depth is -275 feet, so |−275| = 275. Since the helicopter's altitude and submarine's depth are opposites, the helicopter is at +275 feet above sea level. Choice B gives the wrong direction. Choice C incorrectly calculates the final depth as -225. Choice D uses only the initial descent.

Question 25

1. Carmen is correct; the system as written is inconsistent and has no solution regardless of David's suggested change (correct answer)
2. David is correct; Carmen made an arithmetic error and the system actually has infinitely many solutions
3. Both are partially correct; Carmen's method is right but David correctly identifies that the original system contains a typo
4. Neither is correct; the system has exactly one solution that Carmen missed due to calculation errors

Explanation: Carmen's analysis is mathematically sound. The system as given is inconsistent because the left sides of the equations are proportional (the second equation's left side is exactly twice the first), but the right sides are not proportional (24 ≠ 2×12 = 20). This creates a contradiction, so there is no solution. David's suggestion would create a system with infinitely many solutions, but that doesn't mean the original system was wrong - it could intentionally be inconsistent. Choice B misunderstands the mathematics. Choice C assumes there's a typo without justification. Choice D is mathematically impossible given the structure.