SAT Practice Test 3 - SAT Math

A cafeteria recorded the number of sandwiches sold on a Friday, separated by type. The bar graph shows Turkey, Ham, Veggie, and Tuna. According to the graph, which sandwich type accounts for the least number sold, and what is that number?
Answer: Veggie, 12 sandwiches The question asks which sandwich type had the least number sold on Friday and what that number is, based on the bar graph. Examining the bars, Veggie has the shortest bar at 12 sandwiches, while Turkey is at 20, Ham at 15, and Tuna at 18, so identify the lowest height and its label. Confirm by comparing: 12 is less than 15, 18, and 20. A common mistake is misaligning the bar with the y-axis scale, leading to reading 10 or 15 instead. Always read both the category label and bar height carefully to avoid confusing similar bars.
Two fair coins are flipped. What is the probability of getting at least one head?
Answer: $\tfrac{3}{4}$ This question asks for the probability of getting at least one head when flipping two fair coins. The sample space consists of four equally likely outcomes: {HH, HT, TH, TT}. "At least one head" means one or more heads, which includes {HH, HT, TH} - three out of four outcomes. Therefore, P(at least one head) = 3/4. A common approach is using the complement: P(at least one head) = 1 - P(no heads) = 1 - P(TT) = 1 - 1/4 = 3/4. The key insight is that "at least one" is often easier to calculate using the complement of "none."
Lines $p$ and $q$ intersect at point $O$ . One angle at the intersection is $118^{\circ}$ . What is the measure of the vertical angle to this $118^{\circ}$ angle?
Answer: $118^{\circ}$ This problem asks for the measure of a vertical angle when two lines intersect. Vertical angles are the non-adjacent angles formed when two lines intersect, and they are always congruent. Since one angle measures 118°, its vertical angle also measures 118°. The common mistake is confusing vertical angles with adjacent angles (which would be supplementary and sum to 180°). Remember that vertical angles are across from each other at the intersection point and are always equal.
A taxi ride costs a base fee plus a charge per mile. A 4-mile ride costs $17 and a 10-mile ride costs $35. What does the $y$ -intercept represent in this context?
Answer: The base fee (cost at 0 miles) We need to interpret the y-intercept in the context of a taxi ride where cost depends on miles driven. First, let's find the linear equation using points ( $4, 17$ ) and ( $10, 35$ ). The slope is $ (35 - 17)/(10 - 4) = 18/6 = 3 $ dollars per mile. Using point-slope form: $ C - 17 = 3(m - 4) $, which gives $ C = 3m + 5 $. The y-intercept is 5, which represents the cost when $ m = 0 $ miles - this is the base fee charged before any miles are driven. Students often confuse the y-intercept with the slope or misinterpret its meaning. In context problems, always think about what happens when the input variable equals zero.
A taxi fare is $3.50 plus $x$ dollars per mile. If a ride of 6 miles costs $18.50, the situation is modeled by $3.5+6x=18.5$ . What is $x$ ?
Answer: $2.50$ We need to find the cost per mile $x$ from $3.5 + 6x = 18.5$ . Subtract 3.5 from both sides: $6x = 18.5 - 3.5 = 15$ . Divide by 6: $x = 15 \div 6 = 2.5$ . The cost per mile is $2.50. A common mistake is subtracting 6 from 18.5 instead of subtracting the base fare of 3.5. When solving real-world problems, always check that your answer makes sense in context.
A drink mix is combined with water in the ratio 1:7 by volume. To fill a 5.6-liter container, how many milliliters of concentrate are needed?
Answer: 700 Concentrate is 1/8 of the total, so 5.6 L × 1/8 = 0.7 L = 700 mL. 800 mL uses 1/7 instead of 1/(1+7), 4900 mL uses 7/8, and 70 mL mis-converts liters to milliliters.
A cell phone plan charges a fixed monthly fee of $30 plus $0.10 per minute of talk time. Which equation models the total cost $C$ for $m$ minutes of talk time?
