# SAT Math Hard Practice Quiz. 5. How many integers between 10 and 500 begin and end in 3?

Save this PDF as:

Size: px
Start display at page:

Download "SAT Math Hard Practice Quiz. 5. How many integers between 10 and 500 begin and end in 3?"

## Transcription

1 SAT Math Hard Practice Quiz Numbers and Operations 5. How many integers between 10 and 500 begin and end in 3? 1. A bag contains tomatoes that are either green or red. The ratio of green tomatoes to red tomatoes in the bag is4to3. When fivegreentomatoesandfiveredtomatoes are removed, the ratio becomes 3 to 2. How many red tomatoes were originally in the bag? (A) 12 (B) 15 (C) 18 (D) 24 (E) A particular integer N is divisible by two different prime numbers p and q. Which of the following must be true? I. N is not a prime number. II. N is divisible by pq. III. N is an odd integer. 2. If each digit in an integer is greater than the digit to the left, the integer is said to be monotonic. For eample, 12isamonotonicintegersince2 > 1. Howmanypositive two-digit monotonic integers are there? (A) 28 (B) 32 (C) 36 (D) 40 (E) 44 (A) I only (B) II only (C) I and II only (D) I and III only (E) I, II, and III 7. A perfect square is an integer that is the square of an integer. Suppose that m and n are positive integers such that mn > 15. If 15mn is a perfect square, what is the least possible value of mn? a, 2a 1, 3a 2, 4a 3, For a particular number a, the first term in the sequence above is equal to a, and each term thereafter is 7 greater than the previous term. What is the value of the 16 th term in the sequence? 8. M is a set of si consecutive even integers. When the least three integers of set M are summed, the result is. When the greatest three integers of set M are summed, the result is y. Which of the following is true? 4. If p is a prime number, how many factors does p 3 have? (A) One (B) Two (C) Three (D) Four (E) Five (A) y = 18 (B) y = +18 (C) y = 2 (D) y = 2+4 (E) y = 2+6 erikthered.com/tutor pg. 1

2 SAT Math Hard Practice Quiz 9. A three-digit number, XYZ, is formed of three different non-zerodigits X, Y, andz. Anew numberis formedby rearranging the same three digits. What is the greatest possible difference between the two numbers? (For eample, 345 could be rearranged into 435, for a difference of = 90.) 10. An integer is subtracted from its square. The result could be which of the following? (A) A negative integer. (B) An odd integer. (C) The product of two consecutive even integers. (D) The product of two consecutive odd integers. (E) The product of two consecutive integers. erikthered.com/tutor pg. 2

3 SAT Math Hard Practice Quiz Algebra and Functions 1. Let m be an even integer. How many possible values of m satisfy m+7 3? (A) One (B) Two (C) Three (D) Four (E) Five 4. Let m and n be positive integers such that one-third of m is n less than one-half of m. Which of the following is a possible value of m? (A) 15 (B) 21 (C) 24 (D) 26 (E) Let be defined by = +3 for any such that 1 1. Which of the following is equivalent to 1? 5. If a and b are numbers such that (a 4)(b+6) = 0, then what is the smallest possible value of a 2 +b 2? (A) (B) (C) (D) (E) Let f() = a 2 and g() = b 4 for any value of. If a and b are positive constants, for how many values of is f() = g()? (A) None (B) One (C) Two (D) Three (E) Four 3. Let a and b be numbers such that a 3 = b 2. Which of the following is equivalent to b a? (A) b 2/3 (B) b 4/3 (C) b 2 7. Let a and b be numbers such that 30 < a < 40 and 50 < b < 70. Which of the following represents all possible values of a b? (A) 40 < a b < 20 (B) 40 < a b < 10 (C) 30 < a b < 20 (D) 20 < a b < 10 (E) 20 < a b < 30 (D) b 3 (E) b 4 erikthered.com/tutor pg. 3

4 SAT Math Hard Practice Quiz 3 + y 12 = z 8. In the equation shown above,, y, and z are positive integers. All of the following could be a possible value of y EXCEPT 11. Amy is two years older than Bill. The square of Amy s age in years is 36 greater than the square of Bill s age in years. What is the sum of Amy s age and Bill s age in years? (A) 4 (B) 6 (C) 8 (D) 12 (E) 20 y y = f() = m n 9. In the equation above, m and n are integers such that m > n. Which of the following is the value of m? 1 (A) 6 (B) 12 (C) 16 (D) 24 (E) The function f is graphed in its entirety above. If the function g is defined so that g() = f( ), then for what value of does g attain its maimum value? t N(t) (A) 3 (B) 2 (C) 0 (D) 2 (E) The table above shows some values for the function N. If N(t) = k 2 at for positive constants k and a, what is the value of a? (A) 3 (B) 2 (C) If ( + 1) 2 = 4 and ( 1) 2 = 16, what is the value of? (A) 3 (B) 1 (C) 1 (D) 3 (E) 5 (D) 2 (E) 3 erikthered.com/tutor pg. 4

5 SAT Math Hard Practice Quiz On the number line above, the tick marks correspond to consecutive integers. What is the value of? Two cars are racing at a constant speed around a circular racetrack. Car A requires 15 seconds to travel once around the racetrack, and car B requires 25 seconds to travel once around the racetrack. If car A passes car B, how many seconds will elapse before car A once again passes car B? 15. The value of y increased by 12 is directly proportional to the value of decreased by 6. If y = 2 when = 8, what is the value of when y = 16? (A) 8 (B) 10 (C) 16 (D) 20 (E) 28 erikthered.com/tutor pg. 5

6 SAT Math Hard Practice Quiz Geometry a c y y = 2 2 b Note: Figure not drawn to scale. (a,b) y = 2 3. In the figure above, 3 < a < 5 and 6 < b < 8. Which of the following represents all possible values of c? O (A) 0 < c < 3 (B) 1 < c < 3 (C) 0 < c < 13 (D) 1 < c < 13 (E) 3 < c < The curve y = 2 /2 and the line y = /2 intersect at the origin and at the point (a,b), as shown in the figure above. What is the value of b? (A) (B) (C) (D) 1 (E) 2 4. Line l goes through points P and Q, whose coordinates are (0,1) and (b,0), respectively. For which of the following values of b is the slope of line l greater than 1 2? (A) 1 2 (B) 1 B (C) (D) A C (E) In the figure above, AB = 6 and BC = 8. What is the area of triangle ABC? (A) 12 2 (B) 12 3 (C) 24 2 (D) 24 3 (E) 36 3 erikthered.com/tutor pg. 6

