Note: We recommend that this section on hard concepts be given as a review to all students in higher level math courses before their testing date.

Transcription

1 Hard Concepts This section includes math problems that range from difficult to very difficult. They are grouped into units titled Concept # through Concept #. These are the most difficult concepts found on the ST/PST. This particular section was designed to raise the scores of those students who are already doing well on the ST/PST to an even higher level. Many of the problems in this section are representative of the ones on the ST/ PST that are extremely difficult or tricky. Several practice problems are given in order to assist the student in gaining mastery of that type problem. final section called The ig Review provides ten review questions along with explanations of each question. Note: We recommend that this section on hard concepts be given as a review to all students in higher level math courses before their testing date. Copyright 005 by dvanced Placement Strategies 9

2 Math concept # The -step E-Z McSqueezy Method very difficult problem on a PST or ST is one in which all of the answer choices are expressions such as the following example of five choices: () d/g () d/g (C)dg (D) g/d (E) gd/. For this type of question, you can use the famous -step E-Z McSqueezy Method. This method allows you to use arithmetic to solve a difficult algebra problem. The following are the steps. Step : Choose your own value for each variable in the problem. Step : nswer the question using your values for the variables. Step : Insert your values for the variables into each of the five answer choices, and find the answer choice that matches your answer from step. Example : The total cost of g gallons of gas is d dollars. What is the cost of gallons of gas? ll answers are in dollars. (a) d/g (b) d/g (c)dg (d) g/d (e) gd/ Step : Insert your own values for g and d. For example, let g = 0 and d = 0. Step : nswer the question using your chosen values for the variables. If g = 0 and d = 0, then the first sentence of the problem becomes "The total cost of 0 gallons of gas is 0 dollars. Therefore, gallon of gas costs dollars. The question asks for the cost of gallons of gas which would be 8 dollars. Step : Insert our chosen values (0 for g and 0 for d) into each of the five answer choices to see which result matches the answer of 8 from Step. The following are the results: (a) 8 (b) (c) 800 (d) / (e) 00/ The answer is (a) since this matches the answer from Step. Important note: If more than one of the five answer choices matches your answer from Step, you must repeat the process with new values. However, the second time, you just test the answer choices that worked the first time. The answer choice that gives the correct answer (matches your answer from Step ) two times in a row is the answer to the problem. Solve with algebra: Let P be the dollar cost of gallons, and set up the proportion to the right and solve for P: If you are good at algebra, the problem is simple, but if you are unsure of your algebra skills, try the famous -step E-Z McSqueezy method. Even the best algebra students may find it useful for example on the next page. g = d P gp = d P = d/g -- choice (a) Copyright 00 by dvanced Placement Strategies

3 You are the champ if you can do example below using algebra. See what you can do and then try the -step E-Z McSqueezy Method. solution is below the dotted line. Example : The sum of the first n positive integers is s. Which of the following equals the sum of the next n integers in terms of n and s? (a) ns (b) n + s (c) n + s (d) n + s (e) n + s. I will let n =, and therefore, s will equal = 0. My answer to the question, the sum of the next n () integers, is = 6. I now substitute my values for n and s into the answer choices to find which matches my answer of 6. Choice (c) + 0 = 6, my answer. Example : Joe took h hours to drive to Houston. How many hours would his trip have taken if he had increased his average speed by 0%? h 0h h 0h (a) (b) (c) (d) (e) h To do this problem, you must know that distance = (rate)(time) or d = rt. d This means that the time = distance/rate or t = r and rate = distance/time.. Let h = and the distance to Houston be 0 miles. This means that his average speed is 60 miles per hour.. If his speed increases by 0%, it will be 66 miles per hour and the time for the trip will be 0/66. This reduces to 0/. This is my answer.. I now substitute my value for h into the answer choices to find which matches my answer of 0/. Choice (b) 0()/ = 0/, my answer h 0 Example : If y =, which of the following is equal to y? + x (a) (b) + + (c) (d) x x + + x x. Let x =.. Therefore, y = / and y =. is my answer.. I now substitute my value for x into the answer choices to find which matches my answer of. Choice (c) Copyright 005 by dvanced Placement Strategies 95

