Prealgebra Practice Test: Practice Test 12

Question 1

1. 5 hours
2. 6 hours (correct answer)
3. 6.5 hours
4. 7 hours

Explanation: First find the actual distance: $\frac{1.5 \text{ inches}}{60 \text{ miles}} = \frac{8.25 \text{ inches}}{x \text{ miles}}$. Cross multiply: $1.5x = 60 \times 8.25 = 495$, so $x = \frac{495}{1.5} = 330$ miles. Time = $\frac{330 \text{ miles}}{55 \text{ mph}} = 6$ hours. Choice A uses incorrect distance calculation. Choice C results from using 50 mph instead of 55 mph. Choice D comes from arithmetic errors in the proportion setup.

Question 2

1. $x - 2$
2. $x + 2$ (correct answer)
3. $x^2 + 2x + 4$
4. $\frac{x+2}{x-2}$

Explanation: When you see an expression with $x^3 + 8$ in the numerator, recognize this as a sum of cubes since $8 = 2^3$. The sum of cubes formula is $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$. Let's factor the numerator: $x^3 + 8 = x^3 + 2^3 = (x + 2)(x^2 - 2x + 4)$ Now substitute this back into the original expression: $\frac{x^3 + 8}{x^2 - 2x + 4} = \frac{(x + 2)(x^2 - 2x + 4)}{x^2 - 2x + 4}$ Since the factor $(x^2 - 2x + 4)$ appears in both numerator and denominator, they cancel out (assuming $x^2 - 2x + 4 \neq 0$), leaving us with $x + 2$. Looking at the wrong answers: Choice A gives $x - 2$, which would result if you mistakenly thought this was a difference of cubes or made a sign error during factoring. Choice C gives $x^2 + 2x + 4$, which might tempt you if you incorrectly canceled the $(x + 2)$ terms instead of the $(x^2 - 2x + 4)$ terms. Choice D suggests $\frac{x+2}{x-2}$, which could result from confusion about which terms should cancel or from incorrectly factoring the denominator. Study tip: Memorize both cube formulas: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ and $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$. When simplifying rational expressions, always look for common factors that cancel completely.

Question 3

1. Before distributing, there are 3 terms, and after distributing there are 4 terms
2. Before distributing, there are 3 terms, and after distributing there are 5 terms (correct answer)
3. The expression contains exactly 2 factors in its original form before any distribution
4. After simplifying completely, the coefficient of $x$ equals the sum of all constant terms

Explanation: Before distributing, the expression has 3 terms: $2(x + 4)$, $3(x - 1)$, and $5$. After distributing: $2x + 8 + 3x - 3 + 5$, which has 5 terms. Choice A miscounts the terms after distribution. Choice C is incorrect because there are more than 2 factors when considering the expressions in parentheses. Choice D is incorrect because after simplifying to $5x + 10$, the coefficient of $x$ is 5 and the constant term is 10, so they are equal, but this is coincidental and not generally true for such expressions.

Question 4

1. $3(8 - 2y) - y = 1$ (correct answer)
2. $3x - (8 - 2y) = 1$
3. $(8 - 2y) + 2y = 8$
4. $8 - 2y - y = 1$

Explanation: When using substitution, we replace the variable we solved for (x) with its expression (8 - 2y) in the other equation. The second equation is 3x - y = 1. Substituting x = 8 - 2y gives us 3(8 - 2y) - y = 1. This can be simplified to 24 - 6y - y = 1, then 24 - 7y = 1, giving y = 23/7. Choice B incorrectly substitutes the expression for y instead of x. Choice C substitutes into the wrong equation (the first one we already used). Choice D omits the coefficient 3 that multiplies x in the second equation.

Question 5

1. $\frac{10^{10}}{16 \times 625} = \frac{10^{10}}{10000} = 10^6$
2. $\frac{10^{10}}{2^6 \times 5^4} = \frac{10^{10}}{64 \times 625} = \frac{10^{10}}{40000}$
3. $10^4$ (correct answer)
4. $2^4 \times 5^4 = 10^4 \times 16 = 160000$

Explanation: First, recognize that $2^{10} \times 5^{10} = (2 \times 5)^{10} = 10^{10}$. Next, convert the denominator: $4^3 = (2^2)^3 = 2^6$ and $25^2 = (5^2)^2 = 5^4$. So the expression becomes $\frac{10^{10}}{2^6 \times 5^4}$. Since $2^6 \times 5^4 = 2^6 \times 5^4$, we can pair up factors: $\frac{2^{10} \times 5^{10}}{2^6 \times 5^4} = 2^{10-6} \times 5^{10-4} = 2^4 \times 5^4 = (2 \times 5)^4 = 10^4$. Choice A makes an arithmetic error in the denominator calculation. Choice B leaves the expression unsimplified. Choice D incorrectly multiplies instead of using the pattern $(ab)^n = a^n b^n$.