7 SAT Math Hard Practice Quiz C D B 5. In the figure above, AB = 6 and BC = 8. What is the length of segment BD? (A) 2 (B) 12 5 (C) 4 (D) 24 5 A 8. In the figure above, a square is inscribed in a circle. If the area of the square is 36, what is the perimeter of the shaded region? (A) π (B) 6+3π (C) 6+3 2π (D) π (E) 6 (E) 9 2 π 9 6. If four distinct lines lie in a plane, and eactly two of them are parallel, what is the least possible number of points of intersection of the lines? B (A) Two (B) Three (C) Four (D) Five (E) More than five A 7 C Note: Figure not drawn to scale. 7. The perimeter of a particular equilateral triangle is numerically equal to the area of the triangle. What is the perimeter of the triangle? 9. In the figure above, AC = 7 and AB = BC. What is the smallest possible integer value of AB? (A) 3 (B) 4 (C) 4 3 (D) 12 3 (E) 18 3 erikthered.com/tutor pg. 7

8 y (2,a) SAT Math Hard Practice Quiz y O (10, 0) y = +c P 10. In the figure above, two line segments in the -y plane form a right triangle with the -ais. What is the value of a? y = 2 1 (A) 2 2 (B) 4 (C) 5 (D) 4 2 (E) 5 2 O Note: Figure not drawn to scale. 11. The perimeter of square ABCD is, and the perimeter ofisoscelestriangleefgisy. IfAB = EF = FG, which of the following must be true? 12. In the -y plane, the lines y = 2 1 and y = +c intersect at point P, where c is a positive number. Portions of these lines are shown in the figure above. If the value of c is between 1 and 2, what is one possible value of the -coordinate of P? (A) 0 < y < 4 (B) (C) 4 < y < 2 2 < y < (D) < y < 2 (E) 2 < y < 4 erikthered.com/tutor pg. 8

9 SAT Math Hard Practice Quiz Data, Statistics, and Probability 1. The first term of a sequence is the number n, and each term thereafter is 5 greater than the term before. Which of the following is the average (arithmetic mean) of the first nine terms of this sequence? (A) n+20 (B) n+180 (C) 2n (D) 2n+40 (E) 9n Let a, b, and c be positive integers. If the average (arithmetic mean) of a, b, and c is 100, which of the following is NOT a possible value of any of the integers? (A) 1 (B) 100 (C) 297 (D) 298 (E) M is a set consisting of a finite number of consecutive integers. If the median of the numbers in set M is equal to one of the numbers in set M, which of the following must be true? I. The average (arithmetic mean) of the numbers in set M equals the median. II. The number of numbers in set M is odd. 2. The average (arithmetic mean) of a particular set of seven numbers is 12. When one of the numbers is replaced by the number 6, the average of the set increases to 15. What is the number that was replaced? (A) 20 (B) 15 (C) 12 (D) 0 (E) 12 III. The sum of the smallest number and the largest number in set M is even. (A) I only (B) II only (C) I and II only (D) I and III only (E) I, II, and III erikthered.com/tutor pg. 9

10 SAT Math Hard Practice Quiz Answers Numbers and Operations 1. B (Estimated Difficulty Level: 5) The number of green and red tomatoes are 4n and 3n, respectively, for some integer n. In this way, we can be sure that the green-to-red ratio is 4n/3n = 4/3. We need to solve the equation: 4n 5 3n 5 = 3 2. Cross-multiplying, 8n 10 = 9n 15 so that n = 5. There were 3n, or 15, red tomatoes in the bag. Working with the answers may be easier. If answer A is correct, then there were 16 green tomatoes and 12 red tomatoes, in order to have the 4 to 3 ratio. But removing five of each gives 11 green and 7 red, which is not in the ratio of 3 to 2. If answer B is correct, then there were 20 green tomatoes and 15 red tomatoes, since 20/15 = 4/3. Removing five of each gives 15 green and 10 red, and 15/10 = 3/2, so answer B is correct. 4. D (Estimated Difficulty Level: 4) The answer must be true for any value of p, so plug in an easy (prime) number for p, such as 2. The factors of 2 3 = 8 are 1, 2, 4, and 8, so answer D is correct. In general, since p is prime, the only numbers that go into p 3 without a remainder are 1, p, p 2, and p (Estimated Difficulty Level: 4) For the two-digit numbers, only 33 begins and ends in 3. For three-digit numbers, the only possibilities are: 303, 313,..., 383, and 393. We found ten three-digit numbers, and one two-digit number, for a total of 11 numbers that begin and end in 3. Yes, this was a counting problem soon after another counting problem. But this one wasn t so bad, was it? 6. C (Estimated Difficulty Level: 4) 2. C (Estimated Difficulty Level: 4) From 10 to 19, 12 and up (eight numbers) are monotonic. Among the numbers from 20 to 29, seven (23 and up) are monotonic. If you can see a pattern in counting problems like this, you can save a lot of time. Here, the 30s will have 6 monotonic numbers, the 40s will have 5, and so forth. You should find = 36 total monotonic numbers (Estimated Difficulty Level: 5) Since the second term is 7 greater than the first term, (2a 1) a = 7 so that a = 8. The sequence is 8, 15, 22,... You can either continue to write out the sequence until the 16 th term, or realize that the 16 th term is 16a 15 = 16(8) 15 = = 113. This type of SAT math question contains three separate mini-problems. (This kind of question is also known as one of those annoying, long, SAT math questions with roman numerals ). Let s do each mini-problem in order. First, recall that a prime number is only divisible by itself and 1, and that 1 is not a prime number. So, statement I must be true, since a number that can be divided by two prime numbers can t itself be prime. Net, recallthateverynumbercanbewrittenasaproduct of a particular bunch of prime numbers. Let s say that N is divisible by 3 and 5. Then, N is equal to 3 5 p 1 p 2, where p 1, p 2, etc. are some other primes. So, N is divisible by 3 5 = 15. Statement II must be true. Finally, remember that 2 is a prime number. So, N could be 6, since 6 = 2 3. Statement III isn t always true, making C the correct answer. erikthered.com/tutor pg. 10