4 Example 5: Water drips out of a bathtub faucet at a constant rate of drop every n seconds. The bathtub fills at a rate of g gallons per hour. t these rates, how many drops are there in gallon? ng n (a) (b) (c) (d) (e) 60 ng ng g 600n g Example 6: car travels m miles in t minutes. t the same rate, how many hours will it take the car to travel in 0 miles? 0t m m t (a) (b) (c) (d) (e) m t 0t m t m Example 7: The price of a cat, after it was increased by 5%, is D dollars. What was the price of the cat before the increase? (a) $0.80D (b) $0.75D (c) $.0D (d) $.5D (e) $.50D Example 8: The price of 5 pounds of apples is x dollars. The apples weigh an average of pound for every 6 apples. Which of the following expressions is the average price, in cents, of a dozen such apples? x (a) (b) 0x (c) 00x(d) 0 x (e) 0 0x Example 9: EZ uto Rental rents cars at the rate of d dollars per day and c cents per mile. car was rented for 8 days and was driven,00 miles. What amount, in dollars, was owed? (a) 0cd (b) 8(d + 66 / c) 00 (c) 8(d + 66 / c) (d) 6(d + c) (e) 6(d + c) Example 0: fter Q ounces of oil and V ounces of vinegar were put into a jar to make some salad dressing, the jar was / full. Which expression expresses what fractional part of the jar was filled with oil? Q Q Q (a) Q + V (b) QV (c) Q + V (d) 9(Q + V) (e) Q Q + V nswers:. a. c. b. c 5. c 6. e 7. a 8. d 9. e 0. e Copyright 005 by dvanced Placement Strategies 96

5 % Math concept # % nother difficult type of problem that might appear on the PST is one involving percents. difficult problem can be made a lot easier if you insert your own values for the variables in the problem. In fact, if you ever see a variable, you should consider substituting in a value. Example : If r is 5% of s and s is 0% of t, then r is what percent of t? (a) 0% (b) 5% (c) 0% (d).5% (e) 6% Start by choosing values for r and s where r is 5% of s. I will choose 5 for r and 00 for s. Since s is now 00 and is 0% of t, then t would have to be 50. Now, since r is 5 and t is 50, r is 0% of t. So the correct answer is (a) 0%. Of course, this problem may also be done with algebra. Start with r =.5s and solving for s, you get r = s. lso, s =.t and substituting r for s, you get r =.t. r r. Solve r =.t for and you get =, which is equal to /0 or 0%. t t Unless you feel really comfortable doing these problems with algebra, I would suggest that you solve them by substituting your own values for the variables. If you have time, you might even do the problem by both methods to check your work. Example : If r is 0% of s and s is 0% of t, then r is what percent of t? (a) 0% (b) 80% (c) 0% (d) 8% (e) 0% Example : The price of a car is reduced by 0% and the resulting price is then reduced by 0%. These two reductions would be the same as a single reduction of what percent? (a) 60% (b) 0% (c) 5% (d) 8% (e) 80% Do you see that 60% seems to be the correct answer? It's not. egin the problem by setting the original price of the car at $00. Example : The price of oil rose 0% last year and from that point rose 0% this year. These two increases would be the same as a single increase of what percent? (a) 0% (b) 0% (c) 57% (d) % (e) % Copyright 005 by dvanced Placement Strategies 97

6 Example 5: The population of Richardson is 5% of the population of the rest of Dallas County. The population of Richardson is what percent of the entire population of Dallas County? (a) 0% (b) 0% (c) 5% (d) /% (e) /% good start would be to call the population of Richardson 5 and the population of the rest of Dallas County 00. This would make the total population of Dallas County 5. Go from there. Example 6: If x is 50% of y, then y is what percent of x? (a) 50% (b) /% (c) 66 /% (d) 75% (e) 00% Example 7: Yesterday, a store sold apples at 6 pounds for a dollar. Today the store sells apples at pounds for 50. What is the percent of the decrease in the price per pound? (a) /6% (b) /% (c) 6 /% (d) 0% (e) 5% You just have to work out example 7. There are no variables for which you can choose your own value. Example 8: In order to form a rectangle, the lengths of one pair of opposite sides of a square are increased by 0%, and the lengths of the other pair of opposite sides are decreased by 0%. The area of the rectangle is equal to what percent of the area of the square? It always helps to draw diagrams. (a) 6% (b) 90% (c) 50% (d) 8% (e) 60% Example 9: store charges $5 for a video game. This price is 0% greater than the store's cost to buy the game from its supplier. s an incentive to the store's clerks, the manager allows a clerk to purchase the video game for 0% less than the store's cost. How much would a store clerk have to pay for this video game? (a) $5.60 (b) $0.80 (c) $.0 (d) $.76 (e) $6.00 Example 0: three-gallon mixture is / oil and / water. If one gallon of oil is added to the mixture, then approximately what percent of the four-gallon mixture is oil? (a) % (b) % (c) 56% (d) 78% (e) 5% There are no variables in this one, so you must work it out. nswers. a. d. c. e 5. b 6. c 7. e 8. d 9. e 0. b Copyright 005 by dvanced Placement Strategies 98