Answer: $C = 0.10m + 30$ The question asks for an equation giving total cost C for m minutes when the plan charges $30 each month plus $0.10 per minute. The $30 is charged once no matter how much you talk, so it is a constant, while $0.10 is charged per minute, so it multiplies m: C = 30 + 0.10m, which is the same equation as C = 0.10m + 30 with the terms written in the other order. The equation with 0.10m squared squares the minutes, making cost grow faster and faster instead of by a steady ten cents each minute. The equation charging 30m plus 10 reverses the roles by billing $30 per minute, and it misreads $0.10 as $10. And 30m + 0.10 likewise bills $30 per minute while shrinking the monthly fee to a dime.
Given the system of equations below, which of the following provides the solution as an ordered pair $(x,y)$ ? $\begin{cases}3x-2y=1\\ x+y=7\end{cases}$
Answer: (3, 4) Use $x=7-y$ in $3x-2y=1$ : $3(7-y)-2y=1\Rightarrow 21-3y-2y=1\Rightarrow 5y=20\Rightarrow y=4$ and then $x=3$ . Distractors come from swapping $x$ and $y$ or arithmetic mistakes when combining terms.
A snack company compared the number of boxes sold for four flavors. The bar graph shows sales (in boxes) for BBQ, Sour Cream, Salted, and Spicy. The profit per box is $2.00 for BBQ, $3.00 for Sour Cream, $2.50 for Salted, and $1.50 for Spicy. Based on the graph and the profit per box, which flavor produced the greatest total profit?
Answer: Sour Cream Total profit for a flavor is the number of boxes sold times the profit per box. From the graph, BBQ sold 20 boxes, Sour Cream sold 25, Salted sold 15, and Spicy sold 30. BBQ earned 20 boxes at $2.00 each, or $40.00. Sour Cream earned 25 boxes at $3.00 each, or $75.00. Salted earned 15 boxes at $2.50 each, or $37.50. Spicy earned 30 boxes at $1.50 each, or $45.00. Sour Cream produced the greatest total profit, $75.00. Spicy sold the most boxes, but its low profit per box means it earned less than Sour Cream, so the flavor with the most sales is not the flavor with the most profit.
On the coordinate plane, a rectangle has vertices $( -2,1)$ , $(4,1)$ , $(4,6)$ , and $(-2,6)$ . What is the area of the rectangle?
Answer: $30\text{ square units}$ This problem asks for the area of a rectangle with vertices (-2,1), (4,1), (4,6), and (-2,6). The length is the horizontal distance: |4-(-2)| = 6 units. The width is the vertical distance: |6-1| = 5 units. Area = length × width = 6 × 5 = 30 square units. A common error is miscalculating distances when dealing with negative coordinates. Remember that distance between points is always positive, using absolute value of the difference.
A scientist measured the pH of a solution 6 times: 6.8, 6.9, 7.0, 7.0, 7.1, 8.2. If the value 8.2 is a measurement error and is removed, which statement is true about the mean and median of the remaining data?
Answer: Mean decreases; median stays the same. This question examines how removing an outlier (8.2) affects the mean and median of pH measurements. Original data: 6.8, 6.9, 7.0, 7.0, 7.1, 8.2. Mean = 43/6 ≈ 7.17, median = (7.0 + 7.0)/2 = 7.0. After removing 8.2: 6.8, 6.9, 7.0, 7.0, 7.1. New mean = 34.8/5 = 6.96, new median = 7.0. The mean decreases (from 7.17 to 6.96) while the median stays the same (7.0). This illustrates that the median is resistant to outliers while the mean is sensitive to them. When analyzing the effect of removing outliers, calculate both measures before and after to see the actual changes.
In triangle $GHI$ , an exterior angle at vertex $H$ measures $132^{\circ}$ . The remote interior angle at vertex $G$ measures $47^{\circ}$ . What is the measure of the remaining interior angle $\angle I$ ?
