SAT Practice Test 6 - SAT Math

A formula for simple interest is $I = Prt$ . Solve this equation for $r$ in terms of $I$ , $P$ , and $t$ .
Answer: $r=\frac{I}{Pt}$ We need to solve $I = Prt$ for $r$ . To isolate $r$ , we need to divide both sides by everything that's multiplied with $r$ , which is $P$ and $t$ . Dividing both sides by $Pt$ : $\frac{I}{Pt} = \frac{Prt}{Pt} = r$ . Therefore, $r = \frac{I}{Pt}$ . The common mistake is dividing by only one of the factors (just $P$ or just $t$ ) or incorrectly placing variables in the numerator when they should be in the denominator. When solving for a variable, divide by all factors multiplied with that variable.
A bar graph shows the number of books sold by a school club over 4 weeks. According to the graph, how many more books were sold in Week 2 than in Week 3?
Answer: 12 books The question asks how many more books were sold in Week 2 than in Week 3 based on the bar graph. The bar labels indicate Week 2 at 30 books and Week 3 at 18 books. Subtract 18 from 30 to find the difference of 12 books. A key error to avoid is comparing the wrong weeks or misreading the labels, so always confirm the specific categories mentioned. Additionally, since values are provided directly, there's no need to estimate from the graph's scale.
A line passes through $(3,8)$ and has slope $\frac{1}{3}$ . Which equation is the line in point-slope form?
Answer: $y-8=\frac{1}{3}(x-3)$ The question asks for the point-slope form of a line through (3,8) with slope 1/3. The form is y - y1 = m(x - x1), so y - 8 = (1/3)(x - 3). This directly uses the given point and slope. It models the linear equation without expanding. A common error is switching x and y coordinates, like y-3=(1/3)(x-8). Another mistake might be using reciprocal slope like 3. For point-slope, plug in the known point and slope carefully.
A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. One marble is drawn at random. What is the probability that the marble is not blue?
Answer: $\frac{7}{10}$ This problem asks for the probability of NOT drawing a blue marble. The total number of marbles is 5 red + 3 blue + 2 green = 10 marbles. The number of non-blue marbles is 5 red + 2 green = 7 marbles. Therefore, P(not blue) = 7/10. A common error is confusing the complement with the total, thinking P(not blue) = 3/10 (the probability of blue). Remember that P(not A) = 1 - P(A), or you can count the favorable outcomes directly as we did here.
A phone plan costs $30 plus $0.10 per text message. You want to keep the total cost under $45. Let $t$ be the number of texts. What is the greatest whole number of texts you can send?
Answer: $t=149$ We need to find the maximum number of texts while keeping the total cost under $45. The cost is $30 + $0.10t, and we need 30 + 0.10t < 45. Solving: 0.10t < 15, so t < 150. Since we want the greatest whole number less than 150, the answer is t = 149. A common error is using ≤ instead of < when the problem says "under," or forgetting that we need a whole number of texts. When finding maximum values with strict inequalities, remember to use the largest integer that's strictly less than the bound.
A car's value decreases by 18% each year. Its value is $18{,}000$ when purchased (at $t=0$ ). Which expression gives the value $V(t)$ after $t$ years?
Answer: $V(t)=18000(0.82)^t$ A car worth $18,000 decreases by 18% each year. "Decreases by 18%" means it retains 82% (100% - 18% = 82%) of its value each year. In exponential decay, we multiply by the retention factor: V(t) = 18000 $(0.82)^t$ . This gives V(0) = 18000 (initial value) and V(1) = 18000(0.82) = 14,760 (82% of original). A common error is using 1.18 (thinking of 18% as growth) or using 0.18 (the decrease percentage rather than retention percentage). Remember: for p% decrease, multiply by (1 - p/100).
A scientist uses the formula $T=18+2h$ to model temperature $T$ (in °C) at height $h$ (in km). Which equation gives $h$ in terms of $T$ ?