7 vg. Math concept # vg. nother type of problem that can be very difficult is one involving averaging or an average. The main thing to remember in doing one of these is to FIND THE TOTL. Example : student answered problems out of 5 on one test and 8 out of 50 on another test. How many problems must the student answer correctly on a third test of 5 questions in order to have 80% of the questions on all three tests correct? In this problem, what is the total number of questions? = 0. How many questions out of 0 must one answer correctly to score 80%? It would be.80(0), which is equal to 96. How many questions have already been answered correctly? It would be + 8, which is equal to 6. How many questions still need to be answered correctly to get 96 right? It would be 96-6, which is. Example : student answered 0 out of 0 questions correctly on one test and 5 out of 0 correctly on a second test. How many questions must the student answer correctly on a third test of 80 questions in order to have 66% of the questions on all three tests correct? (a) (b) (c) (d) 5 (e) 6 Example : Lolita's usual golf average is y. If she played six rounds of golf and had scores of y -, y +, y, y - 7, y + 5, and y -, what score must she make in a seventh game in order to have an average score of y -? (a) y - (b) y - (c) y - (d) y - (e) y - 5 In example, the total shots for the six given rounds are y- + y+ + y + y-7 + y+5 + y- = 6y - 6y - + 7th round score 7 = y - Copyright 005 by dvanced Placement Strategies 99

8 Example : On a 00-mile trip, a car went 0 miles per hour for the first 00 miles, 50 miles per hour for the second 00 miles, and 60 miles per hour for the third 00 miles. For what fractional part of the time taken for the entire 00-mile trip was the car going 0 miles per hour? (a) /0 (b) / (c) 0/ (d) / (e) / The total is the total time taken for the trip. Example 5: The average weight of children is 55 / pounds. If each child weighs at least 5 pounds, what is the greatest possible weight of one of these children? (a) 55 lbs. (b) 56 lbs. (c) 57 lbs. (d) 58 lbs. (e) 59 lbs. Example 6: The average of three different integers is 7. The sum of the two greater integers is 9. What is the greatest possible value of any of the three integers? (a) 5 (b) 6 (c) 8 (d) 7 (e) It cannot be determined from the information given Remember that there is very little chance that "cannot be determined from the information given" will ever be an answer to a difficult question. Example 7: The average of five numbers is -7. The sum of three of the numbers is 9. What is the average of the other two numbers? (a) 8 (b) (c) - (d) - (e) - Example 8: The average age of Joe, ill and Mary is 6. The average age of Joe and ill is. The average age of ill and Mary is 5. What is the average age of Joe and Mary? (a) 6.5 (b) (c) 8 (d) 6 (e) 0 Example 9: The average of three integers x, y and 8 is 0. Which of the following could not be the value of the product xy? (a) (b) (c) 57 (d) 96 (e) Example 0: (This is not one of the really difficult ones.) If the average of z and 5z is 0, what is the value of z? (a) 5 (b) 6 (c) 0 (d) 5 (e) 0 Copyright 005 by dvanced Placement Strategies 00

9 Example : If m is the average of x and 8, and n is the average of x and, what is the average of m and n? x + x + (a) 6 (b) (c) (d) (e) x + 6 Example : If y > 0 and m is the average of y, y and 0, and n is the average of y and 7, what is the relationship betwen m and n? (a) m = n (b) m > n (c) m < n (d) m could be equal to n, or m could be less than n, or m could be greater than n Try ex. on your own and then look below at two ways to do the problem. Example : If q = r = t, then what is the average of q, r and t in terms of q? 7q q 7q 7q (a) (b) (c) (d) (e) 7q 7 8 First way. Find the total (q + r + t) in terms of q. We must find the average of q, r and t. Therefore, we must find r and t in terms of q. (r = q/ and t = q/) Substitute into the expression q + r + t and get q + q/ + q/. To average this, we must add the three terms up and then divide by. dding them, we get 7q/. Dividing by, we get 7q/ divided by. 7q fter inverting the, we get 7q/ /, which is equal to. Second way. ssign values to q, r and t that will satisfy the equation q = r = t. How about for t, for r, and for q? Your equation will be = () = (). The average of q, r and t will be the average of, and, which is 7/. Now substitute, our value for q, into each of the answer choices, and find which one matches 7/. The choice that matches the 7/ will be the correct answer. If you substitute in for q in choice (e), then you will get which is 8 which is 7. 7() 6 Copyright 005 by dvanced Placement Strategies 0