Answer: $85^{\circ}$ This problem involves an exterior angle of a triangle and asks for angle I. The exterior angle theorem states that an exterior angle equals the sum of the two remote interior angles. The exterior angle at H is 132°, and one remote interior angle (at G) is 47°, so the other remote interior angle (at I) = 132° - 47° = 85°. A common error is confusing which angles are the remote interior angles or trying to use the triangle angle sum instead of the exterior angle theorem. When dealing with exterior angles, identify the two non-adjacent interior angles first.
Which expression is equivalent to $(3x-2)^2$ ? Expand using the perfect square pattern $(a-b)^2=a^2-2ab+b^2$ .
Answer: $9x^2-12x+4$ This question asks for the expanded form equivalent to $(3x - 2)^2$ using the perfect square formula. Recall $(a - b)^2$ = $a^2$ - 2ab + $b^2$ , where a = 3x and b = 2. Thus, $(3x)^2$ = $9x^2$ , -2*(3x)*2 = -12x, and $2^2$ = 4, giving $9x^2$ - 12x + 4. This is the equivalent expression. A common error is getting the middle term's sign wrong, like +12x, leading to $9x^2$ + 12x + 4. Another mistake might be forgetting the middle term entirely, resulting in $9x^2$ - 4 or similar. For squaring binomials, always use the formula and verify by FOIL if unsure.
A right triangle has legs of lengths $10$ cm and $10$ cm. What is the length of the hypotenuse, in centimeters?
Answer: $10\sqrt{2}$ By the Pythagorean theorem the hypotenuse c satisfies $c^2$ = $10^2$ + $10^2$ = 100 + 100 = 200, so c = sqrt(200). Simplify by pulling out the perfect square: sqrt(200) = sqrt(100 2) = 10 sqrt(2), which is about 14.1 cm. The plain value 20 comes from adding the two legs instead of adding their squares. The expression 2 sqrt(100) also works out to 20; it comes from splitting 200 into 2 100 but then taking the square root of only the 100 while leaving the 2 outside, when a radical requires the root of each factor, giving sqrt(2) sqrt(100). And 5 sqrt(2) is about 7.07, which is shorter than either leg and so cannot be a hypotenuse.
A researcher wants to estimate the proportion of all registered voters in a state who support a ballot measure. She samples 1,200 people by calling landline phone numbers listed in a public directory during weekday afternoons. She finds 54% support among those reached. Which statement best describes a potential issue with generalizing to all registered voters?
Answer: The sample may be biased because registered voters without listed landlines or who are unavailable on weekday afternoons could be underrepresented in the estimate. Generalizing from the people reached to all registered voters requires a sample that can reach the whole population, and calling listed landline numbers on weekday afternoons cannot. The correct choice names that coverage gap: voters with no listed landline, including cell-only households, and voters working during weekday afternoons have little chance of being sampled, so 54% may not describe all registered voters. The choice arguing that 1,200 is large enough to rule out bias confuses precision with accuracy, since a big sample drawn from a skewed frame is just a precisely wrong estimate. The choice demanding an exact 0-100 support scale is unnecessary, because a proportion is estimated from yes-or-no responses by definition. The choice restricting the sample to people who voted last election changes the target population from registered voters to past voters.
If a car depreciates by 15% each year and is currently valued at $20,000, what will its value be in 2 years?
Answer: $14,450 This question asks us to calculate the depreciated value of a car after exponential decay over time. Depreciation of 15% per year means the car retains 85% of its value each year, so the decay factor is 0.85. Starting at $20,000, after 1 year: $20,000 × 0.85 = $17,000; after 2 years: $17,000 × 0.85 = $14,450. We can also use the formula: final value = initial × (decay factor)^time = $20,000 × (0.85)² = $20,000 × 0.7225 = $14,450. A common mistake is to subtract 15% twice linearly: $20,000 - $3,000 - $3,000 = $14,000, but this ignores that the second year's depreciation is 15% of the already-reduced value. In exponential decay, each period's change is a percentage of the current amount, not the original.
A taxi charges a flat fee plus a per-mile rate. One ride of 6 miles costs $17, and another ride of 10 miles costs $25. Let $x$ be the flat fee (in dollars) and $y$ be the cost per mile (in dollars). What is the value of $x+2y$ ?