11 SAT Math Hard Practice Quiz Answers (Estimated Difficulty Level: 5) First, note that 15mn = 3 5 mn. We need 3 5 mn to be an integer. We could have, for eample, m = 3 and n = 5 since = = 15, ecept thatthe problemrequiresthat mn > 15. (Thisisahard problem for a reason, after all!) If m = 3 2 and n = 5 2 then 15mn = (3 5)(3 2)(5 2) = That way, 15mn = = 30 is still an integer, making the least possible value of mn equal to 6 10 = (Estimated Difficulty Level: 5) To get the greatest difference, we want to subtract a small number from a large one, so we will need the digit 9 and the digit 1, in order to make a number in the 100 s and a number in the 900 s. The large number will look like 9N1 and the small number will look like 1N9, where N is a digit from 2 to 8. You will find that, no matter what you make N, the difference is B (Estimated Difficulty Level: 4) A good opportunity to plug in real numbers! For eample, suppose set M consists of the integers: 2, 4, 6, 8, 10, and 12. The sum of the least three is 12 and the sum of the greatest three is 30, so answer B is correct. Yousayyouwantanalgebraicsolution? Supposethatn is the first even integer. The remaining integers are then n+2, n+4, n+6, n+8, and n+10. The sum ofthe least threeoftheseintegersis = n+(n+2)+(n+4)= 3n+6, and the sum of the greatest three of these integers is y = (n+6)+(n+8)+(n+10)= 3n+24. So, y = 18, or y = E (Estimated Difficulty Level: 5) Suppose that the integer is n. The result of subtracting n from its square is n 2 n = n(n 1), which is the product of two consecutive integers, so answer E is correct. Notice that if you multiply any two consecutive integers, the result is always even, since it is the product of an even integer and an odd integer. To win an Erik The Red Viking Hat, see if you candetermine why the result is never a negative integer. erikthered.com/tutor pg. 11

13 SAT Math Hard Practice Quiz Answers 8. B (Estimated Difficulty Level: 5) Solve the equation for y. You should get: y = 12z 4. Factor out a 4 from the right-hand side: y = 4(3z ). Since z and are integers, 3z is an integer, so that y is a multiple of 4. Only answer B is not a multiple of 4, so it can t be a possible value of y. You could also combine the fractions on the left-hand side of the equation to get: 4+y 12 = z. For z to be an integer, 4 + y must be a multiple of 12. Try plugging in various integers for and y to get multiples of 12; you should find that y can take on all of the values in the answers ecept for (Estimated Difficulty Level: 4) Let a be Amy s age and b be Bill s age. The problem tells us that: a = b+2 and a 2 = b One way to do this is to plow ahead, substitute for one variable and solve for the other (a bit messy!). But SAT questions are designed to be solved without tedious calculations and/or messy algebra. Let s try doing the problem using the SAT way, not the math teacher way. First, notice that the second equation can be written as: a 2 b 2 = 36. This is a difference of two squares, and is the same as: (a+b)(a b) = 36. The first equation can be written as: a b = 2. This means that the second equation is just (a+b) 2 = 36, so that a+b = 18. We don t know what a and b are, and we don t even care! 12. B (Estimated Difficulty Level: 4) 9. B (Estimated Difficulty Level: 4) Combining the two terms on the left-hand side of the equation gives us 2 72, but that doesn t give us what we need, which is m > n. For the SAT test, you should know how to rewrite and simplify radicals. Here, 72 = 36 2 = 6 2, so the left-hand side is equal to 12 2, making m = 12 and n = 2. You can obtain the graph of y = f( ) by flipping the graph of y = f() across the y-ais. For eample, if the point (3,1) is on the graph of f(), then the point ( 3,1) must be on the graph of f( ), since f( ( 3)) = f(3) = 1. The figure below tells the story. Here, the function g() = f( ) is shown as a dashed line: y g() f() E (Estimated Difficulty Level: 5) Hint: if you see a zero in a table problem like this one, try to use it first! When you plug in 0 for t, you get N(0) = k 2 0 = k 1 = k, which means that k = Net, try plugging in 1 for t: N(1) = a. From the table, N(1) = 16, so that a = 16, or 2 a = 1/2 a = 1/8. Since 8 = 2 3, a = 3. (Your calculator may also help here, but try to understand how to do it without it.) From the graph, g() is maimum when = 2. erikthered.com/tutor pg. 13

14 SAT Math Hard Practice Quiz Answers 13. A (Estimated Difficulty Level: 4) A good skip-the-algebra way to do this problem is to use the answers by plugging them into until the two given equations work. Using answer A, you should find that( 3+1) 2 = ( 2) 2 = 4and( 3 1) 2 = ( 4) 2 = 16, so answer A is correct. You must have the algebraic solution, you say? Try taking the square root of both sides of the equations, but don t forget that there are two possible solutions when you do this. The first equation gives: +1 = ±2 so that = 1 or = 3. The second equation gives 1 = ±4 so that = 5 or = 3. The only solution that works for both equations is = (Estimated Difficulty Level: 5) Since the tick marks correspond to consecutive integers, and it takes four steps to go from /12 to /8, we know that /8 is four greater than /12. (Or, think of the spaces between the tick marks: there are four spaces and each space is length 1, so the distance from /12 to /8 is 4.) In equation form: /2 or 37.5 (Estimated Difficulty Level: 5) Tomakethisproblemmoreconcrete,makeupanumber for the circumference of the racetrack. It doesn t really matter what number you use; I ll use 75 feet. Since speed is distance divided by time, the speed of car A is 75/15 = 5 feet per second, and the speed of car B is 75/25 = 3 feet per second. (I picked 75 mostly because it is divided evenly by 15 and 25.) Every second, car A gains 2 feet on car B. To pass car B, car A must gain 75 feet on car B. This will require 75/2 = 37.5 seconds. You may be thinking, Whoa, tricky solution! Here is the mostly straightforward but somewhat tedious algebraic solution. Once again, I ll use 75 feet for the circumference of the track. Suppose that you count time from when car A first passes car B. Then, car A travels a distance (75/15)t = 5t feet after t seconds. (Remember that distance = speed time.) For eample, after 15 seconds, car A has traveled a distance 5 15 = 75 feet, and after 30 seconds, car A has traveled a distance 5 30 = 150 feet. Similarly, car B travels a distance (75/25)t = 3t feet after t seconds. When the two cars pass again, car A has traveled 75 feet more than car B: 5t = 3t+75. Solving for t gives: 2t = 75, or t = 75/2 = 37.5 seconds. 8 = Multiplying both sides by 24 gives: 3 = so that = B (Estimated Difficulty Level: 5) First, recall that if y is proportional to, then y = k for some constant k. So, y increased by 12 is directly proportional to decreased by 6 translates into the math equation: y + 12 = k( 6). Plugging in y = 2 and = 8 gives 14 = k 2 so that k = 7. Our equation is now: y + 12 = 7( 6). Plugging in 16 for y gives 28 = 7( 6) so that 6 = 4, or = 10. erikthered.com/tutor pg. 14