Answer: $h=\tfrac{T-18}{2}$ This question asks to solve for h in terms of T from $T = 18 + 2h$ . Subtract 18: $T - 18 = 2h$ , then divide by 2: $h = \tfrac{T - 18}{2}$ . This expresses h using inverse operations. A common error is adding instead of subtracting or dividing incorrectly. Another mistake might be solving for T again. In rearranging formulas with two variables, apply operations to isolate the desired variable step-by-step.
A small theater sold $x$ balcony tickets and $y$ floor tickets. The totals are modeled by the system: $\begin{cases} x+y=140\\ 12x+20y=2320 \end{cases}$ What is the number of floor tickets sold?
Answer: $80$ The question asks for the number of floor tickets y sold, given the system x + y = 140 and 12x + 20y = 2320, where x is balcony tickets. I will use substitution: from first, x = 140 - y; plug into second: 12(140 - y) + 20y = 2320, 1680 - 12y + 20y = 2320, 1680 + 8y = 2320, 8y = 640, y = 80. Verify: x = 60, 12*60 + 20*80 = 720 + 1600 = 2320, correct. A key error is misapplying distribution in substitution. In ticket problems, ensure the solution yields non-negative integers that fit the context.
A small factory recorded the number of defective items produced by each machine in one shift. The bar graph shows defects for Machines A–E. If the factory removes 2 defects from Machine C's count after reinspection (they were false alarms), what would be the new total number of defects across all machines?
Answer: 23 defects The question asks for the new total defects after removing 2 false alarms from Machine C. First, I sum all defects from the bar graph: A(4) + B(5) + C(6) + D(3) + E(7) = 25 total defects. After removing 2 from Machine C: 25 - 2 = 23 defects. A common error is only recalculating Machine C's count without updating the total.
Triangles $\triangle ABC$ and $\triangle DEF$ are similar. The sides correspond as $AB \leftrightarrow DE$ , $BC \leftrightarrow EF$ , and $AC \leftrightarrow DF$ . If $AB=8$ , $BC=12$ , $DE=10$ , and $EF=x$ , what is the value of $x$ ?
Answer: 15 This question requires finding x for side EF in similar triangles ABC and DEF, with given sides. Similarity implies corresponding sides are proportional. The ratio AB/DE = 8/10 = 4/5, so BC/EF = 4/5, thus 12/x = 4/5, cross-multiply 4x = 60, x = 15. Corresponding sides must be correctly matched. A common error is inverting the ratio, leading to x = 9.6. Another is adding sides instead. Always list correspondences to set up ratios accurately.
A storage box is a cube with side length $0.5\text{ m}$ . The manufacturer lists volume in liters. What is the volume of the box in liters? (Use $1\text{ m}^3 = 1000\text{ L}$ .)
Answer: $125\text{ L}$ The question asks for the volume of a cubic storage box with side 0.5 m, converted to liters using 1 m³ = 1000 L. The formula needed is volume of cube = side³ in m³, then convert units to liters. Calculate (0.5)³ = 0.125 m³, then 0.125 × 1000 = 125 L. Emphasize handling the unit conversion accurately after finding m³. A key error could be forgetting the conversion factor, leaving it in m³. Another mistake is miscalculating the cube as side² instead. A test-taking strategy is to note conversion factors provided and apply them last.
Two data sets each have 6 values. Set A: 4, 4, 5, 5, 6, 6. Set B: 1, 3, 5, 5, 7, 9. Which statement is true about their centers and spreads? Consider both the mean and the range for each set.
Answer: Same mean, different range This question compares centers and spreads of two data sets. Set A: 4, 4, 5, 5, 6, 6 has mean = (4+4+5+5+6+6)/6 = 30/6 = 5 and range = 6-4 = 2. Set B: 1, 3, 5, 5, 7, 9 has mean = (1+3+5+5+7+9)/6 = 30/6 = 5 and range = 9-1 = 8. Both sets have the same mean (5) but different ranges (2 vs 8). This illustrates that data sets can have identical centers but very different spreads. The key insight is that mean measures center while range measures spread, and these are independent characteristics. Always calculate both measures separately to fully understand a distribution.