Answer: $9$ We need to find the flat fee (x) and per-mile rate (y) for a taxi, then calculate x + 2y. I'll set up the system using the given information: x + 6y = 17 (for 6 miles) and x + 10y = 25 (for 10 miles). Using elimination by subtracting the first equation from the second: (x + 10y) - (x + 6y) = 25 - 17, which gives 4y = 8, so y = 2. Substituting y = 2 into the first equation: x + 6(2) = 17, so x + 12 = 17, giving x = 5. Therefore, x + 2y = 5 + 2(2) = 9. A common error is confusing which variable represents the flat fee versus the per-mile rate, which would lead to an incorrect system setup.
A linear function $f$ satisfies $f(2)=11$ and $f(6)=3$ . What is $f(0)$ ?
Answer: $15$ For a linear function where f(2) = 11 and f(6) = 3, first find the slope: m = (3 - 11)/(6 - 2) = -8/4 = -2. Using point-slope form with (2, 11): f(x) - 11 = -2(x - 2), which gives f(x) = -2x + 4 + 11 = -2x + 15. Therefore, f(0) = -2(0) + 15 = 15. Check: f(2) = -2(2) + 15 = -4 + 15 = 11 ✓; f(6) = -2(6) + 15 = -12 + 15 = 3 ✓. When finding f(0), you're essentially finding the y-intercept of the linear function.
Which expression is equivalent to $2x(3x-4)-5(1-2x)$ ?
Answer: $6x^2+2x-5$ To simplify $2x(3x-4)-5(1-2x)$ , distribute both parts carefully. First: $2x(3x-4) = 6x^2 - 8x$ . Second: $-5(1-2x) = -5 + 10x$ (note the positive $10x$ ). Combining: $6x^2 - 8x - 5 + 10x = 6x^2 + 2x - 5$ . The critical step is correctly distributing $-5$ to get $+10x$ , since $-5 \times (-2x) = +10x$ . Many students incorrectly get $-10x$ , forgetting that negative times negative equals positive.
A rectangle has area $48$ square units. Its length is $2$ units more than its width. Which equation can be used to find the width $w$ (in units)?
Answer: $w^2+2w-48=0$ This question asks for the equation to find the width $w$ of a rectangle with area $48$ and length $2$ units more than the width. Let length = $w + 2$ , so area gives $w(w + 2) = 48$ , expanding to $w^2 + 2w - 48 = 0$ . This is a quadratic equation solvable by factoring: $(w + 8)(w - 6) = 0$ , $w = 6$ (since width $> 0$ ). The quadratic formula could also be used: $w = [-2 \pm \sqrt{4 + 192}]/2 = [-2 \pm \sqrt{196}]/2 = [-2 \pm 14]/2$ , giving $6$ or $-8$ . Key errors include incorrect coefficients on the linear term, as in choices A, B, and C. Always define variables clearly and check by solving to ensure positive dimensions.
A chord is $16$ units long in a circle with radius $10$ . The chord is perpendicular to a radius at its midpoint, forming a right triangle. What is the distance from the center to the chord?
Answer: $6$ A chord of length 16 in a circle with radius 10 is perpendicular to a radius at its midpoint. This creates a right triangle where the hypotenuse is the radius (10), one leg is half the chord (8), and the other leg is the distance from center to chord. Using the Pythagorean theorem: $10^2 = 8^2 + d^2$ , so $100 = 64 + d^2$ , giving $d^2 = 36$ and $d = 6$ . The key insight is that when a radius is perpendicular to a chord, it bisects that chord.
A point $P$ on a number line has coordinate $x$ . If the distance from $P$ to $4$ is $7$ , which equation represents this situation using absolute value, and what are the possible values of $x$ ?
Answer: $|x-4|=7;\ x=11,-3$ The problem requires finding the absolute value equation for a point x where the distance to 4 is 7, and listing possible x values. The distance is represented by |x - 4| = 7. Solving: x - 4 = 7 or x - 4 = -7, so x = 11 or x = -3. This matches the situation described. A common error is confusing the order, like using |x + 4| which shifts the reference point incorrectly. Another mistake is setting the absolute value equal to -7, which is impossible since absolute value is nonnegative. When modeling distances with absolute values, always center the expression on the reference point and check both solutions on the number line.
