SAT Missed Math Questions

The class president chose 200 students at random from each of the junior and senior classes at her high school. Each student was asked how many hours of homework he or she completed in an average school night. The results are shown in the table below. #hours Junior Class Senior Class 1 25 30 2 80 70 3 50 60 4 35 35 5 10 5 There are a total of 600 students in the junior class and 400 students in the senior class. 20) What is the median number of hours of homework in an average night for all the students surveyed? a) 2 b) 3 c) 4 d) 5 21) Based on the survey data, which of the following statements accurately compares the expected total number of members of each class who complete four hours of homework? a) The total number of students who complete four hours of homework in the junior class is 35 more than in the senior class. b) The total number of students who complete four hours of homework in the senior class is 35 more than in the junior class. c) The total number of students who completed four hours of homework is equal in both classes. (a calculator is permitted)

20) There is no 'middle' number from which to select the median (since you have an even number, 400). Half the numbers should be greater than the median, and the other half should be less than the median. Since the president polled 200 students each from the junior and senior classes, a total of 400 students were polled. Therefore, in this case, there should be 200 greater than and 200 less than the median. Therefore, the median is the average of the 200th and 201st numbers. Find the 200th and 201st numbers in the ordered list. In the combined junior of senior classes, there are 80 + 70 = 150 students who complete two hours of homework. Therefore, there must be a total of 55 + 150 = 205 students with one or two hours. Since the tally is greater than 201, the 200th and 201st students must be part of the group of students who complete two hours of homework. Therefore, the average of the 200th and 201st students (i.e., the median) must be 2. The correct answer is a). 21) The question asks how to compare the number of students who complete four hours of homework in the two classes. The number in the table for both classes is 35, which would seem to point to d). However... *AHEM*... THESE NUMBERS DO NOT REPRESENT THE ENTIRETY OF THE TWO CLASSES, but rather, A RANDOM SAMPLE OF 200 FROM EACH CLASS. You need to use proportions to determine the actual expected amounts, since there are different total AMOUNTS of students in the junior class and the senior class. Since there are 600 students in the junior class, do 35/200 (sample #s) = x/600 (ACTUAL total juniors) giving x = 105 students in the junior class who complete 4 hours of homework. NOW, do 35/200 (sample #s) = x/400 (ACTUAL total seniors) giving x = 70 students in the senior class who complete 4 hours of homework. It appears that the junior class has 105 - 70 = 35 more students who complete four hours of homework than does the senior class. The correct answer is a). (PT2 via Princeton Review, S4, Q20+Q21, p. 142)

ω² = ω₀² + 2αθ (angular position - angular velocity) ω = ω₀ + αt (time - angular velocity) θ = ω₀t + (1/2)αt² (time - angular position) A carousel is rotating at an angular velocity of 90 degrees per second. The instant a particular point on the carousel reaches a position θ = 0°, the carousel operator flips a switch, causing the carousel at a constant angular acceleration to slow down and eventually change direction. The equations above describe the constant-acceleration motion of the carousel, where ω₀ represents the initial angular velocity, ω is the angular velocity as it travels, θ is the angular position of the particular point on the carousel, t is the time since the switch was flipped, and α is the constant angular acceleration (-12.6°/s²). 37) To the nearest degree, at what angular position will the carousel change direction? 38) To the nearest second, how long will it take the carousel to come to a complete stop before it changes direction? (a calculator is permitted)

37) The question asks for the angular position at which the carousel will change direction. Angular position is the first equation and is represented by θ. Write down known variables and solve. When the carousel changes direction, the angular velocity is 0. Use the first equation, ω² = ω₀² + 2αθ. Plug in ω = 0 (that's the angular velocity we WANT), ω₀ = 90 (that's the angular velocity right NOW), and α = -12.6 (it's just like m/s², expressing the time rate at which the angular velocity is CHANGING). This gives us 0² = 90² + 2(-12.6)θ. Simplify the right side to get 0² = 8,100 - 25.2θ. Add 25.2θ to both sides to get 25.2θ = 8100. Divide both sides by 25.2 to get θ = 321.4286. Rounded to the nearest degree, the correct answer is 321. 38) The question asks for the time it will take the carousel to completely stop before changing direction. Write down known variables, then choose the equation that gives only time as the unknown. This is the second equation. When the carousel changes direction, the angular velocity is 0. Use the second equation, ω = ω₀ + αt. Plug in ω = 0, ω₀ = 90, and α = -12.6 to get 0 = 90 + (-12.6)t. Simplify the right side to get 0 = 90 - 12.6t. Add 12.6t to both sides to get 12.6t = 90. Divide both sides by 12.6 to get t = 7.1429. Rounded to the nearest second, the correct answer is 7. (PT5 via Princeton Review, S4, Q37-38, p. 425)