Question 6

1. 1 complete portion (correct answer)
2. 2 complete portions
3. 3 complete portions
4. 4 complete portions

Explanation: Calculate $\frac{2}{3} \div \frac{3}{8} = \frac{2}{3} \times \frac{8}{3} = \frac{16}{9} = 1\frac{7}{9}$. Since we need complete portions only, the answer is 1. Choice B results from rounding 1.78 incorrectly. Choice C comes from adding numerators (2+3=5, then estimating). Choice D results from multiplying denominators and misinterpreting the result.

Question 7

1. -1.97°C per hour
2. -3.23°C per hour (correct answer)
3. -2.63°C per hour
4. -19.4°C per hour

Explanation: Temperature change = -3.8 - 15.6 = -19.4°C. Average rate = -19.4 ÷ 6 = -3.233... ≈ -3.23°C per hour. Choice A incorrectly calculates the temperature difference as positive. Choice C uses incorrect division or rounding. Choice D gives the total change without dividing by time.

Question 8

1. $5.7 \times 10^{-6} < 9.2 \times 10^{-7}$ because $5.7 < 9.2$
2. $5.7 \times 10^{-6} > 9.2 \times 10^{-7}$ because $-6 > -7$ (correct answer)
3. $5.7 \times 10^{-6} < 9.2 \times 10^{-7}$ because $-6 < -7$
4. $5.7 \times 10^{-6} > 9.2 \times 10^{-7}$ because $5.7 > 9.2$

Explanation: When comparing numbers in scientific notation with different exponents, the number with the larger exponent is greater (when both are positive or both are negative). Since $-6 > -7$, we have $10^{-6} > 10^{-7}$, so $5.7 \times 10^{-6} > 9.2 \times 10^{-7}$. Converting to decimal form: $5.7 \times 10^{-6} = 0.0000057$ and $9.2 \times 10^{-7} = 0.00000092$. Choice A incorrectly compares only coefficients. Choice C incorrectly states $-6 < -7$. Choice D has wrong reasoning about coefficients.

Question 9

1. 1 decimal place
2. 2 decimal places (correct answer)
3. 3 decimal places
4. 4 decimal places

Explanation: Testing different decimal places: At 1 decimal place: 2.4056→2.4, 2.405→2.4, 2.4→2.4 (all three equal, not just two). At 2 decimal places: 2.4056→2.41, 2.405→2.41, 2.4→2.40 (two equal: 2.4056 and 2.405 both round to 2.41). This satisfies Sarah's claim with the minimum number of decimal places. Choice A makes all three equal, not two. Choice C (3 places) works but isn't the minimum. Choice D (4 places) makes all numbers different.

Question 10

1. $0.10t - 25 = 32.50$
2. $25t + 0.10 = 32.50$
3. $25 + 0.10t = 32.50$ (correct answer)
4. $25(t + 0.10) = 32.50$

Explanation: When you encounter word problems about costs or pricing, your goal is to translate the real-world situation into a mathematical equation by identifying the fixed costs and variable costs. In this cell phone plan, there are two components: a fixed monthly fee of $25 (this stays the same regardless of usage) and a variable cost of $0.10 per text message (this depends on how many texts are sent). The total bill equals the fixed cost plus the variable cost, so: Total Bill = Fixed Fee + (Cost per text × Number of texts). Substituting the given values: $$32.50 = 25 + 0.10t$ $, which can be rewritten as$ $25 + 0.10t = 32.50$ $. This matches answer choice C. Let's examine why the other options are incorrect. Choice A ($$0.10t - 25 = 32.50$$) subtracts the monthly fee instead of adding it, which doesn't make sense since fees increase your bill. Choice B ($$25t + 0.10 = 32.50$$) incorrectly multiplies the monthly fee by the number of texts and treats the per-text cost as a fixed amount. Choice D ($$25(t + 0.10) = 32.50$$) uses distribution incorrectly, suggesting the monthly fee is multiplied by both the number of texts and the per-text rate. Study tip: In cost problems, always identify what stays constant (fixed costs) and what changes based on usage (variable costs). Write the equation as: Total = Fixed + (Rate × Variable). This pattern appears frequently in pre-algebra word problems involving phone plans, taxi fares, and rental costs.