A school tracks the number of students who joined four clubs this year. The bar graph shows the number of members in Art, Drama, Robotics, and Chess. According to the graph, how many more students joined Robotics than Drama?
Answer: 10 students The question asks how many more students joined the Robotics club than the Drama club based on the bar graph. Looking at the graph, the bar for Robotics reaches 25 students, while the bar for Drama reaches 15 students, and the y-axis is scaled in increments of 5 students, so ensure you align the bar tops accurately with the grid lines. To find the difference, subtract the number for Drama from Robotics: 25 - 15 = 10 students. A common error is misreading the scale, such as thinking increments are 10 instead of 5, which could lead to incorrect differences; always confirm the axis labels. Another mistake might be comparing the wrong clubs, so double-check the category labels on the x-axis.
A line has equation $y=-4x+9$ . Which of the following is the $y$ -intercept of the line?
Answer: $(0,9)$ The y-intercept of a line is the point where the line crosses the y-axis, which occurs when x = 0. For the equation y = -4x + 9, substituting x = 0 gives y = -4(0) + 9 = 9. Therefore, the y-intercept is the point (0, 9). Remember that the y-intercept is always of the form (0, b) where b is the constant term in the slope-intercept form y = mx + b. Don't confuse this with the x-intercept, which would have y = 0.
A shipping company uses the formula $C=15+2.5w$ for cost $C$ in dollars and weight $w$ in pounds. Solve for $w$ in terms of $C$ . Your final answer should isolate $w$ and not leave extra constants on the right side.
Answer: $w=\frac{C-15}{2.5}$ We need to solve $C=15+2.5w$ for $w$ in terms of $C$ . First, subtract 15 from both sides: $C-15=2.5w$ . Then divide both sides by 2.5: $w=\frac{C-15}{2.5}$ . The key is to isolate $w$ by undoing the operations in reverse order: first subtract the constant, then divide by the coefficient. A common error is to divide $C$ by 2.5 before subtracting 15.
Two dot-plot-like frequency summaries are shown as counts in each value. Distribution A has values 1(1 time), 2(2 times), 3(3 times), 4(2 times), 5(1 time). Distribution B has values 1(1 time), 2(1 time), 3(1 time), 4(1 time), 9(1 time). Which distribution has the larger standard deviation, and why?
Answer: Distribution B, because its values are more spread out from the mean. The question asks which distribution has the larger standard deviation and why, comparing two frequency distributions. Standard deviation measures average distance from the mean; for A, mean = 3 with values clustered nearby, yielding low spread. For B, mean ≈ 3.8, but the value 9 is far out, increasing the average deviation significantly. Thus, B has larger standard deviation due to greater spread from the mean. A key error is confusing larger range with larger SD without considering distribution shape. Compute means first, then assess deviations to compare SD qualitatively.
A line passes through the points $(4,-1)$ and $(-2,11)$ . Which equation gives the line in slope-intercept form?
Answer: $y=-2x+7$ The question asks for the slope-intercept form of the line passing through points (4, -1) and (-2, 11). First, calculate the slope m = (11 - (-1))/(-2 - 4) = 12/(-6) = -2. Then, use one point, say (4, -1), to find the intercept: -1 = -2*4 + b, so -1 = -8 + b, b = 7, giving y = -2x + 7. This form highlights the slope as the rate of change and the y-intercept as the starting value. A common error is incorrect sign handling in slope calculation, leading to a positive instead of negative value. Another mistake might be plugging into the equation incorrectly when solving for b. A good strategy is to verify by plugging both original points back into the derived equation.
A model car is built to a scale of 1:25. If the actual car is 10 meters long, how long is the model?