Pₜ₊₁ = Pₜ + 0.3(Pₜ)(1 - [Pₜ/C]) A certain species of deer on an isolated island has a current population of 4,200. The estimated population of deer next year, Pₜ₊₁, is related to the population this year, Pₜ, by the formula above. In this formula, the constant C represents the maximum number of deer the island is capable of supporting. 37) Suppose that environmental conditions on the island changed suddenly, and there was a resultant decrease in the maximum number of deer the island is capable of supporting. If the number of deer increased from 4200 this year to 4704 next year, what would be the maximum number of deer the island is capable of supporting? 38) If C = 10,500, and the given formula is accurate, what will the population of deer be 2 years from now? (Round your answer to the nearest whole number). (a calculator is permitted)

37) The question asks for the maximum number of deer the island can support, given the populations for two years. According to the question, Pₜ = 4,200 and Pₜ₊₁ = 4,704. Plug these values into the equation to get 4,704 = 4,200 + (0.3)(4,200)(1 - [4,200/C]). Simplify the right side of the equation to get 4,704 = 4,200 + 1,260(1 - [4,200/C]). Subtract 4,200 from both sides to get 504 = 1,260(1 - [4,200/C]). Divide the equation by 1,260 to get 0.4 = 1 - (4,200/C). Subtract 1 from both sides to get -0.6 = -(4,200/C), then multiply both sides by C to get -0.6C = -4,200. Divide both sides by -0.6 to get C = 7,000. The correct answer is 7,000. 38) The question asks for the deer population 2 years from now. According to the equation, C = 10,500 and Pₜ = 4,200. Therefore, Pₜ₊₁ = 4,200 + (0.3)(4,200)(1 - [4,200/10,500]). Solve the equation to get Pₜ₊₁ = 4,200 + 1,260(1 - 0.4) = 4,200 + 756 = 4,956. That's the deer population after one year, but the question asks for the population after two years, so do it again. Plug 4,956 into the equation as the new Pₜ to get the deer population two years from now: Pₜ₊₁ = 4,956 + 0.3(4,956)(1 - [4,956/10,500]). Solve the equation to get 4,956 + 1,486.8(1 - 0.472) = 4,956 + 785.03 ≈ 5,741. Only round at the last step to make sure the answer is as accurate as possible. The correct answer is 5,741. (PT2 via Princeton Review, S4, Q37+Q38, p. 149)

Which of the following is the equation of a circle in the xy-plane with center (0,4) and a radius with endpoint (4/3, 5)? a) x² + (y - 4)² = 25/9 b) x² + (y + 4)² = 25/9 c) x² + (y - 4)² = 5/3 d) x² + (y + 4)² = 5/3 (a calculator is permitted)

Based on the given coordinates for the center, we know the left will look like (x - 0)² + (y - 4)² = x² + (y - 4)². To find the radius, we need to plug the center and endpoint values into the distance formula, d = √(x₂ - x₁) + (y₂ - y₁). This gives d = √(4/3)² + (1)² = √(16/9) + (9/9) = √(25/9) = 5/3. So we just do x² + (y - 4)² = 5/3, right? WRONG. The standard form of the equation of a circle is (x - h)² + (y - k)² = r². We need to write the equation such that r is still squared. Therefore, the correct expression is x² + (y - 4)² = 25/9. The correct answer is a). (PT1 via Khan Academy, S4, Q24)

x = √(30 + x) Which of the following includes all solutions to the equation above? a) There are no values of x that satisfy the given equation. b) -6 and 5 c) -6 d) 5

If you work it all out, the roots are -6 and 5. Did you say -6 and 5? WRONG. On the SAT, the result of a square root is only positive, so this is not true. The value of x = -6 is an extraneous solution, so eliminate b) and c). We know that 5 satisfies the equation, so eliminate a). That leaves d). The correct answer is d). You can access a fuller explanation in the book. (PT5 via Princeton Review, S3, Q14, p. 408)

G = (ab)/d² The gravitational force, G, between an object of mass a and an object of mass b is given by the formula above, where d represents the distance between the two objects. Objects k and m have the same masses, respectively, as do objects a and b. If the gravitational forces between k and m is 9 times the gravitational force between a and b, then the distance between k and m is what fraction of the distance between a and b? a) 1/243 b) 1/81 c) 1/9 d) 1/3 (a calculator is permitted)