A right triangle is formed by the points $P(-3,4)$ , $Q(-3,-2)$ , and $R(5,-2)$ . What is the length of hypotenuse $PR$ ?
Answer: $10$ First, verify this forms a right triangle by checking the sides. PQ has length |4-(-2)| = 6 (vertical), QR has length |5-(-3)| = 8 (horizontal), making a right angle at Q. The hypotenuse PR can be found using the distance formula: √[(5-(-3))² + (-2-4)²] = √[8² + (-6)²] = √[64 + 36] = √100 = 10. This is the 6-8-10 Pythagorean triple. Always sketch coordinate geometry problems to visualize the right angle.
If $ax^2+bx+9=(x+3)^2$ , what are the values of $a$ and $b$ ?
Answer: $a=1,\ b=6$ Given $ax^2+bx+9=(x+3)^2$ , we expand the right side to find $a$ and $b$ . Expanding: $(x+3)^2=x^2+2(x)(3)+3^2=x^2+6x+9$ . Comparing $ax^2+bx+9=x^2+6x+9$ , we can match coefficients: the coefficient of $x^2$ gives us $a=1$ , the coefficient of $x$ gives us $b=6$ , and the constants already match (both are 9). A common error is to expand $(x+3)^2$ as $x^2+3x+9$ , forgetting to double the middle term. Remember that $(x+c)^2=x^2+2cx+c^2$ .
A lab report uses the linear formula $m=\frac{y-b}{a}$ where $m$ , $y$ , $a$ , and $b$ are variables and $a \neq 0$ . Solve the equation for $y$ in terms of $m$ , $a$ , and $b$ .
Answer: $y=am+b$ We need to solve $m = \frac{y-b}{a}$ for $y$ in terms of $m$ , $a$ , and $b$ . First, multiply both sides by $a$ : $am = y - b$ . Then add $b$ to both sides: $y = am + b$ . A common error is to incorrectly distribute or to get the order wrong, resulting in $y = \frac{m+b}{a}$ or $y = am - b$ . When solving for a variable in the numerator of a fraction, first eliminate the denominator by multiplication.
Two models are proposed for a quantity $Q$ after $t$ days: Model 1: $Q=200+30t$ . Model 2: $Q=200(1.15)^t$ . If the quantity increases by a constant amount each day (not a constant percent), which model should be used?
Answer: Model 1, because it is linear We need to choose between Model 1: Q = 200 + 30t (linear) and Model 2: Q = 200 $(1.15)^t$ (exponential) for a quantity that increases by a constant amount each day. The key phrase is "constant amount" which indicates additive change, not multiplicative. Linear models add the same amount each time period, while exponential models multiply by the same factor. Model 1 adds 30 each day: day 0: 200; day 1: 230; day 2: 260 (constant difference of 30). Model 2 multiplies by 1.15: day 0: 200; day 1: 230; day 2: 264.5 (increasing differences). Since we need constant amount increase, Model 1 (linear) is correct.
Solve the system. What is the solution $(x, y)$ ? $\begin{cases} 7x-2y=20\\ 3x+2y=4 \end{cases}$
Answer: $(\tfrac{12}{5},\,-\tfrac{8}{5})$ We need to solve $7x - 2y = 20$ and $3x + 2y = 4$ . Since the y terms have opposite signs, I'll use elimination by adding the equations: $7x - 2y + 3x + 2y = 20 + 4$ , which gives $10x = 24$ , so $x = 24/10 = 12/5$ . Substituting into the second equation: $3(12/5) + 2y = 4$ , so $36/5 + 2y = 4$ . Converting 4 to fifths: $20/5$ , we get $2y = 20/5 - 36/5 = -16/5$ , so $y = -8/5$ . The solution is $(12/5, -8/5)$ . The warning about subtracting instead of adding refers to incorrectly doing $7x - 2y - (3x + 2y) = 20 - 4$ , which would give $4x - 4y = 16$ . When coefficients have opposite signs, adding equations eliminates that variable efficiently.