15 SAT Math Hard Practice Quiz Answers Geometry 1. C (Estimated Difficulty Level: 4) To determine where two curves intersect, set the equations equal to one another and solve for. (Hint: know this for the SAT!) In this question, we need to figure out which values of satisfy: 2 /2 = /2. If = 0, this equation works, but we need the solution when 0. Dividing both sides of the equation by gives: /2 = 1/2 so that = 1. By plugging in = 1 to either of the two curves, you should find that y = 1/2. So, the point of intersection is (1,1/2), making answer C the correct one. 2. B (Estimated Difficulty Level: 5) This is the kind of problem that would be too hard and/or require things you aren t epected to know for the SAT (such as trigonometry), unless youdraw a constructionlineinthefigure. (Thisisaveryhardquestion anyway.) In this case, you want to draw a line from A perpendicular to the opposite side of the triangle: A B This forms a triangle whose hypotenuse has length 6. Now, use the triangle diagram given to you at the beginning of each SAT math section: The length of the side opposite the 30 angle is 3, and the length of the side opposite the 60 angle (the dashed line) is 3 3. So finally, if the base of the triangle is segment BC, then the dashed line is the height of the triangle, and the area of the triangle is (1/2) = C 3. D (Estimated Difficulty Level: 5) You need to know the third-side rule for triangles to solve this question: The length of the third side of a triangle is less than the sum of the lengths of the other two sides and greater than the positive difference of the lengths of the other two sides. Applied to this question, the first part of the rule says that the value of c must be less than a +b. Since we are interested in all possible values of c, we need to know the greatest possible value of a + b. With a < 5 and b < 8, a + b < 13 so that c must be less than 13. For the second part of the rule, c must be greater than b a. (Note that b is always bigger than a, so that b a is positive.) We are interested in all possible values of c, so we need to know the least possible value of b a. The least valueoccurswhen b is assmallas possibleand a is as large as possible: b a > 6 5 = 1. Then, c must be greater than 1. Putting this together, 1 < c < 13, making answer D the correct one. 4. E (Estimated Difficulty Level: 4) First, calculate the slope ofline l using the given points: slope = rise run = 0 1 b 0 = 1 b. At this point, a good approach is to work with the answers by plugging them into the epression for slope above until you get a value greater than 1/2. For eample, using answer A gives a slope of 1/(1/2) = 2, which is not greater than 1/2, so answer A is incorrect. You should find that answer E is the correct one, since 1/(5/2) = 2/5 is greater than 1/2. (Knowing the decimal equivalents of basic fractions will really help speed this process up.) Here is the algebraic solution: 1 b > b < 1 2 b > 2. (Remember to flip the inequality when multiplying by negative numbers or when taking the reciprocal of both sides.) Only answer E makes b > 2. erikthered.com/tutor pg. 15

16 SAT Math Hard Practice Quiz Answers 5. D (Estimated Difficulty Level: 5) Since ABC isarighttriangle, the length ofsegmentac is = 10. (Hint: you will see and triangles a lot on the SAT.) The area of triangle ABC is (1/2) b h, where b is the base and h is the height of triangle ABC. The key thing to remember for this problem is that the base can be any of the three sides of a triangle, not just the side at the bottom of the diagram. If the base is AB, then the height is BC and the area of the triangle is (1/2)(6)(8) = 24. If the baseis AC, then the height is BD and the area of the triangle is still 24. This means that (1/2)(AC)(BD) = (1/2)(10)(BD) = 24 so that BD = 24/5. 7. D (Estimated Difficulty Level: 5) One formula that a good math student such as yourself may want to memorize for the SAT is the area of an equilateral triangle. If the length of each side of the triangle is s, then the area is 3s 2 /4. The perimeter of this triangle is 3s. Now, since the perimeter equals the area for this triangle, we have: 3s = 3s 2 /4 so that 3 = 3s/4 and s = 12/ 3 = 4 3. The perimeter is then 12 3, making answer D the correct one. (Did you get s = 4 3 and then choose answer C? Sorry about that.) 8. A (Estimated Difficulty Level: 5) 6. B (Estimated Difficulty Level: 4) Draw a diagram for this problem! With eactly two parallel lines, the other two lines cannot be parallel to themselves or to the first two lines. Your diagram may seem to suggest five points of intersection; however, the point of intersection of the two non-parallel lines can overlapwithapointofintersectionononeoftheparallel lines: For many difficult SAT questions, it can be very helpful to know some etra math along with the required math. First, when a square is inscribed in a circle, the diagonals are diameters of the circle. Second, the diagonals of a square meet at right angles. Third, a diagonal of a square is 2 times as long as the length of one of the sides. (A diagonal of a square makes a triangle with two sides.) For this question, the length of each side of the square is 6 (since the area is 6 2 = 36), and the length of a diagonal is 6 2, so the radius of the circle is 3 2, as shown below: From the figure, the least possible number of intersection points is then three. Afinalpiece ofneeded math: the arclength ofaportion of a circleis the circumferencetimes the centralangle of the arc divided by 360. Here, the central angle is 90, so the needed arc length (shown darkened in the figure above) is just 1/4 times the circle s circumference. The arc length is then 2πr/4 = 2π 3 2/4 = 3π 2/2 and the perimeter of the shaded region is 6+3π 2/2. erikthered.com/tutor pg. 16