One method is, how can I manipulate the original distance (d) to raise the G by a factor of 9? (Remember, the masses have to stay the same). Well, I can make the denominator 1/9 of what it was before, so do 1/9(d²). So is it just 1/9? WRONG. What would you have to do to each individual distance term? You'd have to do 1/3 of each one. (1/3)(d)(1/3)(d) = 1/9(d²), raising G by a factor of 9. The correct answer is d). Here's how Princeton did it: The question asks for the relationship between the distances of two pairs objects with the same mass and one pair with 9 times the gravitational force of the other pair. Start by putting in the same mass for k, m, a, and b. Let k = m = a = b = 2. The gravitational force between k and m is 9 times the force between a and b, so plug in Gₖₘ = 81 and Gₐb = 9. Square numbers will work well here, since the distance is squared in the formula. Use these values in the formula to find the distance between the given objects. The gravitational force for k and m becomes 81 = (2)(2)/(dₖₘ)², so 81 (dₖₘ)² = 4/81. Taking the square root of both sides gives dₖₘ = 2/9. Follow the same steps to find the gravitational force for a and b: 9 = (2)(2)/(dₐb)², then 9(dₐb)² = 4, so (dₐb)² = 4/9 and dₐb = 2/9. Now make the fraction: (dₖₘ)/(dₐb) = (2/9)(2/3) = (2/9) x (3/2) = 1/3. The correct answer is d). (PT2 via Princeton Review, S4, Q24, p. 143)

The graph in the xy-plane of the function g has the property that y is always greater than or equal to -2. Which of the following could be g? a) g(x) = x² - 3 b) g(x) = (x - 3)² c) g(x) = |x| - 3 d) g(x) = (x - 3)³

Since you can't use a calculator, try out some values of x instead and use Process of Elimination. Usually, zero is a number to avoid, since it messes things up, so the goal here is to find what could be true, so messing things up helps. Choices a), c), and d) all yield y-values that are less than -2, so those values don't work. Plugging x = 0 in for b) gives x² = 0, so that one works. The correct answer is b). NOTE: Remember that any squared real number is positive, so the output of y = f(x) = x² will always be positive! Remembering that will speed up the process significantly. (PT2 via Princeton Review, S3, Q11, p. 131)

y < -x + a y > x + b In the xy-plane, if (0,0) is a solution to the system of inequalities above, which of the following relationships between a and b must be true? a) a > b b) a < b c) |a| > |b| d) a = -b (a calculator is permitted)

Substituting (0,0) into the system gives us 0 < a and 0 > b, which means a is positive and b is negative. Since positive numbers are always greater than negative numbers, a > b must be true. The correct answer is a). (PT1 via Khan Academy, S4, Q18)

(80x² + 84x - 13)/(kx - 4) = -16x - 4 - [29/(kx - 4)] (you might want to rewrite that as a fraction on your paper) The equation above is true for all values of x ≠ (4/k), where k is a constant. What is the value of k? a) -5 b) -2 c) 2 d) 5

The question asks for a value of constant k in the equation. This is a complicated algebra equation, so look for a way to use an actual number instead. Since no calculator is allowed on this section, it is especially important to pick an easy number. Try x = 1. If x = 1, then (80[1]² + 84(1) - 13)/[k(1) - 4] = -16(1) - 4 - 29/[k(1) - 4]. Simplify to get 151/(k - 4) = -20 - 29/(k - 4). Add 29/(k - 4) to both sides to get 151/(k - 4) + 29/(k - 4) = -20. Since the fractions on the right have the same denominator, add both the numerators to get 180/(k - 4) = -20. Multiply both sides by (k - 4) to get 180 = -20k + 80. Distribute on the right side to get 180 = -20k + 80. Subtract 80 from both sides to get 100 = -20k. Divide both sides by -20 to get k = -5. The correct answer is a). (PT3 via Princeton Review, S3, Q14, p. 225)

x g(x) 0 2 1 5 3 -1 7 0 The function g is defined by a polynomial. Some of the values of x and g(x) are shown in the table above. Which of the following must be a factor of g(x)? a) x - 1 b) x - 2 c) x - 3 d) x - 7