10 Example : Joe traveled ten miles to work at 0 miles per hour and traveled the same distance home at 60 miles per hour. What is his average speed? (a) 50 mph (b) mph (c) 5 mph (d) 8 mph (e) 5 mph What is the total? To find the average speed, you need the total distance divided by the total time. For instance, if you traveled 00 miles in hours, your average speed would be 50 mph (00/ = 50). The total distance in this problem is 0 miles, 0 miles to work and 0 miles home. To get to work it takes / of an hour to drive the 0 miles at 0 mph. To get home it takes /6 of an hour to drive the 0 miles at 60 mph. His total time driving is / + /6, which is 5/ of an hour. To find his average speed, divide the 0 miles (total distance) by the 5/ (total time). You should get 8 mph. If you are having trouble with this, you can still make an educated guess if you can answer this question. If a person drives 0 mph to a place and 0 mph returning over the same distance, is he going to spend more time driving 0 mph or 0 mph? Hopefully, you think 0 mph. If he spent the exact same time driving 0 mph as 0 mph, his average speed would be 5 mph. Since he is going 0 mph for a longer period of time than 0 mph, his average speed would be less than 5 mph. Using this logic in the above problem, you can determine that his average speed must be less than 50 mph, and you can eliminate answers (a) and (e). Example 5: Joe traveled to work at 0 miles per hour and traveled the same distance home at 60 miles per hour. What is his average speed? (a) 50 mph (b) mph (c) 5 mph (d) 8 mph (e) 5 mph Notice that this problem is just like example except the distance to work is not given. Will this make a difference? No, since the distances going and returning are equal regardless of the total length. Therefore, you need to make up any convenient distance and do the problem. good distance would be 0 miles. It would take Joe hours to get to work and hours to get home, or a total of 5 hours for 0 miles. Notice that by dividing the total miles by the total time (0/5), you get 8 mph. Example 6: Jenny spends a total of one hour driving to work and back. She averages 0 miles per hour going to work and 50 miles an hour returning. What is the total number of miles in a round trip? (a) 0 (b) / (c) 7 / (d) 0 (e) 50 What is your best guess? To work it, call the time going (t) and the time returing (-t). Solve 0t = 50( - t) and go from there. nswers.. c. d. c 5. d 6. b 7. d 8. e 9. b 0. c. e. d. e. d 5. d 6. c Copyright 005 by dvanced Placement Strategies 0

11 Math concept # (x + y) = x + xy + y (x - y) = x - xy + y (x + y)(x - y) = x - y You can make some very difficult problems quite easy if you make sure that you learn three common factor patterns. Notice that there are six expressions in the three equations. They are: (x + y) (x - y) (x + y)(x - y) x + xy + y x - xy + y x - y Whenever you see any of these six in any problem, you should immediately write the expression that it is equal to. So, if you see (x + y), you write x + xy + y ; if you see x - y, you write (x + y)(x - y). Example : If x - y = and x - y =, what is the value of x + y? () 0 () / (C) (D) / (E) I hope that after the lecture you started by writing x - y = (x + y)(x - y). nswers to the examples follow the last problem. Copyright 005 by dvanced Placement Strategies 0