Question 11

1. Student A's plot will appear to show a stronger association than Student B's plot (correct answer)
2. Student B's plot will appear to show a stronger association than Student A's plot
3. Both plots will show the same apparent strength since they use identical data points
4. The association strength cannot be compared because different scales change the actual correlation

Explanation: The correct answer is A. Student A's narrower scale (60°F to 100°F) will compress the x-axis, making the data points appear more spread out vertically relative to horizontally, creating the visual impression of a stronger association. Student B's wider scale (0°F to 120°F) will make the same points appear more compressed vertically relative to horizontally, making the association appear weaker. Choice B reverses this relationship. Choice C incorrectly assumes visual appearance is unaffected by scale. Choice D incorrectly suggests the actual correlation changes, when only the visual impression changes.

Question 12

1. 300 square feet (correct answer)
2. 400 square feet
3. 375 square feet
4. 350 square feet

Explanation: Using the Pythagorean theorem: $15^2 + l^2 = 25^2$, so $225 + l^2 = 625$, which gives $l^2 = 400$, so $l = 20$ feet. The area is $15 × 20 = 300$ square feet. Choice B uses the incorrect length (25) instead of finding the actual length. Choice C incorrectly calculates $15 × 25$. Choice D represents a computational error in the Pythagorean calculation.

Question 13

1. 16
2. 23 (correct answer)
3. 9
4. 30

Explanation: First, find $a$ by solving $f(a) = 8$: $3a - 7 = 8$, so $3a = 15$ and $a = 5$. Then $f(2a) = f(10) = 3(10) - 7 = 23$. Choice A results from calculating $2 \cdot 8 = 16$. Choice C comes from finding $a$ correctly but calculating $f(a+a)$ incorrectly. Choice D results from $f(2) \cdot f(a) = -1 \cdot 8$.

Question 14

1. Calculator to compute ratios between consecutive terms, since consistent ratios confirm geometric sequences (correct answer)
2. Graph paper to plot term number vs. term value, since exponential patterns show characteristic curves
3. Table format to organize term positions and values, since patterns become clear in organized data
4. Calculator to find differences between consecutive terms, since arithmetic patterns show constant differences

Explanation: When you encounter a sequence like 2, 6, 18, 54, 162, you need to identify what type of pattern it follows. Sequences can be arithmetic (with constant differences) or geometric (with constant ratios), and choosing the right analysis tool is crucial for accurate pattern recognition. Looking at this sequence, let's check for a geometric pattern by examining ratios between consecutive terms: 6÷2 = 3, 18÷6 = 3, 54÷18 = 3, 162÷54 = 3. The consistent ratio of 3 confirms this is a geometric sequence where each term is multiplied by 3. A calculator makes these ratio calculations quick and accurate, allowing you to verify the pattern definitively and predict future terms (486, 1458, 4374). Option B suggests graphing, but while exponential curves do show characteristic shapes, plotting points is time-consuming and less precise than calculating exact ratios. The visual pattern doesn't give you the specific multiplier needed for predictions. Option C recommends using a table format. While organization helps, simply arranging data doesn't reveal the underlying mathematical relationship—you still need to perform calculations to identify the pattern. Option D focuses on finding differences between consecutive terms (4, 12, 36, 108), but these differences aren't constant, which would indicate this isn't an arithmetic sequence. This approach leads you away from the correct geometric pattern. Study tip: When analyzing sequences, always test for geometric patterns first by calculating ratios between consecutive terms. If ratios are constant, you've found a geometric sequence; if differences are constant, it's arithmetic.

Question 15

1. The prediction is reliable because linear models maintain accuracy over any time period
2. The prediction may be unreliable because external factors could change the business environment significantly over 60 months (correct answer)
3. The prediction is unreliable because the model shows negative profits in early months
4. The prediction is reliable because the positive slope indicates consistent growth will continue

Explanation: Extrapolating 60 months into the future using a linear model is potentially unreasonable because many external factors (market conditions, competition, economic changes) could alter the relationship over such a long period. Choice A incorrectly assumes linear models are always accurate for extrapolation. Choice C confuses early negative profits (which may be realistic for a startup) with model reliability. Choice D ignores the fundamental issue that long-term extrapolation assumes conditions remain constant.