A taxi charges a base fare plus a per-mile rate. A ride of 3 miles costs $11.50, and a ride of 7 miles costs $21.50. If $C$ is the cost (in dollars) and $m$ is miles, what does the $y$ -intercept of the graph of $C$ versus $m$ represent?
Answer: The base fare (cost at 0 miles) The question asks what the y-intercept of the graph of cost C versus miles m represents, given rides of 3 miles for $11.50$ and 7 miles for $21.50$ . First, find the slope (per-mile rate): $(21.50 - 11.50) / (7 - 3) = 10 / 4 = 2.50$ . Then, find the y-intercept using one point: $11.50 = 2.50 \times 3 + b$ , $b = 11.50 - 7.50 = 4$ , which is the base fare. Thus, the y-intercept represents the base fare, the cost at 0 miles. A common error is confusing the intercept with the slope or total cost. Another mistake might be incorrect subtraction in slope calculation. A strategy is to interpret the intercept in context after finding the linear equation.
If $2(x-3)+k$ is equivalent to $2x+5$ for all values of $x$ , what is the value of $k$ ?
Answer: $11$ The question requires finding k such that $2(x - 3) + k$ is equivalent to $2x + 5$ for all values of $x$ by matching coefficients. First, distribute the 2: $2x = 2x$ and $2(-3) = -6$ , so the left side is $2x - 6 + k$ . For equivalence, the constant terms must match: $-6 + k = 5$ , so $k = 5 + 6 = 11$ . This makes both expressions identical. A key error is distributing as $2x + 6$ instead of $-6$ , leading to $6 + k = 5$ and $k = -1$ . Another mistake might be ignoring distribution and setting k directly without adjusting constants, giving incorrect values like 5 or 3. When equating expressions, expand fully and compare like terms separately for accuracy.
What is the axis of symmetry for the quadratic function $f(x) = x^2 - 6x + 8$ ?
Answer: $x = 3$ This problem asks for the axis of symmetry of $f(x) = x^2 - 6x + 8$ . The axis of symmetry for a quadratic function $f(x) = ax^2 + bx + c$ is the vertical line $x = -\frac{b}{2a}$ . Here, $a = 1$ and $b = -6$ , so the axis of symmetry is $x = -\frac{(-6)}{2(1)} = \frac{6}{2} = 3$ . This line passes through the vertex and divides the parabola into two symmetric halves.
A circle has radius $12$ cm. What is the length of an arc that subtends a central angle of $150^\circ$ , in centimeters? (Use $\pi$ in your answer.)
Answer: $10\pi$ This question asks for the arc length given the radius and central angle. The arc length formula is L = (θ/360°) × 2πr, where θ is the central angle in degrees. Substituting θ = 150° and r = 12 cm: L = (150/360) × 2π(12) = (5/12) × 24π = 10π cm. A common mistake is forgetting to use the circumference (2πr) as the base for the calculation. Remember that arc length is a fraction of the circumference, not the area.
A landscaper uses two formulas for the same boundary: $y=-2x+5$ and $y=x^2-4x+5$ . The system is $\begin{cases} y=-2x+5\\ y=x^2-4x+5 \end{cases}$ What are all solutions $(x,y)$ ?
Answer: $(0,5)$ and $(2,1)$ We need all solutions where the line $y = -2x + 5$ meets the parabola $y = x^2 - 4x + 5$ . This is a linear-nonlinear system solved by substitution. Setting equal: $-2x + 5 = x^2 - 4x + 5$ , which simplifies to $0 = x^2 - 2x$ , or $x(x - 2) = 0$ , giving $x = 0$ or $x = 2$ . When $x = 0$ , $y = -2(0) + 5 = 5$ ; when $x = 2$ , $y = -2(2) + 5 = 1$ . The solutions are $(0, 5)$ and $(2, 1)$ . Always verify by substituting back into both original equations to avoid calculation errors.