Answer: 0.4 m This scale problem asks us to find the model length when given the actual length and scale ratio. The scale 1:25 means 1 unit on the model represents 25 units in reality. We set up the proportion 1:25 = x:10, where x is the model length in meters. Cross-multiplying gives us 25x = 1 × 10 = 10, so x = 10 ÷ 25 = 0.4 meters. This means the 10-meter actual car becomes a 0.4-meter (or 40 cm) model. A common error is inverting the scale relationship or forgetting that the model should be smaller than the actual object.
A conical paper cup has radius $4\text{ cm}$ and height $9\text{ cm}$ . What is the volume of the cup? Use $\pi \approx 3.14$ and round to the nearest tenth of a cubic centimeter.
Answer: $150.7\text{ cm}^3$ The question asks for the volume of a conical paper cup with radius 4 cm and height 9 cm, rounded to the nearest tenth of a cubic centimeter. The formula needed is V = (1/3) π r² h, selecting it for cones and noting cm³ units. Substitute r=4 cm, h=9 cm, π≈3.14: r²=16, π r²=3.14×16=50.24, then 50.24×9=452.16, and (1/3)×452.16=150.72, rounding to 150.7 cm³. Perform multiplications step by step for accuracy. A key error is using the full cylinder formula without the 1/3 factor. Another mistake could be confusing radius and height. A strategy is to recall that cone volume is one-third of a cylinder with same base and height.
Which expression is equivalent to $\dfrac{(x-4)(x+1)}{x-4}$ for $x\ne 4$ ?
Answer: $x+1$ To simplify $\frac{(x-4)(x+1)}{x-4}$ for $x≠4$ , we can cancel the common factor $(x-4)$ from both numerator and denominator. This leaves us with $x+1$ . The restriction $x≠4$ remains because the original expression is undefined when $x=4$ (division by zero). A common error is to expand the numerator first, getting $\frac{x^2-3x-4}{x-4}$ , which makes the problem unnecessarily complex. When factors appear in both numerator and denominator, cancel them directly without expanding.
A science experiment uses the formula $T=\frac{5}{9}(F-32)$ to convert Fahrenheit $F$ to Celsius $T$ . Solve this equation for $F$ in terms of $T$ .
Answer: $F=\frac{9}{5}T+32$ We need to solve $T=\frac{5}{9}(F-32)$ for $F$ in terms of $T$ . First, multiply both sides by $\frac{9}{5}$ to get $\frac{9}{5}T = F - 32$ . Then add 32 to both sides: $F = \frac{9}{5}T + 32$ . A common error is to distribute the fraction incorrectly, getting $F = \frac{9}{5}T - 32$ or $F = \frac{5}{9}T + 32$ by confusing which fraction to use. When solving for a variable inside parentheses, first eliminate the coefficient outside the parentheses.
What is the annual percentage growth rate of the stock price?
Answer: 4% This question asks for the annual growth rate from the equation P = 50 $(1.04)^t$ . In exponential functions of the form y = $a(1 + r)^t$ , the growth rate r is found by subtracting 1 from the base. Here, the base is 1.04, so the growth rate is 1.04 - 1 = 0.04 = 4%. The coefficient 50 represents the initial stock price, while 1.04 is the growth factor indicating 4% annual increase. Students might confuse this with 40% by misinterpreting the decimal 0.04, or think it's 2% by some other calculation error. When identifying growth rates from exponential equations, always subtract 1 from the base and convert the decimal result to a percentage.
A video gets shared online. The number of views is 2,000 at $t=0$ days and increases by 25% each day. Which equation models views $V(t)$ , and about how many views are expected after 3 days?
Answer: $V=2000(1.25)^t$ ; 3906.25 This problem describes viral growth where views increase by 25% each day, indicating exponential growth. Starting with V(0) = 2000 and growing by 25% means multiplying by 1.25 each day, giving V(t) = 2000 $(1.25)^t$ . After 3 days: V(3) = 2000(1.25)³ = 2000(1.953125) = 3906.25 views. The growth factor 1.25 represents a 25% increase (100% + 25% = 125% = 1.25). A common error is using 0.25 as the base instead of 1.25 - remember that percentage increase means adding to 100%, not using the percentage alone.