The question asks for an expression that must be a factor of polynomial function g. A factor of a polynomial is used to find a solution, or a value of x for which the corresponding value of the function is 0. When a function in factored form is set equal to 0, each factor can be set equal to 0 to get each solution (e.g., solutions of (x - 7)(x + 5) are 7 and -5 respectively). Since, according to the table, g(7) = 0, x = 7 is one solution of g. Therefore, it must also be the solution to an equation made by setting one of the factors equal to 0. To find this factor, get the equation x = 7 into the form of an equation with one side equal to 0 (since you're trying to find the ROOTS, or solutions, of g[x]). Subtract 7 from both sides to get x - 7 = 0. Therefore, x - 7 is one of the factors of g. The correct answer is d). (PT3 via Princeton Review, S3, Q8, p. 222)

Mathias saves an average of d dollars per month, where d > 300. The actual amount he saves per month varies, but is always within $20 of the average amount. If Mathias saved k dollars this month, which of the following inequalities expresses the relationship between k, the amount he saved this month, and d, the average amount he saves per month? a) d - k < 20 b) d + k < 20 c) -20 < d - k < 20 d) -20 < d + k < 20 (a calculator is permitted)

The question asks for an inequality to represent the relationship between the amount Mathias saved this month and the amount he usually saves on average. Try using some numbers for the variables. For example, let d = 400 and k = 390. Plug these values into the answer choices to see which one works. Choice a) becomes 10 < 20, so keep a). Choice b) becomes 790 < 20, so eliminate b). Choice c) becomes -20 < 10 < 20, so keep c). Choice d) becomes -20 < 790 < 20, so eliminate d). Now plug in some different numbers to try and eliminate a) or c). Try, for example, d = 390 and k = 400. Choice a) becomes -10 < 20, and choice c) becomes -20 < -10 < 20. Both of these are still true. When that happens, try using some numbers that don't work - the correct answer will prove false and an incorrect answer may be true. Try d = 400 and k = 450, which don't work because they are more than $20 apart. Choice a) becomes -50 < 20, which is still true (even though we have the wrong conditions) so eliminate. it. The correct answer is c). (PT2 via Princeton Review, S4, Q19, p. 141)

At a certain food truck, hamburgers are sold for $5 each and hot dogs are $3 each. If Martina buys one hamburger and 'h' hot dogs, and spends at least $20 and no more than $25, what is one possible value of 'h'? (a calculator is permitted)

The question asks for one possible value of h, the number of hot dogs Martina buys. Martina spends between $20 and $25, inclusive, and she buys one hamburger at a cost of $5. This would leave her at least $20 - $5 = $15 and at most $25 - $5 = $20 for hot dogs. In the first case, $15 total divided by $3 per hot dog would get her 5 hot dogs, so 5 is one possible value for h. If she spent up to $20 on hot dogs, she could get $20 divided by $3 per hot dog for 6.67 hot dogs. She can only by whole hot dogs, so 6 is another possible value of h. Therefore, the two possible correct answers are 5 and 6. (PT5 via Princeton Review, S4, Q31, p. 424)

y ≤ 20x + 3,500 y ≤ -8x The graph in the xy-plane of the solution set of the system of inequalities above contains the point (j,k). What is the greatest possible value of k? (a calculator is permitted)

The question asks for the greatest value of k, the y-coordinate of a point in the solution to a system of inequalities. Draw a rough sketch of the graph of this system of inequalities to figure out what is going on here (the one they did is attached). The area included in both equalities represents the solution to the system. The question asks for the greatest value of k, which is the y-coordinate, so it would happen as close to the top of the graph as possible. For the area of overlap representing the solution, this happens at the point of intersection of the two lines. Find this point of intersection by setting the two equations, 'y = 20x + 3,500' and 'y = -8x', equal to each other to get 20x + 3,500 = -8x. Add 8x to both sides to get 28x + 3500 = 0, so then 28x = 3500, right? WRONG. Subtract 3,500 from both sides, giving 28x = -3,500. Dividing both sides by 28 results in x = -125. This is the value for j, and it can be plugged back into either equation to get the value of k, the y-coordinate at that point. Use the easier equation, y = -8x, to get k = -8(-125) = 1,000. The correct answer is 1,000. (PT3 via Princeton Review, S4, Q35, p. 242)

The cost C, in dollars, that a catering company charges to cater a wedding is given by the function C = 20wt + 300, where w represents the number of workers catering the wedding and t represents the total time, in hours, it will take to cater the wedding using w workers. Which of the following is the best explanation of the number 20 in the function? a) A minimum of 20 workers will cater the wedding. b) The cost of every wedding will increase by $20 per hour. c) The catering company charges $20 per hour for each worker. d) There will be 20 guests at the wedding.