Convert 2.5 gallons to quarts. If 1 gallon equals 4 quarts, how many quarts are in 2.5 gallons?
Answer: 10 quarts We need to convert 2.5 gallons to quarts using the given conversion factor. Set up the conversion using dimensional analysis: 2.5 gallons × (4 quarts / 1 gallon). The gallons cancel out, leaving us with: 2.5 × 4 quarts = 10 quarts. A common error is dividing by 4 instead of multiplying when converting from gallons to quarts. Since quarts are smaller units than gallons, we expect more quarts than gallons.
A line has equation $y=-4x+12$ . Which statement is true?
Answer: The line passes through $(0,12)$ The question asks which statement is true for the line y=-4x+12. The slope is -4, y-intercept is 12, so it passes through (0,12). Also, at x=3, y=-12+12=0, so passes (3,0). Statement C is true about (0,12). A key error is confusing slope with intercept, thinking slope is 12. Another mistake might be solving for x-intercept wrongly. In analyzing equations, identify slope and intercept directly to evaluate statements.
A store marks up a wholesale price by adding $7.50$ and then doubling the result. The final price is $41.50$ . This is modeled by $2(x+7.5)=41.5$ , where $x$ is the wholesale price. What is the value of $x$ ?
Answer: $13.25$ The equation $2(x+7.5)=41.5$ models the store's pricing where $x$ is the wholesale price. To solve, first divide both sides by 2: $x + 7.5 = 20.75$ . Then subtract 7.5 from both sides: $x = 20.75 - 7.5 = 13.25$ . The key error students make is trying to distribute the 2 instead of dividing by it first, or making arithmetic errors when subtracting decimals. When an entire expression in parentheses is multiplied by a constant, dividing by that constant first often simplifies the problem.
A map uses a scale of 1 inch to 100 miles. If the distance between two cities on the map is 3 inches, what is the actual distance between the cities?
Answer: 300 miles This question asks for the actual distance between cities given a map scale and the measured distance on the map. The scale 1 inch to 100 miles means each inch on the map represents 100 miles in reality. We can set up the proportion: map distance/actual distance = 1/100, so 3/x = 1/100. Cross-multiplying gives us x = 3 × 100 = 300 miles actual distance.
A rumor spreads so that the number of people who have heard it is multiplied by 1.25 each day. If 160 people have heard it on day 0, how many have heard it after 3 days?
Answer: 312.5 This calculation involves exponential growth over multiple time periods. Starting with 160 people and multiplying by 1.25 each day, after 3 days we have 160 $(1.25)^3$ . Calculate $(1.25)^3$ = 1.953125, then 160 × 1.953125 = 312.5 people. This represents compound growth where each day's increase is 25% of the previous day's total, not the original amount. Students sometimes multiply 160 × 1.25 × 3 = 600, incorrectly treating it as simple interest. For exponential growth, always raise the growth factor to the power of time periods, then multiply by the initial value.
A line passes through the points $(2,-1)$ and $(8,11)$ . What is the slope of the line?
Answer: $2$ We need to find the slope of a line passing through points $(2,-1)$ and $(8,11)$ . The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$ , where we must use the same order for both coordinates. Using $(8,11)$ as point 2 and $(2,-1)$ as point 1: $m = \frac{11 - (-1)}{8 - 2} = \frac{12}{6} = 2$ . A common error is to mix up the order, such as calculating $(11 - (-1))/(2 - 8) = 12/(-6) = -2$ , which gives the wrong sign. Always subtract coordinates in the same order for both numerator and denominator.
A cylindrical water tank has radius $2.5\text{ m}$ and height $6\text{ m}$ . What is the volume of the tank? Use $\pi\approx 3.14$ .