Question 16

1. $3$
2. $2$
3. $1$ (correct answer)
4. $-1$

Explanation: When you encounter an equation with fractions, your goal is to eliminate them by finding a common denominator and then clearing the fractions entirely. This transforms the equation into a simpler form where you can easily identify coefficients. Let's work through $\frac{x + 2}{4} - \frac{x - 1}{6} = 1$ step by step. First, find the least common denominator of 4 and 6, which is 12. Rewrite each fraction: $\frac{x + 2}{4} = \frac{3(x + 2)}{12} = \frac{3x + 6}{12}$ $\frac{x - 1}{6} = \frac{2(x - 1)}{12} = \frac{2x - 2}{12}$ Now your equation becomes: $\frac{3x + 6}{12} - \frac{2x - 2}{12} = 1$ Combine the fractions: $\frac{(3x + 6) - (2x - 2)}{12} = 1$ $\frac{3x + 6 - 2x + 2}{12} = 1$ $\frac{x + 8}{12} = 1$ To clear the fraction, multiply both sides by 12: $x + 8 = 12$ The coefficient of $x$ is 1, making C correct. Choice A (3) likely comes from mistakenly using just the coefficient from the first fraction. Choice B (2) might result from using the coefficient from the second fraction. Choice D (-1) could occur if you incorrectly handled the subtraction when combining fractions. Remember: when clearing fractions, always multiply the entire equation by the common denominator. The resulting linear equation will have integer coefficients that are much easier to work with than the original fractions.

Question 17

1. $A = 2p^2 - 6p + p - 3$
2. $A = 2p^2 - 5p - 3$
3. $A = 2p(p - 3) + 1(p - 3)$ (correct answer)
4. $A = (2p + 1)(p) - (2p + 1)(3)$

Explanation: When you see a problem asking for the "first step" of applying the distributive property, you need to set up the multiplication without actually completing it. The distributive property breaks down multiplication of expressions like $(a + b)(c + d)$ into separate parts. Here, you're finding the area using $A = \text{base} \times \text{height} = (2p + 1)(p - 3)$. The first step of the distributive property is to distribute each term in the first expression to the entire second expression. This means taking $2p$ and multiplying it by $(p - 3)$, then taking $1$ and multiplying it by $(p - 3)$. So the correct setup is $A = 2p(p - 3) + 1(p - 3)$, which is choice C. Choice A shows what happens after you've already distributed and partially simplified—this skips the "first step" the question asks for. Choice B goes even further by completely combining like terms, making it a final answer rather than a first step. Choice D attempts to distribute but incorrectly breaks apart the second expression $(p - 3)$ into separate multiplications with $p$ and $3$, which isn't how the distributive property works. Remember that when a question asks for the "first step" of the distributive property, you're looking for the setup that shows each term from the first expression being distributed to the complete second expression. Don't be tempted by answers that show partial or complete simplification—stick to what the question specifically requests.

Question 18

1. It is not a function because different formulas are used for different values of x
2. It is not a function because at x = 0, the value changes from one formula to another
3. It is a function because each value of x in the domain corresponds to exactly one value of y (correct answer)
4. It is a function only if the pieces connect continuously at the boundary points

Explanation: This piecewise relation is a function because for every possible input value of x, there is exactly one corresponding output value. For x < 0, use 2x + 1; for x = 0, use 3; for x > 0, use -x + 4. No x-value is assigned to multiple y-values. Choice A is incorrect because using different formulas for different intervals doesn't prevent something from being a function. Choice B is wrong because the change in formula doesn't create multiple outputs for any single input. Choice D is incorrect because continuity is not required for a relation to be a function (though it may be required for other properties).

Question 19

1. 12 inches (correct answer)
2. 6 inches
3. 18 inches
4. 24 inches

Explanation: Given SA = 4πr² = 144π. Dividing both sides by 4π: r² = 36, so r = 6 inches. Since diameter = 2r, the diameter is 2(6) = 12 inches. Choice B gives the radius, not the diameter. Choice C (18) would correspond to r = 9, giving SA = 4π(81) = 324π ≠ 144π. Choice D (24) would correspond to r = 12, giving SA = 4π(144) = 576π ≠ 144π.

Question 20

1. Generate 3 numbers; if at least 2 are between 1-6, count as success. Repeat many times and calculate the proportion of successes. (correct answer)
2. Generate 3 numbers; if at least 2 are between 1-4, count as success. Repeat many times and calculate the proportion of successes.
3. Generate 10 numbers; if at least 6 are between 1-6, count as success. Repeat many times and calculate the proportion of successes.
4. Generate 1 number; if it's between 1-6, count as success. Repeat many times and calculate the proportion of successes.

Explanation: For a 60% success rate using numbers 1-10, we need 6 out of 10 numbers to represent success (1-6). Since we want to simulate 3 free throws with at least 2 successes, we generate 3 numbers and count success when at least 2 fall in the range 1-6. Choice B uses wrong range (40% instead of 60%). Choice C simulates wrong scenario (6 out of 10 throws). Choice D simulates only 1 throw instead of 3.