The question asks for the meaning of the number 20 in the equation. Label the pieces of the equation, and use Process of Elimination. The w represents the number of workers, and t represents the total time in hours. Therefore, the 20 must have something to do with the cost of the workers' hourly wage. Eliminate a) because nothing is mentioned about a minimum number of workers. Eliminate b) because it is unrelated to the workers. Choice c) relates to the hourly wage, so keep it, but check d) just in case. Choice d) can be eliminated because the number of wedding guests is unrelated to the cost and unrelated to the workers and their time. The correct answer is c). (PT2 via Princeton Review, S3, Q1, p. 129)

A recipe for making lemonade states that one ounce of sugar is sufficient to make 30 imperial pints of lemonade. If an imperial pint is equal to 1 and 1/4 U.S. pints, approximately how many U.S. pints of lemonade can be made with 17 ounces of sugar? a) 515 b) 640 c) 1015 d) 1280 (a calculator is permitted)

The question asks for the number of U.S. pints of lemonade that can be made using 17 ounces of sugar. First, calculate the number of imperial pints that can be made with 17 ounces of sugar. Set up the following proportion: (1 oz/30 I. Pints) = (17 oz/x). Cross-multiply to get x = 510 imperial pints. Next, calculate the number of U.S. pints that are equivalent to 510 imperial pints by setting up the following proportion: (1 I. Pint/1.25 U.S. pints) = (510 I. pints/x). Simplify the left side of the equation to (1/[5/4]) = (510/x), or (4/5) = (510/x). Cross-multiply, resulting in 4x = 2550. Divide both sides by 4 to get x = 637.5. The closest approximation for 637.5 is 640. The correct answer is b). (PT3 via Princeton Review, S4, Q20, p. 236)

Mrs. Warren has b boxes of Girl Scout cookies that she wants to distribute to members of her troop. If she gives each girl 4 boxes, she will have 11 boxes left over. If she wanted to give each student 5 boxes, she would need an additional 12 boxes. How many girls are in Mrs. Warren's Girl Scout troop? a) 12 b) 23 c) 27 d) 32 (a calculator is permitted)

The question asks for the number of girls in the Girl Scout troop. This is a specific value and the answers are numbers, so try out the answers to see which one works. Start with b). If there are 23 girls in the troop, then Mrs. Warren currently has 23(4) + 11 = 103 boxes. If she were to give 5 boxes to each girl, she would need 23(5) = 115. Therefore, she is 115 - 103 = 12 boxes short. This matches the information given in the problem. The correct answer is b). (PT3 via Princeton Review, S4, Q22, p. 237)

Jessica owns a store that sells only laptops and tablets. Last week, her store sold 90 laptops and 210 tables. This week, the sales, in number of units, of laptops increased by 50 percent, and the sales, in number of units, of tablets increased by 30 percent. By what percentage did total sales, in units, in Jessica's store increase? a) 20 percent b) 25 percent c) 36 percent d) 80 percent (a calculator is permitted)

The question asks for the percent increase in total sales. Since the number of laptops and the number of tables are different, don't just add the two percent increased. Thus, d) is a trap answer. A percent change is always equal to the expression (difference/original) x 100. The original is the total number of units sold last week, which is 90 + 210 = 300. To get the difference, get the increase in laptops and the increase in tablets separately and add. There is a fifty percent increase in laptop sales, so the increase is (50/100) x 90 = 45. There is a thirty percent increase in tablet sales, so the increase is (30/100) x 210 = 63. Therefore, the total difference is 45 + 63 = 108, and the percent increase is (108/300) x 100 = 36%. The correct answer is c). (PT5 via Princeton Review, S4, Q21, p. 419)

===== Decaffeinated ===== Caffeinated Tea _____________________ _________________ Coffee __________________ _________________ ============28=================116===== The partially completed (and poorly reproduced) table above shows all the drinks that were sold on one day at a coffee shop. The shop sold 3 times as many cups of caffeinated tea as it did decaffeinated tea, and it sold 5 times as many cups of caffeinated coffee as it did decaffeinated coffee. If 28 cups of decaffeinated beverages and 116 cups of caffeinated beverages were sold, and one cup is selected at random out of all the caffeinated beverages that were sold, which of the following is closest to the probability that this cup contains coffee? a) 0.508 b) 0.583 c) 0.672 d) 0.690 (a calculator is permitted)