12 Example : (x - y) + xy =? (a) x - y + xy (b) (x - y) (c) (x - y) (d) (x + y) (e) (x + y) Example : If x - 9 = r, x + = s, and rs = 0, then x - =? (a) r + s (b) r - s (c) r + 9 s (d) rs (e) r s Example : If x - y = and x - y = 65, then x =? (a) (b) 9 (c) (d) 5 (e) 6 Example 5: For this rhombus (x + y)(x - y) =? x y (a) 0 (b) x (c) x (d) x + xy + y (e) Example 6: If r = s + and s = t - t, what is r in terms of t? (a) (t - ) (b) (t + ) (c) t - t (d) t + (e) t - t t + Example 7: (5) + (5)(75) + (75) =? (a) 6,550 (b) 0,000 (c) 00 (d) 9,900 (e),950 Example 8: If x + y = and x - y = 6, then what is x - y? (a) (b) 0 (c) 0 (d) - (e) Example 9: If x + y = 6 and x - y = 8, then what is x - y? (a) (b) 5 (c) 8 (d) (e) 9 Example 0: If x - y = 6 and x - y =, then what is x + y? (a) / (b) (c) 8 (d) 7 (e) 6 Example : If (x + y) = 00 and xy =, then what is x + y? (a) 00 (b) 5 (c) 76 (d) 0 (e) 576 Copyright 005 by dvanced Placement Strategies 0

13 Example : If (x - y) =6 and xy = 60, what is x + y? (a) 9 (b) (c) 66 (d) 76 (e) 6 Example : If xy = / and x + y = 0/9, what is (x + y)? (a) /9 (b) 0/7 (c) 0/9 (d) 6/9 (e) 00/8 Example : If x + y = 00 and xy = 8, what is (x - y)? (a) 0 (b) 7 (c) (d) 0-7 (e) (0-7) Example 5: ( tough one) If a + b = x and a - b = y, find ab in terms of x and y. x - y x - y 8 (a) xy (b) xy (c) (d) (e) x - y One way to start is to square both equations. This will result in: a + ab + b = x and a - ab + b = y Now, subtract the second equation from the first. This will result in: 8ab = x - y Now, solve for ab and the answer is: ab =, which is choice (d). nother way to solve the problem (and I think a lot easier) is to insert your own values for a and b, find x and y, find ab, and then substitute your values for x and y in the answer choices to see which choice gives you your ab. Try this now. y Example 6: If (y + ) = 50, then + y =? (a) 7 (b) 8 (c) 50 (d) 98 (e) 99 y x - y 8 Example 7: m + n = z and m - n = /z, assuming z = 0, m - n =? (a) (b) (c) z (d) z (e) 0 Copyright 005 by dvanced Placement Strategies 05

14 The following are student-produced response questions. You will enter your answer by marking the ovals in a grid like the one below. 8. If xy = 05, and x + y = 7, what is the value of (x + y)? 9. If xy = 8, x + y = 00, and (x - y) > 0, what is the value of (x - y)? 0. If x + y = 0, and x - y = 5, what is the value of x - y? If x - y = 0, and x - y =, what is the value of x + y?. If x - y =, and x + y = 8, what is the value of x?. If x + y = 6, and xy = 0, what is the value of (x - y)? Review Whenever you see any of these six expressions in a problem, immediately write the expression that it is equal to. (x + y) (x - y) (x + y)(x - y) x + xy + y x - xy + y x - y So, if you see (x + y), you write x + xy + y ; if you see x - y, you write (x + y)(x - y). nswers. c. d. e. c 5. a 6. a 7. b 8. e 9. c 0. b. b. e. d.c 5. d 6. b 7. a / or Copyright 005 by dvanced Placement Strategies 06

15 --5 a Math concept #5 + b = c The Pythagorean Theorem and Special Triangles You need to be able to apply the Pythagorean theorem and you should know Pythagorean triples. Given one side, you also should be able to find the lengths of the other two sides of triangles and triangles. The Pythagorean theorem, a + b = c, means that the length of one leg of a right triangle squared plus the length of the other leg squared is equal to the length of the hypotenuse squared. There are certain ratios, called Pythagorean triples, that will always work for right triangles and need to be memorized. These are: In addition, there are certain relationships in triangles and in triangles that would be helpful to know. For a triangle, a 5 c 5 a. c = a c. a = For a triangle, b 60 c 0 **Note** a It is very, very likely that you will have one or more problems that involve either the Pythagorean theorem or sides of these special triangles. You should definitely learn the triples and at least the three equations to the right that are circled.. a =. c =. c = b. b = 5. a = b 6. b = c a c a Copyright 005 by dvanced Placement Strategies 07