Answer: $117.75\text{ m}^3$ The question seeks the volume of a cylindrical tank with dimensions in meters, answered in cubic meters. The formula is volume = π × radius² × height. Substitute to get 3.14 × (2.5 m)² × 6 m = 3.14 × 6.25 × 6 = 117.75 m³. Common errors include using diameter instead of radius, squaring incorrectly, or omitting π. Another pitfall is confusing volume with surface area formulas. Always double-check unit consistency in three-dimensional calculations; a strategy is to estimate by rounding π to 3 and radius to 2 or 3 for a rough check against choices.
A runner's mile times (in minutes) over 7 practices were: 7.2, 7.1, 7.0, 7.3, 7.1, 7.0, 7.1. What is the mode of the data?
Answer: 7.1 This question asks for the mode of mile times: 7.2, 7.1, 7.0, 7.3, 7.1, 7.0, 7.1. The mode is the value that appears most frequently. Counting occurrences: 7.0 appears 2 times, 7.1 appears 3 times, 7.2 appears 1 time, 7.3 appears 1 time. Therefore, the mode is 7.1 minutes since it appears most often (3 times). A common error is confusing mode with median (middle value) or mean (average). When finding the mode, always count how many times each distinct value appears. If multiple values tie for the highest frequency, the data set can have multiple modes.
A rectangle's perimeter is given by $P = 2L + 2W$ . Solve this equation for $W$ in terms of $P$ and $L$ .
Answer: $W=\frac{P-2L}{2}$ We need to solve $P = 2L + 2W$ for $W$ . First, subtract $2L$ from both sides: $P - 2L = 2W$ . Then divide both sides by 2: $\frac{P - 2L}{2} = W$ . Therefore, $W = \frac{P - 2L}{2}$ . The common mistake is dividing only $P$ by 2 or forgetting to subtract $2L$ before dividing, which would give $\frac{P}{2} - 2L$ . When solving for a variable, perform operations in the correct order: first isolate the term containing the variable, then divide.
A survey recorded the number of books read last month and the amount of money spent at a bookstore this month for 10 people. The scatterplot shows Books Read (books) versus Spending (dollars), and a line of best fit is drawn. Which prediction is an example of extrapolation using the line of best fit?
Answer: Predict spending when books read is 15, if the largest x-value shown is 10. This question asks us to identify extrapolation, which means making predictions outside the range of observed data. If the largest x-value in the data is 10 books, then predicting spending for 15 books would be extrapolation because 15 is beyond the observed range. Options A and B describe interpolation (predictions within the data range), while option D describes identifying association type, not making predictions. Extrapolation is risky because the relationship might change outside the observed range - always note when predictions extend beyond your data.
A science class measured plant height each week. The line graph shows height (in cm) for Weeks 0, 1, 2, 3, and 4. If the trend from Week 3 to Week 4 continues for one more week at the same weekly increase, what height would you predict for Week 5? Use the last observed increase as the rate for one more week.
Answer: 20 cm The question asks for a Week 5 prediction based on the Week 3 to Week 4 trend. From the graph, Week 3 shows 16 cm and Week 4 shows 18 cm, an increase of 2 cm. If this continues, Week 5 would be 18 + 2 = 20 cm. Students often use the wrong weeks to establish the trend or forget to add the increase to the Week 4 value.
A bookstore membership costs $12$ upfront and then reduces the price of each book by $2$ . Let $x$ be the number of books purchased and $S$ be the total savings compared to nonmembers. Which linear equation gives $S$ ?
Answer: $S=2x-12$ This problem involves finding the savings equation for a bookstore membership. The membership costs $12$ upfront but saves $2$ per book. If we buy $x$ books, we save $2x$ on books but paid $12$ for membership, so net savings is $S = 2x - 12$ . This will be negative (a net cost) until we buy enough books to break even at $x = 6$ books. Common errors include adding instead of subtracting the membership cost or misunderstanding what "savings" means. Think carefully about whether costs should be added or subtracted from savings.
Consider the system $\begin{cases}2x+3y=7\\4x+6y=10\end{cases}$ How many solutions does the system have?