The question asks for the probability that a randomly selected caffeinated beverage will be coffee, but the table is incomplete. To fill in the table, write a system of equations using the information given. Call the number of decaffeinated teas t, so the number of caffeinated teas is 3t. Call the number of decaffeinated coffees c, so the number of caffeinated coffees is 5c. The two equations that can be written from this information and the table are t + c = 28, and 3t + 5c = 116. When dealing with systems of equations, look for a way to stack and add the equations to eliminate one variable and solve for another (girl please. See p. 275 to see what they did, I don't have that kind of time. 3(28 - c) + 5c = 84 - 3c + 5c = 84 + 2c = 116, 2c = 32, c = 16). Use the values to fill in the chart, giving c = 16, t = (28 - c) = 12, 3t = 3(12) = 36, and 5c = 5(16) = 80. Now find the probability that a caffeinated beverage chosen at random is a coffee. Divide the number of caffeinated coffees, 80, by the total number of caffeinated beverages, 116, to get a probability of 0.69. The correct answer is d). (PT3 via Princeton Review, S4, Q28, p. 238)

The line y = cx + 6, where c is a constant, is graphed in the xy-plane. If the point (r,s) lies on that line, where r ≠ 0 and s ≠ 0, what is the slope of the line, in terms of r and s? a) (r - 6)/s b) (6 - s)/r c) (6 - r)/s d) (s - 6)/r

The question asks for the slope of the line in terms of r and s, the coordinates of a point on the line. A line whose equation is in the form 'y = mx + b' has slope m and y-intercept b. In the equation 'y = cx + 6', the slope is c. Plug the point (r,s) in the equation to get 's = cr + 6'. To find the slope, solve for c. First, subtract 6 from both sides to get 's - 6 = cr'. Now, divide both sides by r to get c = (s - 6)/r. The correct answer is d). (PT3 via Princeton Review, S3, Q9, p. 223)

cx - 6y = 8 3x - 7y = 5 In the system of equations shown above, c is a constant and x and y are variables. For what value of c will the system of equations have no solution? a) (24/5) b) (18/7) c) -(18/7) d) -(24/5)

The question asks for the value of c that will cause the system of equations to have no solution. Since there are no exponents of x or y in either equation, the equations are linear. A system of linear equations has no solution if the two lines represented by the solutions are parallel. Two lines are parallel when the have the same slope. To determine the slope of the lines, get each line into slope-intercept form: 'y = mx + b'. Start with the second equation, 3x - 7y = 5. Subtract 3x from both sides to get -7y = -3x + 5. Divide both sides by -7 to get y = (3/7)x - (5/7). In slope-intercept form, the slope is equal to m, so the slope of this line is (3/7). Now get the slope of the other line, cx - 6y = 8. Subtract cx from both sides to get -6y = -cx + 8. Divide both sides by -6 to get y = (c/6)x - (8/6), so the slope of this line is (c/6). Since these two slopes have to be equal, set (3/7) = (c/6). Cross multiply to get 7c = 18. Divide both sides by 7 to get c = (18/7). The correct answer is b). (PT3 via Princeton Review, S3, Q7, p. 223)

In the xy-plane, the line determined by the points (c,3) and (27,c) intersects the origin. Which of the following could be the value of c? a) 0 b) 3 c) 6 d) 9

The question asks for the value of c, a coordinate in two points on a line. The line intersects the origin as well as the points (c,3) and (27,c). Questions about lines in the xy-plane often involve slope, so determine the slope of this line. Any two points can be used to find the equation of a line, as confirmed by Euclid's 1st postulate (including the slope). Note that since the line intersects the origin, it intersects point (0,0) as well as the other two points. Use points (0,0) and (c,3) to calculate the slope: (y₂ - y₁)/(x₂ - x₁) = (3 - 0)/(c - 0) = (3/c). The slope can also be determined using points (0,0) and (27,c): (c - 0)/(27 - 0) = (c/27). Since these two slopes must be equal, (3/c) = (c/27). Cross multiply to get c² = 81. Take the square root of both sides to get c = ±9. Only 9 is a choice. Therefore, the correct answer is d). (PT5 via Princeton Review, S4, Q25, p. 420)

In triangle UVW, the measure of ∠U is 90°, WV = 39, and UV = 36. Triangle XYZ is similar to triangle UVW, where ∠X, ∠Y, and ∠Z correspond to ∠U, ∠V, and ∠W, respectively. If each side of triangle XYZ is 3/5 the length of its corresponding side of triangle UVW, what is the value of cos Z?