16 Example : In circle X, radius XC = and C =. Find. (a) (b) / (c) 5 (d) 5 - (e) C X Example : In the following right triangle, find x. (a) 6 (b) 8 (c) (d) 9 (e) 9 x 0 Example : CD is a rectangle. EF GH DC. The diagonal E of the rectangle is 6. F =, G FH = /, and HC = /. Find D the area of quadrilateral EFHG. (a) 6 (b) 9 / (c) 6 (d) 5 (e) 65 F H C Example : D is the midpoint of C, and E is the midpoint of C. = 9 and EC = 7.5. What is the area of the shaded region? ll answers are in square units. (a) 8 (b) 7 (c) 6 (d) 0.5 (e) 67.5 E D C Example 5: man is standing on a pier and is holding a rope that is attached to the front of a raft right at the water level. The rope is 5 feet long, and he is holding it 5 feet above the water level. If he pulls in 8 feet of rope, how many feet will the boat move? ll answers are in feet. (a) 8 (b) 0 (c) (d) (e) 5 Example 6: The legs of a right triangle are 6 and 8. The hypotenuse of the right triangle is also a leg of an isosceles right triangle. What is the length of the hypotenuse of the isosceles right triangle? (a) 0 (b) 5 (c) 0 (d) 0 (e) 5 Example 7: Given the following diagram, which of the following statements is true? a 9 b (a) a + b = c (b) a + b < c (c) a + b > c (d) a + b = c (e) a + b < c c Copyright 005 by dvanced Placement Strategies 08

17 Example 8: Given the following diagram, what are the coordinates of point Y? (a) (,0) (b) (,0) (c) (,0) (d) (,0) (e) ( 0,0) X(,) Y Example 9: In the given figure, C =, C = 5, D = 8x, and D = 8x. What is the length of D? D (a) 7 (b) 8 (c) 9 (d) (e) C E Example 0: Given the following diagram, find a + b. (a) 0 (b) 0 (c) (d) 00 (e) 60 5 a b Example : In the given diagram, m C = 90. Find the value of (C) + (C). (a) 6 (b) 5 (c) 9 (d) 5 (e) 9 C Example : What is the area of the given figure? (a) (b) 6 + (c) 6 (d) 6 + (e) It cannot be determined from the given information. 5 Copyright 005 by dvanced Placement Strategies 09

18 Example : In the given figure, what is the length of C? (a) 7 (b) 7 / (c) 8 (d) 9 (e) C 6 D Example : What is the perimeter of C? (a) 9 (b) (c) (d) 5 + (e) 8 / C Example 5: Given the following figure, what is the length of D? 5 (a) 5 (b) (c) 6 (d) 7 (e) 5 D C Example 6: In the given figure, quadrilateral CD is a square. Under which of the following conditions is the area of square CD equal to four times the area of right EC? (a) if x > y (b) if x = y (c) if x < y (d) if the perimeter of the square is equal to twice the perimeter of the triangle (e) It is not possible for the area of the square to be equal to four times the area of the. D C y x E Example 7: In the given figure, DC is a square, E = and E =. What is the length of ED? E (a) (b) 6 (c) 9 (d) (e) 5 C D Copyright 005 by dvanced Placement Strategies 0

19 The following examples deal with the sides of and triangles. Example 8: (a really tough one) rectangular piece of paper is inches long. What is the least possible width that the paper could have in order for one to cut out circles, each with a radius of inches? Choices are in inches. (a) 8 (b) + (c) 6 (d) + (e) 7 Example 9: C is an equilateral triangle. M is the midpoint of C. Quadrilateral TMXC is a rectangle. If =, then what is the area of TMC? T M (a) (b) (c) (d) (e) C X Example 0: In the given figure, = 6, and C = DC. Find the ratio of C to DC. 0 (a) (b) (c) (d) (e) D C Example : Use the figure and the given information in example 0 to find the ratio of C to D (a) (b) (c) (d) (e) 6 Example : What is the area of the smallest equilateral triangle with sides whose lengths are whole numbers? (a) (b) (c) (d) (e) Example : rc XY is / the circumference of a circle. If XY = 8, what is the length of arc XY? (a) π (b) 8 π (c) π (d) π (e) 8π X 8 Y Copyright 005 by dvanced Placement Strategies