Question 21

1. The same amount of flour as $\frac{3}{4}$ cup, expressed as 6 parts out of 8 equal parts (correct answer)
2. Six separate $\frac{1}{8}$-cup measurements that equal more flour than needed
3. A different amount of flour that requires 6 divisions into 8 parts each
4. The result of adding $\frac{3}{4}$ cup and $\frac{1}{8}$ cup together for extra flour

Explanation: $\frac{3}{4} = \frac{6}{8}$ represents the same quantity expressed with different denominators. Both fractions represent the division of the same amount into different-sized equal parts: $\frac{3}{4}$ means 3 out of 4 equal parts, while $\frac{6}{8}$ means 6 out of 8 equal parts of the same whole. Choice B misunderstands equivalent fractions as different amounts. Choice C confuses the division interpretation. Choice D incorrectly suggests addition rather than equivalence.

Question 22

1. 5:2 (correct answer)
2. 20:8
3. 12:8
4. 3:2

Explanation: When you encounter ratio problems involving categories, you need to carefully classify the items before comparing. This question asks you to compare mammals to non-mammals among the pet choices. First, identify which pets are mammals and which aren't. Dogs and cats are both mammals, while birds and fish are not mammals. So you have:

• Mammals: 12 students (dogs) + 8 students (cats) = 20 students
• Non-mammals: 5 students (birds) + 3 students (fish) = 8 students

Question 23

1. $\frac{1}{5}$ of the original length (correct answer)
2. $\frac{4}{15}$ of the original length
3. $\frac{1}{15}$ of the original length
4. $\frac{2}{15}$ of the original length

Explanation: The total must equal 1 (the whole board). We have: first piece + second piece + third piece + sawdust = 1. So: $\frac{2}{5} + \frac{1}{3} + \text{third piece} + \frac{1}{15} = 1$. Converting to common denominator 15: $\frac{6}{15} + \frac{5}{15} + \text{third piece} + \frac{1}{15} = 1$. This gives us: $\frac{12}{15} + \text{third piece} = 1$. Therefore: third piece = $1 - \frac{12}{15} = \frac{15}{15} - \frac{12}{15} = \frac{3}{15} = \frac{1}{5}$. Choice A is correct. Choice B ($\frac{4}{15}$) would result if students forgot to account for sawdust. Choice C ($\frac{1}{15}$) might result from calculation errors. Choice D ($\frac{2}{15}$) could result from misreading the sawdust amount.

Question 24

1. $10\%$
2. $7.5\%$
3. $9\%$ (correct answer)
4. $12\%$

Explanation: When you encounter mixture problems involving percentages, focus on tracking the actual amount of the substance (salt) and the total volume separately, then find the new percentage. Start with what you know: $3.6$ liters of $15\%$ salt solution contains $3.6 \times 0.15 = 0.54$ liters of pure salt. When you add $2.4$ liters of water, you're adding zero salt but increasing the total volume. The new mixture has $0.54$ liters of salt in a total volume of $3.6 + 2.4 = 6.0$ liters. The new percentage is $\frac{0.54}{6.0} = 0.09 = 9\%$, making C correct. Looking at the wrong answers: A) $10\%$ likely comes from incorrectly assuming the percentage drops proportionally to the volume increase ($15\% \times \frac{3.6}{6.0} = 9\%$ would be correct, but students might miscalculate). B) $7.5\%$ results from the common error of thinking the percentage is simply halved because you're roughly doubling the volume ($15\% \div 2 = 7.5\%$). D) $12\%$ might come from subtracting the percentage of water added relative to original solution ($15\% - 3\% = 12\%$), which is mathematically meaningless. For mixture problems, always calculate the actual amount of the key substance first, then find the new total, and finally compute the new percentage. Don't try to work directly with percentages—track the concrete quantities.

Question 25

1. $A = 4y^2 + 3y - 8y - 6$
2. $A = 4y(y) + 4y(-2) + 3(y) + 3(-2)$
3. $A = 4y^2 - 8y + 3y - 6$
4. $A = 4y(y - 2) + 3(y - 2)$ (correct answer)

Explanation: The distributive property can be applied by factoring out each term in the first expression: $(4y + 3)(y - 2) = 4y(y - 2) + 3(y - 2)$. This correctly shows the first step of distribution. Choice A shows an incorrect intermediate step with wrong signs. Choice B shows the complete distribution but isn't the clearest first step. Choice C shows terms after distribution but with incorrect arrangement.