The question asks for the value of cos Z. By definition, corresponding angles in similar triangles are congruent. Since ∠Z corresponds to ∠W in a similar triangle, ∠Z is congruent to ∠W. Since congruent angles have equal cosines, cos Z = cos W. Therefore, this question can be answered by ignoring triangle XYZ and working exclusively with triangle UVW to determine the value of cos W, and thus the value of cos Z. Draw triangle UVW, filling in WV = 39 and UV = 36. (got it? great.) By definition, cos = adj/hyp. The hypotenuse is 39, but the adjacent side isn't given. The adjacent side can be solved for using the Pythagorean theorem, but this is difficult with numbers this large and no calculator. Instead, look for a Pythagorean triple. The ratio 36:39 can be reduced by a factor of 3 to 12:13, so this is a 5:12:13 right triangle. Therefore, the missing side, UW, must have a length of 5 x 3 = 15. Thus, the adjacent side is 15, and cos W = 15/39 = 5/13. The correct answer is 5/13. (PT3 via Princeton Review, S3, Q19, p. 227)

y = rx² + s y = -2 In the system of equations above, r and s are constants. For which of the following values of r and s does the system have exactly two real solutions? a) r = -2, s = -1 b) r = -1, s = -2 c) r = 2, s = -2 d) r = 3, s = 1 (a calculator is permitted)

The question asks for the values of constants r and s in a system of equations that will result in two real solutions to the system. The solutions to a system of equations are the points the equations share. The line y = -2 is the straightforward equation, so y must be -2 in the other equation. Plug the values given in each answer choice for r and s, along with -2 for y, and solve for x. The answer that yields two real solutions for x will be the correct answer. Choice a) becomes -2 = -2x² - 1. Add 1 to both sides to get -1 = -2x². Divide both sides by -2 to get 1/2 = x². Take the square root of both sides of the equation to get ±√(1/2) = x. Therefore, the equation for the values in a) has two real solutions: (-√1/2, -2) and (√1/2, -2). The correct answer is a). (PT2 via Princeton Review, S4, Q30, p. 146)

A survey of 130 that randomly selected workers in a particular metropolitan area was conducted to gather information about average daily commute times. The data is shown in the table below. ================= Commutes with PT=====Doesn't commute with PT===Total Less than 1 hour===========22=========================46================68 At least 1 hour=============29=========================33================62 Total======================51==========================79=================130 Based on the data, how many times more likely is it for a person with a commute with less than 1 hour NOT to commute by public transit than it is for a person with a commute of at least one hour NOT to commute by public transit? (Round the answer to the nearest hundredth.) a) 1.39 times as likely b) 1.27 times as likely c) 0.78 times as likely d) 0.72 times as likely. (a calculator is permitted)

The question asks how many times more likely it is for a commuter whose average daily commute is less than 1 hour NOT to take public transit than it is for a commuter whose average daily commute is at least one hour NOT to take public transit. The term 'more likely' refers to probability, so determine the probability of each. Go to the (poorly drawn) table and find the number of commuters who commute less than 1 hour and do NOT commute using public transit. According to the table, there were 46 people in this category, and the total number of commuters who commute less than 1 hour is 68. Therefore, the probability is (46/68). Now do the same for the probability that someone who commutes at least 1 hour does NOT take public transit. According to the table, the number under 'Doesn't Commute with PT' is 33 and the number under 'Total' is 62, so the probability is (33/62). The question asks HOW MANY TIMES MORE LIKELY is the first probability than the second. In other words, P₂ = P₁(x). Set up the equation (46/68) = (33/62)x. Divide both sides by (33/62) to get x ≈ 1.27. The correct answer is b). (PT5 via Princeton Review, S4, Q17, p. 417)

At the end of a card game, Eve has a pile of red and blue chips that is worth $120. If red chips are worth $5 and blue chips are worth $20, and Eve has at least one blue chip, what is one possible number of red chips Eve has?

This question asks for the number of red chips Eve has in a game. Rather than writing equations, try out different numbers. Start by testing whether a single red chip could work. If Eve has one red chip, then she has $120 = $5 = $115 worth of blue chips, which does not divide by 20 for an integer number of blue chips. If Eve had 2 red chips, she would have $110 worth of blue chips, which still won't work. If Eve had $20 worth of red chips, though, she would have $100 left for exactly 5 blue chips. She would need 4 red chips to equal $20, so 4 is one possible answer. Any number of red chips that gave Eve a multiple of $20 would also work, so 8, 12, 16, and 20 are also correct answers. (PT2 via Princeton Review, S3, Q18, p. 134)

Matthew constructs a fence around a patch of grass in his backyard. The patch has a width that is 8 feet more than 4 times the length. What is the perimeter of the fence if Matthew's patch of grass has an area of 5,472 square feet? a) 364 feet b) 376 feet c) 396 feet d) 400 feet (a calculator is permitted)