Answer: $0$ We need to determine how many solutions the system $2x + 3y = 7$ and $4x + 6y = 10$ has. I'll check if the equations are proportional by comparing ratios. The second equation has coefficients that are exactly double the first: $4x = 2(2x)$ and $6y = 2(3y)$ . However, the constant term $10$ is not double $7$ (which would be $14$ ). Since the left sides are proportional but the right sides are not, these represent parallel lines with no intersection. The key insight is that when equations have proportional coefficients but non-proportional constants, the lines are parallel. Always check both the variable coefficients AND the constants when determining the number of solutions.
A projectile is launched with an initial velocity that creates a parabolic path described by $h(t) = -16t^2 + 64t + 5$ , where $h$ is the height in feet and $t$ is the time in seconds. At what time does the projectile reach its maximum height?
Answer: $t = 2$ This problem asks when a projectile reaches maximum height given by $h(t) = -16t^2 + 64t + 5$ . For a quadratic function in the form $f(t) = at^2 + bt + c$ , the maximum occurs at $t = -\frac{b}{2a}$ . Here, $a = -16$ and $b = 64$ , so the time of maximum height is $t = -\frac{64}{2(-16)} = -\frac{64}{-32} = 2$ seconds.
The function $f(x)=x^2+4x+k$ has exactly one real solution to $f(x)=0$ . What is the value of $k$ ?
Answer: $k=4$ For $f(x)=x^2+4x+k$ to have exactly one real solution, the discriminant must equal zero. The discriminant is $b^2-4ac$ where $a=1$ , $b=4$ , $c=k$ . Setting it to zero: $(4)^2-4(1)(k)=0$ , so $16-4k=0$ , which gives $4k=16$ and $k=4$ . This makes the equation $x^2+4x+4=0$ , which factors as $(x+2)^2=0$ with the single solution $x=-2$ . A common error is setting the discriminant to 1 instead of 0, but remember: $discriminant = 0$ means one repeated root, $discriminant > 0$ means two distinct roots, $discriminant < 0$ means no real roots.
Which of the following expressions is equivalent to $12a^2b-18ab^2$ for all values of $a$ and $b$ ?
Answer: $6ab(2a-3b)$ Factor out the greatest common factor from $12a^2b-18ab^2$ : the GCF of the coefficients 12 and 18 is 6, and both terms share a factor of $ab$ , so the GCF is $6ab$ . Dividing each term by $6ab$ gives $12a^2b\div 6ab=2a$ and $-18ab^2\div 6ab=-3b$ , so $12a^2b-18ab^2=6ab(2a-3b)$ . Choice B reverses the sign inside the parentheses, giving $6ab(3b-2a)=-12a^2b+18ab^2$ , the negative of the target. Choice C only factors out part of the GCF and introduces an incorrect sign. Choice D factors out an extra factor of $ab$ , changing the degree of the expression entirely.
A shipping company charges a flat fee plus a per-pound fee. The cost is modeled by $C=7+1.2w$ , where $C$ is the cost in dollars and $w$ is the weight in pounds. What is the weight $w$ when the cost is $19$ dollars?
Answer: $10$ We need to find w when C = 19 in the equation C = 7 + 1.2w. Substituting C = 19: 19 = 7 + 1.2w. Subtracting 7 from both sides: 12 = 1.2w. Dividing by 1.2: w = 12/1.2 = 10. A common error is dividing 19 by 1.2 without first subtracting the flat fee. When solving cost equations with fixed and variable components, always account for the fixed fee before solving for the variable quantity.
Triangle $XYZ$ has side lengths $XY=7$ , $YZ=10$ , and $XZ=x$ . Which of the following values of $x$ is possible?