17 SAT Math Hard Practice Quiz Answers 9. 4 (Estimated Difficulty Level: 5) You need to know half of the third-side rule for triangles to solve this question: The length of the third side of a triangle is less than the sum of the lengths of the other two sides. For this question, we will make AC the third side. 11. C (Estimated Difficulty Level: 5) Make a diagram, and fill it in with the information that is given. (You should do this for any difficult geometry question without a figure.) Since the perimeter of square ABCD is, each side of the square has length /4, so your figure should look something like this: Now, suppose that the length of each of the other two sides of the triangle is, so that AB = BC =. Then, the third-side rule says that AC is less than the sum of AB and BC: 7 < +. Simplifying gives: 2 > 7 so that > 3.5. The smallest possible integer value for is 4. B 4 A 4 4 C 4 D 4 E F 4 G 10. B (Estimated Difficulty Level: 5) One way to do this question is to use the fact that the product of the slopes of two perpendicular lines (or line segments) is 1. The slope of the line segment on the left is (a 0)/(2 0) = a/2. The slope of the line segment on the right is (0 a)/(10 2) = a/8. The two slopes multiply to give 1: a 2 a 8 = a2 16 = 1. Solving for a gives a 2 = 16 so that a = 4. A messier way to do this problem is to use the distance formula and the Pythagorean theorem. The length of the line segment on the left is 2 2 +a 2, and the length of line segment on the right is (10 2) 2 +(0 a) 2. Then, the Pythagorean theorem says that: ( 22 +a 2) 2 + ( (10 2)2 +(0 a) 2) 2 = Simplifying the left-hand side gives: 2a = 100 so that 2a 2 = 32. Then, a 2 = 16, making a = 4. Now, use the third-side rule for triangles: The length of the third side of a triangle is less than the sum of the lengths of the other two sides and greater than the positive difference of the lengths of the other two sides. When the rule is applied to EG as the third side, we get: 0 < EG < /2. Ify isthe perimeterofthe triangle, then y = /4+/4+EG = /2+EG. Solving for EG gives EG = y /2. Substituting into the inequality gives 0 < y /2 < /2 so that /2 < y <, making answer C the correct one. To make this problem less abstract, it may help to make up a number for the perimeter of the square. (A good choice might be 4 so that = 1. You ll find 1/2 < y < 1, the same as answer C when = 1.) < < 3 (Estimated Difficulty Level: 5) In order to determine at what point two lines intersect, set the equations of the lines equal to one another. In this case, we have: 2 1 = + c so that = c + 1. In other words, = c+1 is the -coordinate of P, the point where the lines intersect. Now, if c is between 1 and 2, then c+1 is between 2 and 3. Any value for the -coordinate of P between 2 and 3 is correct. erikthered.com/tutor pg. 17

18 SAT Math Hard Practice Quiz Answers Data, Statistics, and Probability 1. A (Estimated Difficulty Level: 4) The first nine terms of the sequence are: n, n+5, n+10, n+15,..., n+40. (You should probably write all nine terms out to avoid mistakes.) Adding these terms up gives: 9n+180. The average is the sum (9n+180) divided by the number of terms (9). The average is then: (9n+180)/9 = n E (Estimated Difficulty Level: 4) Using the definition of average gives: a+b+c 3 = 100 so that a+b+c = 300. Since a, b, and c are all positive, the smallest possible value for any of the numbers is 1. The largest possible value of one of the three numbers then occurs when the other two numbers are both 1. In this case, the numbers are 1, 1, and 298, so that the largest possible value is 298. Answer E can not be a possible value, so it is the correct answer. 2. B (Estimated Difficulty Level: 5) The average of a set of numbers is the sum of the numbers divided by the number of numbers: average= sum N. We can solve this equation for the sum: sum = average N. Here, since there are 7 numbers and the average is 12, the sum of the numbers is 7 12 = 84. The sum of the new set of numbers is 7 15 = 105. Now, suppose that the seven numbers are a, b, c, d, e, f, and g, and that g gets replaced with the number 6. Then, we have: and a+b+c+d+e+f +g = 84, a+b+c+d+e+f +6 = 105. The second equation says that a+b+c+d+e+f = 99. Substituting into the first equation gives 99+g = 84 so that g = E (Estimated Difficulty Level: 5) Plug in real numbers for set M to make this problem concrete. For eample, if M is the set of consecutive integers from 1 to 5, then the median and average are both 3. If M is the set of consecutive integers from 1 to 4, then the median and average are both 2.5. From these eamples, we can see that the number of numbers in set M needs to odd, otherwise the median is not an integer. Choice II must be true. Also, if the number of numbers in a set of consecutive integers is odd, then when the first number is odd, the last number is odd. Or, when the first number is even, the last number is even. This is because the difference of the largest number and the smallest number will be even when the number of numbers is odd. Choice III must then be true, since the sum of two odd numbers or two even numbers is an even number. At this point, the only answer with choices II and III is answer E, so that must be the correct answer. Why is choiceialsocorrect? Theaverageofasetofconsecutive integers is equal to the average of the first and the last integers in the set. The average of two integers that are both odd or both even is the integer halfway between the two, which is also the median of the set. Whew! erikthered.com/tutor pg. 18

### SAT Math Medium Practice Quiz (A) 3 (B) 4 (C) 5 (D) 6 (E) 7

SAT Math Medium Practice Quiz Numbers and Operations 1. How many positive integers less than 100 are divisible by 3, 5, and 7? (A) None (B) One (C) Two (D) Three (E) Four 2. Twice an integer is added to

### SAT Math Facts & Formulas Review Quiz

Test your knowledge of SAT math facts, formulas, and vocabulary with the following quiz. Some questions are more challenging, just like a few of the questions that you ll encounter on the SAT; these questions

### SAT Math Strategies Quiz

When you are stumped on an SAT or ACT math question, there are two very useful strategies that may help you to get the correct answer: 1) work with the answers; and 2) plug in real numbers. This review

### PowerScore Test Preparation (800) 545-1750

Question 1 Test 1, Second QR Section (version 1) List A: 0, 5,, 15, 20... QA: Standard deviation of list A QB: Standard deviation of list B Statistics: Standard Deviation Answer: The two quantities are

### If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C?

Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question

### Mathematics & the PSAT

Mathematics & the PSAT Mathematics 2 Sections 25 minutes each Section 2 20 multiple-choice questions Section 4 8 multiple-choice PLUS 10 grid-in questions Calculators are permitted - BUT You must think

### SUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills

SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)

### Skill Builders. (Extra Practice) Volume I

Skill Builders (Etra Practice) Volume I 1. Factoring Out Monomial Terms. Laws of Eponents 3. Function Notation 4. Properties of Lines 5. Multiplying Binomials 6. Special Triangles 7. Simplifying and Combining

### A Quick Algebra Review

1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals

Advanced GMAT Math Questions Version Quantitative Fractions and Ratios 1. The current ratio of boys to girls at a certain school is to 5. If 1 additional boys were added to the school, the new ratio of

### ACT Math Vocabulary. Altitude The height of a triangle that makes a 90-degree angle with the base of the triangle. Altitude

ACT Math Vocabular Acute When referring to an angle acute means less than 90 degrees. When referring to a triangle, acute means that all angles are less than 90 degrees. For eample: Altitude The height

### Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

### Section P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities

Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.