20 Example : In the given figure, quadrilateral CD is a square, and arc XD is a semicircle. If D =, what is the area of figure CDX? 8 (a) π + (b) π + (c) π + X 8 (d) π + (e) π + D C Example 5: In the given figure, which of the following statements is true? (a) m > 5 (b) m = 5 (c) m < 5 (d) m + m C > m (e) No statements are true. C Example 6: Circle is inscribed in a square that has a perimeter of 8. What is the length of C? (a) (b) (c) - (d) (e) - C Example 7: In the given diagram, there are three squares and = EF = GH. What is the ratio length of C? length of EK (a) (b) (c) (d) (e) D E C J F G H K nswers. d. c. c. d 5. c 6. c 7. b 8. e 9.c 0. d. e. b. e. d 5. d 6. b 7. c 8. d 9. b 0. d. e. b. d. a 5. b 6. e 7. a Copyright 005 by dvanced Placement Strategies

21 Weird αβχδφγηιϕκλµνιπθρτ Math concept #6 αβχδφγηιϕκλµνιπθρτ very tricky type of problem, until one is used to it, is one with an unfamiliar symbol. little practice will make these quite a bit easier. Symbols Example : Let **Y be defined for all Y by the equation **Y = Y +. For example, **6 will equal (6) +, which is 9. If (**) (**Y) = Y, what is Y? (a) (b) (c) (d) (e) Since **Y = Y +, you should begin the problem by saying, "I need to multiply times whatever follows the ** and add." It helps if you formulate a sentence that shows you the meaning of the symbol. Example : X signifies the smallest integer greater than or equal to X. For example,. = 5 and - = -. What is -. +.? (a) - (b) 0 (c) (d) 8 (e) 9 Example : Let X Y be defined as ( Y ) for all nonzero numbers X and Y. What is - ( - )? 8 8 (a) 6 (b) (c) (d) - (e) 0 Example : For all real numbers, (a, b, c, d)θ(k, l, m, n) is equal to ak + bl + cm + dn. What is (,, 5, )θ(, -,, 6) (-, -, -, )θ(5, -,, )? (a) -0 (b) 0 (c) (d) - (e) 0 Example 5: If X is a positive even number, let X@ equal the sum of all positive even numbers from to X. For example, 0@ = = 0. Which of the following statements is false? (a) X@ is always even. (b) If X is divisible by 6, then X@ is divisible by 6. (c) (X+)@ - X@ = X +. (d) If X is divisible by, then X@ is divisible by. X Copyright 005 by dvanced Placement Strategies

22 Example 6: The operation is defined on ordered pairs of numbers in the following way: (a,b) (c,d) = (ac + bd, ad + bc). If (a,b) (m,n) = (a + b, a + b), what is (m,n)? (a) (0,0) (b) (0,) (c) (,0) (d) (,-) (e) (,) Example 7: For all numbers X, *X = X(X-)(X-). What is? *6 (e careful.) (a) * (b) * (c) *6 (d) *8 (e) * Example 8: n operation Ω is defined for all real numbers r and s, and the equation r Ω s = rs. For which of the following is r Ω s = s Ω r? I. r = 0, s = II. r = s III. r = -s (a) I only (b) I and II only (c) II only (d) II and III only (e) I, II and III Example 9: For all real numbers X and Y except zero, X!$! Y = X. Y If = 0, which of the following is (are) always true? I. X!$! Y = Y!$! X II. (X!$! Y)(!$! ) = (X)!$! (Y) III. (X!$! Y) (!$! ) = - X Y (a) I only (b) II only (c) III only (d) II and III only (e) I, II and III - *0 nswers. b. c. e. d 5. d 6. e 7. a 8. e 9. b Copyright 005 by dvanced Placement Strategies

23 Math concept #7 You will have to do some very tricky math problems that many students miss due to carelessness or because the problems defy your natural intuition. good example: Which is greater, x or x? It seems that it would be x, but what if x is or zero or a fraction? Therefore, be careful! Example : If y is an even number, what is the sum of the next two even numbers greater than y? (a) 6y + (b) 6y + (c) 6y + 6 (d) 6y + 8 (e) 6y +0 5 C Example : If = and =, which of the following is (are) true? I. = 5 II. /C = - III. ( + C) = 0 (a) none (b) I only (c) II only (d) II and III only (e) I, II and III -5 Example : Eight girls went to the movies together. Every ten minutes a pair of the girls would go to the snack bar together. If no two girls ever made more than one trip to the snack bar together, and trips to the snack bar were only made in pairs, what is the maximum number of trips that the group could send to the snack bar during the movie? (a) 7 (b) 8 (c) 8 (d) 5 (e) 56 Example : If 5 = x m, then what will 5x equal? (a) x m+5 (b) x m+ (c) x m+ (d) x m (e) 5x m Copyright 005 by dvanced Placement Strategies 5