This question asks for the perimeter of the fence Matthew will need around his patch of grass. The question says that the width is 8 feet more than 4 times the length, so translate this scenario into the equation 'w = 4l + 8'. The question also says that the area is 5472. The area of a rectangle can be found by using the formula A = lw. Substitute 'A = 5,472' and 'w = 4l + 8' to get 5,472 = l(4l + 8). Distribute the l to get 5472 = 4l² + 8l. Since this is a quadratic equation, get one side equal to 0 by subtracting 5,472 from both sides to get 4l² + 8l - 5,472 = 0. This is a difficult quadratic to factor, so use the quadratic formula where a = 4, b = 8, and c = -5,472. I'm not writing it all out, but you should end up with (-8 ± 296)/8 = 288/8 = 36. (Since we're talking about length, the negative answer is an extraneous solution). If l = 36, then w = 4l + 8 = 4(36) + 8 = 152. To find the perimeter, use P = 2l + 2w = 2(36) + 2(152) = 376. The correct answer is b). (PT5 via Princeton Review, S4, Q24, p. 420)

Alma bought a laptop computer at a store that gave a 20 percent discount off its original price. The total amount she paid to the cashier was p dollars, including an 8 percent sales tax on the discounted price. Which of the following represents the original price of the computer in terms of p? a) 0.88p b) p/0.88 c) (0.8)(1.08)p d) p/(0.8)(1.08) (a calculator is permitted)

Using O to represent the original price of the laptop computer, the discounted price is (1 - 0.2)O = 0.8O, and Alma paid (0.08)(0.8O) in taxes. The total amount she paid, p, is therefore: 0.08(0.8O) + 0.8O = (0.8)(1.08)O. Since p = (0.8)(1.08)O, rewriting the equation to isolate O gives us O = p/(0.8)(1.08). The correct answer is d). (PT1 via Khan Academy, S4, Q20)

The annual budget for agriculture/natural resources in Kansas (in thousands) was $358,708 in 2008. The annual budget for agriculture/natural resources in Kansas (in thousands) in 2010 was $488,106. Which of the following best approximates the average rate of change in the annual budget for agriculture/natural resources in Kansas from 2008 to 2010? a) $50,000,000 per year b) $65,000,000 per year c) $75,000,000 per year d) $130,000,000 per year (a calculator is permitted)

We can rewrite $358,708 as $358,708,000 and $488,106 as $488,106,000. The difference between these two budgets, in dollars, is: $488,106,000 - $358,708,000 = $129,398,000. Since this change took place over the course of 2010 - 2008 = 2 years, the average rate of change, in dollars per year, is: $129,398,000/2 = $64,699,000. Of the choices provided, $65,000,000 is closest to $64,699,000 per year. The correct answer is b). (PT1 via Khan Academy, S4, Q22)

Which of the following numbers is NOT a solution of the inequality 3x − 5 ≥ 4x − 3? a) -1 b) -2 c) -3 d) -5 (a calculator is permitted)

We can solve the linear inequality and identify the choice that is not in the solution set. 1) 3x − 5 ≥ 4x − 3 2) 3x - 5 - 3x ≥ 4x - 3 - 3x 3) -5 ≥ x - 3 4) x ≤ -2 Since x must be less than or equal to -2, choice a), −1, is not part of the solution set. (PT1 via Khan Academy, S4, Q11)

A square field measures 10 meters by 10 meters. Ten students mark off a randomly selected region of a field; each region is square and has side lengths of 1 meter, and no two regions overlap. The students count the earthworms contained in the soil to a depth of 5 centimeters beneath the ground's surface in each region. The results are shown in the table below. Region/# of earthworms A 107 B 147 C 146 D 135 E 149 F 141 G 150 H 154 I 176 J 166 Which of the following is a reasonable approximation of the number of earthworms to a depth of 5 centimeters beneath the ground's surface in the entire field? A) 150 B) 1,500 C) 15,000 D) 150,000 (a calculator is permitted)

We need to find a reasonable approximation of the number of earthworms to a depth of 5 centimeters for a 1 square meter region, then apply this approximation to the area of the entire field. The values of this region are for 10 regions each with an area of 1² = 1 square meter. Since the square field measures 10 meters by 10 meters, the area of the field, in square meters, is 10² = 100. Since there are approximately 150 earthworms (calculate the average, or just look at the data values) in each individual square meter of the field, there should be approximately 150 x 100 = 15,000 earthworms in all 100 square meters of the field. Be careful not to select choice a), which gives a reasonable approximation for the number of earthworms in 1 square meter of the field. The correct answer is c). (PT1 via Khan Academy, S4, Q27)