### EXPONENTS. To the applicant: KEY WORDS AND CONVERTING WORDS TO EQUATIONS

To the applicant: The following information will help you review math that is included in the Paraprofessional written examination for the Conejo Valley Unified School District. The Education Code requires

### Review of Intermediate Algebra Content

Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6

### How do you compare numbers? On a number line, larger numbers are to the right and smaller numbers are to the left.

The verbal answers to all of the following questions should be memorized before completion of pre-algebra. Answers that are not memorized will hinder your ability to succeed in algebra 1. Number Basics

### Pre-AP Geometry Summer Assignment (Required Foundational Concepts)

Name: Pre-AP Geometry Summer Assignment (Required Foundational Concepts) Pre-AP Geometry is a rigorous critical thinking course. Our epectation is that each student is fully prepared. Therefore, the following

### Objectives. By the time the student is finished with this section of the workbook, he/she should be able

QUADRATIC FUNCTIONS Completing the Square..95 The Quadratic Formula....99 The Discriminant... 0 Equations in Quadratic Form.. 04 The Standard Form of a Parabola...06 Working with the Standard Form of a

### TSI College Level Math Practice Test

TSI College Level Math Practice Test Tutorial Services Mission del Paso Campus. Factor the Following Polynomials 4 a. 6 8 b. c. 7 d. ab + a + b + 6 e. 9 f. 6 9. Perform the indicated operation a. ( +7y)

### Geometry Unit 7 (Textbook Chapter 9) Solving a right triangle: Find all missing sides and all missing angles

Geometry Unit 7 (Textbook Chapter 9) Name Objective 1: Right Triangles and Pythagorean Theorem In many geometry problems, it is necessary to find a missing side or a missing angle of a right triangle.

### Factoring Polynomials

UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can

### Answer: Quantity A is greater. Quantity A: 0.717 0.717717... Quantity B: 0.71 0.717171...

Test : First QR Section Question 1 Test, First QR Section In a decimal number, a bar over one or more consecutive digits... QA: 0.717 QB: 0.71 Arithmetic: Decimals 1. Consider the two quantities: Answer:

### Answers to Basic Algebra Review

Answers to Basic Algebra Review 1. -1.1 Follow the sign rules when adding and subtracting: If the numbers have the same sign, add them together and keep the sign. If the numbers have different signs, subtract

### In a triangle with a right angle, there are 2 legs and the hypotenuse of a triangle.

PROBLEM STATEMENT In a triangle with a right angle, there are legs and the hypotenuse of a triangle. The hypotenuse of a triangle is the side of a right triangle that is opposite the 90 angle. The legs

### The majority of college students hold credit cards. According to the Nellie May

CHAPTER 6 Factoring Polynomials 6.1 The Greatest Common Factor and Factoring by Grouping 6. Factoring Trinomials of the Form b c 6.3 Factoring Trinomials of the Form a b c and Perfect Square Trinomials

### SAT Math Easy Practice Quiz (A) 6 (B) 12 (C) 18 (D) 24 (E) 36 (A) 1 (B) 2 (C) 4 (D) 6 (E) 8. (A) a (B) b (C) c (D) d (E) e

Numbers and Operations 1. Aubrey can run at a pace of 6 miles per hour. Running at the same rate, how many miles can she run in 90 minutes? (A) 4 (B) 6 (C) 8 (D) 9 (E) 12 2. Which of the following is a

### Core Maths C2. Revision Notes

Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...

### Core Maths C1. Revision Notes

Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the

### MATHCOUNTS TOOLBOX Facts, Formulas and Tricks

MATHCOUNTS TOOLBOX Facts, Formulas and Tricks MATHCOUNTS Coaching Kit 40 I. PRIME NUMBERS from 1 through 100 (1 is not prime!) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 II.

### Park Forest Math Team. Meet #5. Algebra. Self-study Packet

Park Forest Math Team Meet #5 Self-study Packet Problem Categories for this Meet: 1. Mystery: Problem solving 2. Geometry: Angle measures in plane figures including supplements and complements 3. Number

### CSU Fresno Problem Solving Session. Geometry, 17 March 2012

CSU Fresno Problem Solving Session Problem Solving Sessions website: http://zimmer.csufresno.edu/ mnogin/mfd-prep.html Math Field Day date: Saturday, April 21, 2012 Math Field Day website: http://www.csufresno.edu/math/news

### The GED math test gives you a page of math formulas that

Math Smart 643 The GED Math Formulas The GED math test gives you a page of math formulas that you can use on the test, but just seeing the formulas doesn t do you any good. The important thing is understanding

### Mathematics 31 Pre-calculus and Limits

Mathematics 31 Pre-calculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals

### 2015 Chapter Competition Solutions

05 Chapter Competition Solutions Are you wondering how we could have possibly thought that a Mathlete would be able to answer a particular Sprint Round problem without a calculator? Are you wondering how

### MATH 21. College Algebra 1 Lecture Notes

MATH 21 College Algebra 1 Lecture Notes MATH 21 3.6 Factoring Review College Algebra 1 Factoring and Foiling 1. (a + b) 2 = a 2 + 2ab + b 2. 2. (a b) 2 = a 2 2ab + b 2. 3. (a + b)(a b) = a 2 b 2. 4. (a

### What are the place values to the left of the decimal point and their associated powers of ten?

The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything

4th Grade Competition Solutions Bergen County Academies Math Competition 19 October 008 1. Before taking the AMC, a student notices that he has two bags of Doritos and one bag of Skittles on his desk.

### 2010 Solutions. a + b. a + b 1. (a + b)2 + (b a) 2. (b2 + a 2 ) 2 (a 2 b 2 ) 2

00 Problem If a and b are nonzero real numbers such that a b, compute the value of the expression ( ) ( b a + a a + b b b a + b a ) ( + ) a b b a + b a +. b a a b Answer: 8. Solution: Let s simplify the

### ALGEBRA READINESS DIAGNOSTIC TEST PRACTICE

1 ALGEBRA READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the examples, work the problems, then check your answers at the end of each topic. If you don t get the answer given, check your work and

### Expression. Variable Equation Polynomial Monomial Add. Area. Volume Surface Space Length Width. Probability. Chance Random Likely Possibility Odds

Isosceles Triangle Congruent Leg Side Expression Equation Polynomial Monomial Radical Square Root Check Times Itself Function Relation One Domain Range Area Volume Surface Space Length Width Quantitative

### Solutions Manual for How to Read and Do Proofs

Solutions Manual for How to Read and Do Proofs An Introduction to Mathematical Thought Processes Sixth Edition Daniel Solow Department of Operations Weatherhead School of Management Case Western Reserve

### Introduction to the Practice Exams

Introduction to the Practice Eams The math placement eam determines what math course you will start with at North Hennepin Community College. The placement eam starts with a 1 question elementary algebra