Given the system $6x - 5y = 9$ and $7x + 4y = 40$ , what is the value of $y$ ?
Answer: 3 Eliminate $y$ by forming $24x - 20y = 36$ and $35x + 20y = 200$ , so $59x = 236$ gives $x = 4$ and substituting back yields $y = 3$ . Other values come from sign errors or mixing up variables.
A parabola has vertex $(2,-7)$ and passes through the point $(0,1)$ . Which equation represents the parabola in vertex form?
Answer: $y=2(x-2)^2-7$ We need the equation of a parabola with vertex $(2,-7)$ passing through $(0,1)$ . The vertex form is $y=a(x-2)^2-7$ for some value $a$ . To find $a$ , substitute the point $(0,1)$ : $1=a(0-2)^2-7=a(4)-7=4a-7$ . Solving: $1+7=4a$ , so $8=4a$ and $a=2$ . The equation is $y=2(x-2)^2-7$ . Common mistakes include using $(x+2)^2$ instead of $(x-2)^2$ (remember: vertex form uses $(x-h)$ where $h$ is the x-coordinate) or calculating the wrong value of $a$ . Always verify by checking both the vertex and the given point.
The number of gallons of water $g$ remaining in a tank after $t$ minutes is modeled by $g=120-3t$ . Which equation gives $t$ in terms of $g$ ?
Answer: $t=\frac{120-g}{3}$ We need to solve g = 120 - 3t for t in terms of g. Start by isolating the term with t: subtract 120 from both sides to get g - 120 = -3t. Divide both sides by -3: t = (g - 120)/(-3) = -(g - 120)/3 = (120 - g)/3. This matches choice C. A common error is sign mistakes when dividing by negative numbers or incorrectly distributing the negative sign. When solving for a variable, be especially careful with negative coefficients and verify by substituting back.
A coordinate plane shows a curve that crosses the $y$ -axis at $(0,3)$ and passes through $(1,6)$ and $(2,12)$ . Which equation best represents the curve?
Answer: $y=3\cdot 2^x$ We need to find an equation that passes through (0,3), (1,6), and (2,12). Notice that the y-values form the pattern 3, 6, 12, where each value doubles from the previous one - this suggests exponential growth. Testing y = 3· $2^x$ : when x = 0, y = 3· $2^0$ = 3·1 = 3; when x = 1, y = 3· $2^1$ = 6; when x = 2, y = 3· $2^2$ = 3·4 = 12. All three points match perfectly. The doubling pattern is the key indicator of an exponential function with base 2. When points show a constant multiplicative relationship, think exponential rather than linear or other function types.
Triangle $XYZ$ has side lengths $XY=7$ , $YZ=10$ , and $XZ=x$ . If the triangle inequality must be satisfied, which of the following could be the value of $x$ ?
Answer: 12 This problem tests the triangle inequality theorem, which states that the sum of any two sides must be greater than the third side. With XY = 7 and YZ = 10, we need: (1) 7 + 10 > x, so x < 17; (2) 7 + x > 10, so x > 3; (3) 10 + x > 7, which is always true for positive x. Therefore, 3 < x < 17. Looking at the options: 2 is too small, 3 is not valid (must be greater than 3), 17 is not valid (must be less than 17), but 12 satisfies 3 < 12 < 17. A common error is using ≤ or ≥ instead of strict inequalities. Remember that the triangle inequality uses strict inequalities: the sum must be greater than, not equal to, the third side.
A $45^\circ-45^\circ-90^\circ$ triangle has hypotenuse $14$ . What is the area of the triangle?
Answer: $49$ In a 45°-45°-90° triangle with hypotenuse 14, each leg has length 14/√2 = 14/√2 × √2/√2 = 14√2/2 = 7√2. The area = (1/2) × leg × leg = (1/2) × 7√2 × 7√2 = (1/2) × 49 × 2 = 49. A common error is forgetting that both legs are equal in an isosceles right triangle. Another approach: if hypotenuse = 14, then leg² + leg² = 196, so 2(leg²) = 196, giving leg² = 98 and area = (1/2) × 98 = 49.