Answer: $12$ The question asks which value of x is possible for side XZ in triangle XYZ with XY=7, YZ=10, and XZ=x. The relevant property is the triangle inequality theorem, which requires the sum of any two sides to be greater than the third. Check: x < 7 + 10 = 17, x > 10 - 7 = 3, and other inequalities confirm 3 < x < 17 for a non-degenerate triangle. Among choices, 12 satisfies, while 2 ≤ 3, 3 = 3 (degenerate), and 17 = 17 (degenerate) do not. A key error is using equality instead of strict inequality, allowing degenerate cases. Another mistake is misapplying the inequalities to the wrong sides. When testing triangle sides, apply all three inequalities and exclude equalities for proper triangles.
A right triangle has legs of lengths $6$ and $8$ . A second right triangle is similar to the first and has hypotenuse $25$ . What is the perimeter of the second triangle?
Answer: $60$ This problem involves similar triangles and scaling. The first triangle has legs 6 and 8, so its hypotenuse is √(6² + 8²) = √(36 + 64) = √100 = 10. The second triangle has hypotenuse 25, which is 2.5 times larger (25 ÷ 10 = 2.5). Since the triangles are similar, all corresponding sides are scaled by the same factor: the legs of the second triangle are 6 × 2.5 = 15 and 8 × 2.5 = 20. The perimeter is 15 + 20 + 25 = 60. A key insight is that in similar triangles, the ratio of corresponding sides is constant. When given one side of a similar triangle, find the scale factor first to determine all other dimensions.
A lab culture doubles every 3 hours. At time $t=0$ , there are 200 cells. Which equation models the number of cells $N$ after $t$ hours?
Answer: $N=200(2)^{t/3}$ This problem involves modeling exponential growth with a non-unit time factor. Since the culture doubles every 3 hours, after t hours it has gone through t/3 doubling periods. Starting with 200 cells, the model is N = 200 $(2)^{t/3}$ , which correctly accounts for the 3-hour doubling time. For example, after 6 hours (2 doubling periods), N = 200 $(2)^2$ = 800 cells. Students often write N = 200 $(2)^t$ , forgetting to adjust for the doubling period. When the growth period doesn't match the time unit, always include the appropriate fraction in the exponent. Test your model at easy values like t = 3 to verify correctness.
A right triangle has an acute angle $\theta$ such that $\sin(\theta)=\frac{5}{13}$ . What is $\cos(\theta)$ ?
Answer: $\frac{12}{13}$ Given $\sin(\theta) = \frac{5}{13}$ , we need to find $\cos(\theta)$ using the Pythagorean identity. Since $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{5}{13}$ , the opposite side is 5 and hypotenuse is 13. Using the Pythagorean theorem: $\text{adjacent}^2 = 13^2 - 5^2 = 169 - 25 = 144$ , so adjacent = 12. Therefore, $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{12}{13}$ . The most common error is forgetting that $\sin^2(\theta) + \cos^2(\theta) = 1$ , which gives $\cos(\theta) = \sqrt{1 - \left( \frac{5}{13} \right)^2} = \sqrt{\frac{144}{169}} = \frac{12}{13}$ . When given one trig ratio, use the Pythagorean identity or visualize the triangle to find others.
Let $p(x)=x^3+ax^2+bx+6$ have zeros at $x=-2$ and $x=1$ , and suppose $p(0)=6$ . What is the remaining zero?
Answer: $3$ The question asks for the remaining zero of $p(x) = x^3 + a x^2 + b x + 6$ with zeros at $-2$ and $1$ , and $p(0) = 6$ . Since it's cubic and monic, factor as $(x + 2)(x - 1)(x - k)$ . The constant term is $( - (-2) ) ( -1 ) ( -k ) = - (2) (-1) (-k)$ wait, better: constant = $-$ (product of roots)$$ = 6, so product = $-6$ ; $(-2)(1)k = -2k = -6$ , k=3. Alternatively, expand $(x + 2)(x - 1)(x - 3) = x^3 - 2x^2 - 5x + 6$ , but wait, constant 6 yes. This uses the factor theorem and root product, connecting coefficients to zeros. A common error assumes the third zero is 0 due to the constant. Some misapply Vieta's formulas with signs. Strategically, use the product of roots for quick constant-based solving.

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