### SAT Subject Math Level 2 Facts & Formulas

Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Arithmetic Sequences: PEMDAS (Parentheses

### Math Content

2012-2013 Math Content PATHWAY TO ALGEBRA I Unit Lesson Section Number and Operations in Base Ten Place Value with Whole Numbers Place Value and Rounding Addition and Subtraction Concepts Regrouping Concepts

### Section 5.0A Factoring Part 1

Section 5.0A Factoring Part 1 I. Work Together A. Multiply the following binomials into trinomials. (Write the final result in descending order, i.e., a + b + c ). ( 7)( + 5) ( + 7)( + ) ( + 7)( + 5) (

### Solutions to Exercises, Section 5.1

Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle

### Grade 7/8 Math Circles Greek Constructions - Solutions October 6/7, 2015

Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Greek Constructions - Solutions October 6/7, 2015 Mathematics Without Numbers The

### PRIMARY CONTENT MODULE Algebra I -Linear Equations & Inequalities T-71. Applications. F = mc + b.

PRIMARY CONTENT MODULE Algebra I -Linear Equations & Inequalities T-71 Applications The formula y = mx + b sometimes appears with different symbols. For example, instead of x, we could use the letter C.

### Polynomial Degree and Finite Differences

CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial

### Prep for College Algebra

Prep for College Algebra This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet

### MAT 0950 Course Objectives

MAT 0950 Course Objectives 5/15/20134/27/2009 A student should be able to R1. Do long division. R2. Divide by multiples of 10. R3. Use multiplication to check quotients. 1. Identify whole numbers. 2. Identify

### 2. Simplify. College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses

College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key 1. Multiply 2 3 5 1 Use the distributive property to remove the parentheses 2 3 5 1 2 25 21 3 35 31 2 10 2 3 15 3 2 13 2 15 3 2

### Fifth Grade. Scope & Sequence of Lessons. by lesson number

Scope & Sequence of Lessons by lesson number PLACE VALUE AND COUNTING Place value 1 Recognizing numbers less than a million 65 Recognizing tenths and hundredths places 80 Recognizing numbers up through

### Higher Education Math Placement

Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication

### Quick Reference ebook

This file is distributed FREE OF CHARGE by the publisher Quick Reference Handbooks and the author. Quick Reference ebook Click on Contents or Index in the left panel to locate a topic. The math facts listed

### Looking for Pythagoras: Homework Examples from ACE

Looking for Pythagoras: Homework Examples from ACE Investigation 1: Coordinate Grids, ACE #20, #37 Investigation 2: Squaring Off, ACE #16, #44, #65 Investigation 3: The Pythagorean Theorem, ACE #2, #9,

### 1. By how much does 1 3 of 5 2 exceed 1 2 of 1 3? 2. What fraction of the area of a circle of radius 5 lies between radius 3 and radius 4? 3.

1 By how much does 1 3 of 5 exceed 1 of 1 3? What fraction of the area of a circle of radius 5 lies between radius 3 and radius 4? 3 A ticket fee was \$10, but then it was reduced The number of customers

### SAT Math Easy Practice Quiz #2 (A) 8 (B) 10 (C) 12 (D) 14 (E) 16

SAT Math Easy Practice Quiz #2 Numbers and Operations 1. A meter is a measure of length, and 10 decimeters is equal in length to one meter. How many decimeters are equal in length to 12.5 meters? (A) 1250

### Algebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123

Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from

### Answer: The relationship cannot be determined.

Question 1 Test 2, Second QR Section (version 3) In City X, the range of the daily low temperatures during... QA: The range of the daily low temperatures in City X... QB: 30 Fahrenheit Arithmetic: Ranges

### Whole Numbers and Integers (44 topics, no due date)

Course Name: PreAlgebra into Algebra Summer Hwk Course Code: GHMKU-KPMR9 ALEKS Course: Pre-Algebra Instructor: Ms. Rhame Course Dates: Begin: 05/30/2015 End: 12/31/2015 Course Content: 302 topics Whole

### North Carolina Community College System Diagnostic and Placement Test Sample Questions

North Carolina Community College System Diagnostic and Placement Test Sample Questions 01 The College Board. College Board, ACCUPLACER, WritePlacer and the acorn logo are registered trademarks of the College

### Equations and Inequalities

Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.

### Algebra Geometry Glossary. 90 angle

lgebra Geometry Glossary 1) acute angle an angle less than 90 acute angle 90 angle 2) acute triangle a triangle where all angles are less than 90 3) adjacent angles angles that share a common leg Example:

### Pilot Flyskole AS Hangarveien 13 N-3241 Sandefjord Tlf Epost Preparatory Course.

Pilot Flyskole AS Hangarveien 13 N-3241 Sandefjord Tlf +47 9705 6840 Epost post@pilot.no www.pilot.no Preparatory Course Mathematics Pilot Flight School 2014 Order of operations Operations means things

### 4. An isosceles triangle has two sides of length 10 and one of length 12. What is its area?

1 1 2 + 1 3 + 1 5 = 2 The sum of three numbers is 17 The first is 2 times the second The third is 5 more than the second What is the value of the largest of the three numbers? 3 A chemist has 100 cc of

### Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

### Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson

### Grade 6 Math Circles March 24/25, 2015 Pythagorean Theorem Solutions

Faculty of Mathematics Waterloo, Ontario NL 3G1 Centre for Education in Mathematics and Computing Grade 6 Math Circles March 4/5, 015 Pythagorean Theorem Solutions Triangles: They re Alright When They

### 10-4 Inscribed Angles. Find each measure. 1.

Find each measure. 1. 3. 2. intercepted arc. 30 Here, is a semi-circle. So, intercepted arc. So, 66 4. SCIENCE The diagram shows how light bends in a raindrop to make the colors of the rainbow. If, what

### 1) (-3) + (-6) = 2) (2) + (-5) = 3) (-7) + (-1) = 4) (-3) - (-6) = 5) (+2) - (+5) = 6) (-7) - (-4) = 7) (5)(-4) = 8) (-3)(-6) = 9) (-1)(2) =

Extra Practice for Lesson Add or subtract. ) (-3) + (-6) = 2) (2) + (-5) = 3) (-7) + (-) = 4) (-3) - (-6) = 5) (+2) - (+5) = 6) (-7) - (-4) = Multiply. 7) (5)(-4) = 8) (-3)(-6) = 9) (-)(2) = Division is

### If a question asks you to find all or list all and you think there are none, write None.

If a question asks you to find all or list all and you think there are none, write None 1 Simplify 1/( 1 3 1 4 ) 2 The price of an item increases by 10% and then by another 10% What is the overall price