A ramp rises to a platform 6 ft above the ground. If the ramp makes a $24^\circ$ angle with the ground, what is the length of the ramp (nearest tenth)?
Answer: 14.7 ft We need to find the length of a ramp that rises 6 ft at a 24° angle with the ground. The 6 ft height is the side opposite the 24° angle, and the ramp length is the hypotenuse, so we use sine: sin(24°) = 6/ramp length. Solving: ramp length = 6/sin(24°) = 6/0.407 ≈ 14.7 ft. A common mistake is using cosine or tangent instead of sine when the opposite side and hypotenuse are involved. Remember that sine relates the opposite side to the hypotenuse in SOH-CAH-TOA.
A road trip is $165$ miles long. How many yards is this trip? Use $1\text{ mile}=5280\text{ ft}$ and $3\text{ ft}=1\text{ yd}$ .
Answer: 290,400 yd We need to convert 165 miles to yards using two conversion steps. First, convert miles to feet: 165 miles × 5280 ft/mile = 871,200 feet. Then convert feet to yards: 871,200 ft ÷ 3 ft/yd = 290,400 yards. The dimensional analysis shows: 165 miles × (5280 ft/1 mile) × (1 yd/3 ft) = 165 × 5280/3 = 290,400 yards. A common error is multiplying by 3 instead of dividing when converting feet to yards. Remember that yards are larger than feet, so you get fewer yards.
On the coordinate plane, the line $\ell$ passes through the points $(-4,6)$ and $(2,-6)$ . Which of the following points lies on line $\ell$ ?
Answer: $(-1,0)$ The question asks which of the given points lies on the line passing through (-4, 6) and (2, -6). First, find the slope m = (-6 - 6)/(2 - (-4)) = -12/6 = -2. Then, derive the equation using point-slope form: y - 6 = -2(x + 4), simplifying to y = -2x - 2. Testing point C (-1, 0): 0 = -2*(-1) - 2 = 2 - 2 = 0, which matches. A common error is miscalculating the slope by forgetting the negative change in y. Another potential mistake is incorrect simplification when finding the equation. For efficiency in tests, derive the equation first, then plug in choices to check which satisfies it.
Consider the system of equations $y = 2x + 3$ and $y = x^2 + 1$ . How many solutions does the system have?
Answer: 2 This question asks us to find the number of solutions to a system with one linear equation and one quadratic equation. Since we have a line ( $y = 2x + 3$ ) and a parabola ( $y = x^2 + 1$ ), we solve by setting the equations equal: $2x + 3 = x^2 + 1$ . Rearranging gives $x^2 - 2x - 2 = 0$ , and using the quadratic formula yields $x = \frac{2 \pm \sqrt{4 + 8}}{2} = \frac{2 \pm \sqrt{12}}{2} = 1 \pm \sqrt{3}$ . Since we get two distinct real solutions for $x$ , there are exactly 2 intersection points. Remember that a line and parabola can intersect at 0, 1, or 2 points depending on whether the discriminant is negative, zero, or positive.
Simplify the expression $\sqrt{72}-2\sqrt{8}+\sqrt{18}$ . Combine like radicals after simplifying each term. Which expression is the simplified result?
Answer: $5\sqrt{2}$ The question asks to simplify the expression $\sqrt{72} - 2\sqrt{8} + \sqrt{18}$ by combining like radicals after simplifying each term. Start by simplifying each square root: $\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$ , $\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$ , and $\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$ . Substitute these back into the expression: $6\sqrt{2} - 2(2\sqrt{2}) + 3\sqrt{2} = 6\sqrt{2} - 4\sqrt{2} + 3\sqrt{2}$ . Combine the coefficients: $6 - 4 + 3 = 5$ , resulting in $5\sqrt{2}$ . A key error to avoid is attempting to combine unlike radicals or forgetting to distribute the -2 to the simplified $\sqrt{8}$ term. Always check your simplification by verifying that all radicals have the same radicand before combining. As a test-taking strategy, simplify each term individually before operating to minimize mistakes.

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