### Welcome to Math 19500 Video Lessons. Stanley Ocken. Department of Mathematics The City College of New York Fall 2013

Welcome to Math 19500 Video Lessons Prof. Department of Mathematics The City College of New York Fall 2013 An important feature of the following Beamer slide presentations is that you, the reader, move

### Algebra I Pacing Guide Days Units Notes 9 Chapter 1 ( , )

Algebra I Pacing Guide Days Units Notes 9 Chapter 1 (1.1-1.4, 1.6-1.7) Expressions, Equations and Functions Differentiate between and write expressions, equations and inequalities as well as applying order

### MATH 65 NOTEBOOK CERTIFICATIONS

MATH 65 NOTEBOOK CERTIFICATIONS Review Material from Math 60 2.5 4.3 4.4a Chapter #8: Systems of Linear Equations 8.1 8.2 8.3 Chapter #5: Exponents and Polynomials 5.1 5.2a 5.2b 5.3 5.4 5.5 5.6a 5.7a 1

### Summer Math Exercises. For students who are entering. Pre-Calculus

Summer Math Eercises For students who are entering Pre-Calculus It has been discovered that idle students lose learning over the summer months. To help you succeed net fall and perhaps to help you learn

### Functions and their Graphs

Functions and their Graphs Functions All of the functions you will see in this course will be real-valued functions in a single variable. A function is real-valued if the input and output are real numbers

### Geometry Notes PERIMETER AND AREA

Perimeter and Area Page 1 of 17 PERIMETER AND AREA Objectives: After completing this section, you should be able to do the following: Calculate the area of given geometric figures. Calculate the perimeter

### SECTION 0.11: SOLVING EQUATIONS. LEARNING OBJECTIVES Know how to solve linear, quadratic, rational, radical, and absolute value equations.

(Section 0.11: Solving Equations) 0.11.1 SECTION 0.11: SOLVING EQUATIONS LEARNING OBJECTIVES Know how to solve linear, quadratic, rational, radical, and absolute value equations. PART A: DISCUSSION Much

### Prep for Calculus. Curriculum

Prep for Calculus This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet curricular

### REVISED GCSE Scheme of Work Mathematics Higher Unit 6. For First Teaching September 2010 For First Examination Summer 2011 This Unit Summer 2012

REVISED GCSE Scheme of Work Mathematics Higher Unit 6 For First Teaching September 2010 For First Examination Summer 2011 This Unit Summer 2012 Version 1: 28 April 10 Version 1: 28 April 10 Unit T6 Unit

### 16. Let S denote the set of positive integers 18. Thus S = {1, 2,..., 18}. How many subsets of S have sum greater than 85? (You may use symbols such

æ. Simplify 2 + 3 + 4. 2. A quart of liquid contains 0% alcohol, and another 3-quart bottle full of liquid contains 30% alcohol. They are mixed together. What is the percentage of alcohol in the mixture?

### The problems in this booklet are organized into strands. A problem often appears in multiple strands. The problems are suitable for most students in

The problems in this booklet are organized into strands. A problem often appears in multiple strands. The problems are suitable for most students in Grade 11 or higher. Problem E Mathletes in Action Four

### SAT Subject Math Level 1 Facts & Formulas

Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences: PEMDAS (Parenteses

### 13. Write the decimal approximation of 9,000,001 9,000,000, rounded to three significant

æ If 3 + 4 = x, then x = 2 gold bar is a rectangular solid measuring 2 3 4 It is melted down, and three equal cubes are constructed from this gold What is the length of a side of each cube? 3 What is the

### Algebra Concept-Readiness Test, Form A

Algebra Concept-Readiness Test, Form A Concept : The Distributive Property Study the concept, and then answer the test questions on the net page. You can use the distributive property to simplify an epression

### Junior Math Circles November 18, D Geometry II

1 University of Waterloo Faculty of Mathematics Junior Math Circles November 18, 009 D Geometry II Centre for Education in Mathematics and Computing Two-dimensional shapes have a perimeter and an area.

### Higher Mathematics Homework A

Non calcuator section: Higher Mathematics Homework A 1. Find the equation of the perpendicular bisector of the line joining the points A(-3,1) and B(5,-3) 2. Find the equation of the tangent to the circle

### Math Foundations IIB Grade Levels 9-12

Math Foundations IIB Grade Levels 9-12 Math Foundations IIB introduces students to the following concepts: integers coordinate graphing ratio and proportion multi-step equations and inequalities points,

### Winter 2016 Math 213 Final Exam. Points Possible. Subtotal 100. Total 100

Winter 2016 Math 213 Final Exam Name Instructions: Show ALL work. Simplify wherever possible. Clearly indicate your final answer. Problem Number Points Possible Score 1 25 2 25 3 25 4 25 Subtotal 100 Extra

### Centroid: The point of intersection of the three medians of a triangle. Centroid

Vocabulary Words Acute Triangles: A triangle with all acute angles. Examples 80 50 50 Angle: A figure formed by two noncollinear rays that have a common endpoint and are not opposite rays. Angle Bisector:

### Mathematics Placement

Mathematics Placement The ACT COMPASS math test is a self-adaptive test, which potentially tests students within four different levels of math including pre-algebra, algebra, college algebra, and trigonometry.

### ALGEBRA I A PLUS COURSE OUTLINE

ALGEBRA I A PLUS COURSE OUTLINE OVERVIEW: 1. Operations with Real Numbers 2. Equation Solving 3. Word Problems 4. Inequalities 5. Graphs of Functions 6. Linear Functions 7. Scatterplots and Lines of Best

### The Not-Formula Book for C1

Not The Not-Formula Book for C1 Everything you need to know for Core 1 that won t be in the formula book Examination Board: AQA Brief This document is intended as an aid for revision. Although it includes

### Individual Round Arithmetic

Individual Round Arithmetic (1) A stack of 100 nickels is 6.25 inches high. To the nearest \$.01, how much would a stack of nickels 8 feet high be worth? 8 feet = 8 12 inches. Dividing 96 inches by 6.25

### GRADES 7, 8, AND 9 BIG IDEAS

Table 1: Strand A: BIG IDEAS: MATH: NUMBER Introduce perfect squares, square roots, and all applications Introduce rational numbers (positive and negative) Introduce the meaning of negative exponents for

PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

### Parallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular.

CONDENSED L E S S O N. Parallel and Perpendicular In this lesson you will learn the meaning of parallel and perpendicular discover how the slopes of parallel and perpendicular lines are related use slopes