24 Example 5: X is the set of 0 consecutive odd integers whose sum is 0. Y is the set of 6 consecutive odd integers whose sum is 8. How many elements of X are also elements of Y? (a) none (b) (c) (d) 5 (e) 6 Example 6: If 8 = 5 and 7 =, what is in terms of T? T (a) 5 - (b) (c) (d) (e) T T 5 T T 5T Example 7: If ( 7-6 )( 5 - ) = n, what is n? (Don't work it out the long way.) (a) (b) (c) 6 (d) 8 (e) 0 Example 8: ( + / + / + /8 + /6 + /) (a) / (b) 7/8 (c) 5/6 (d) / (e) 6/6 ( + / + / + /8 + /6 + /) = Example 9: pet store sells only dogs and cats. There are twice as many cats as dogs in the store. If /8 of the cats are males and /6 of the dogs are males, what fraction of the animals are males? (Making your own problem with your own values is probably easier than solving with algebra.) (a) /8 (b) 7/ (c) 5/6 (d) 7/6 (e) 5/8 Example 0: hockey team is going to play three games. It will get 0 points for a win, points for a tie, and 0 points for a loss. How many different point possibilities are there that the team can have after completing the three games? (For example, if the team could end up with only 0 points, 9 points or 0 points, then there would be possibilities.) (a) (b) 7 (c) 9 (d) 0 (e) 0 Copyright 005 by dvanced Placement Strategies 6

25 Example : If two odd numbers are not multiples of each other, which of the following must always be true? (a) The sum of the integers is odd. (b) Twice the smaller is greater than the larger. (c) The smaller will divide evenly into three times the larger. (d) The product cannot be negative. (e) The product is an odd integer. Example : fter Joe completed high school, he had made 's for every 's and 6 C's for every 5 's. What was his ratio of C's to 's? (a) It is a wise person who knows the difference between the next two problems, and, on top of that, can do them both correctly. (b) 5 8 (c) 5 (d) 8 5 (e) 7 5 Example : Emmitt gains as much yardage in three carries as arry does in four. arry gains as much in five carries as Thurman does in six. What is the ratio of Thurman's yards per carry to Emmitt's yards per carry? (a) (b) 8 5 (c) 5 (d) 5 8 (e) 5 7 Example : There are eight men in a tennis league. If each man plays two matches against every other man, how many total matches are played in the league? (a) 5 (b) 6 (c) 9 (d) 56 (e) 6 Example 5: If X and Y are positive integers and Y > X, which of the following statements is always true about the value of the following expression? ( X ) _ X Y Y (a) The value is always zero. (b) The value is always greater than zero. (c) The value is always less than zero. (d) The value is sometimes positive and sometimes negative. (e) The value is always undefined. Copyright 005 by dvanced Placement Strategies 7

26 Example 6: There are 75 girls in a row. The first three have red ribbons in their hair, and the next two have blue ribbons in their hair. The next three have red, and the next two have blue. This pattern continues all of the way down the row. What color ribbon do girls 68, 69 and 70 have in their hair? (a) red, red, red (b) red, red, blue (c) red, blue, blue (d) blue, blue, red (e) blue, red, red Example 7: In all except one of the following, at least one of the coordinates of the ordered pair, when squared, is equal to the reciprocal of the other coordinate. For which ordered pair does this not hold true? (a) (,) (b) (,) (c) (,) (d) (,9) (e) (,-) 9 Example 8: In a junior high school with seventh and eighth grades, there are the same number of girls as boys. The eighth grade has 80 students, and there are boys for every 5 girls. In the seventh grade there are boys for every girls. How many girls are in the seventh grade? (a) 0 (b) 00 (c) 70 (d) 60 (e) 80 Example 9: X and Y are positive integers and X > Y. Which of the following must be less than X? (a) X + Y (b) X Y (c) X - 0. Y - 0. Y (d) ( ) X Y (e) X + Y + Example 0: What fraction of the number of integer multiples of between and 9 are also integer multiples of? (a) (b) (c) 6 5 (d) 8 5 (e) 8 9 Example : What is the remainder when is divided by 0? (a) (b) (c) (d) 6 (e) 8 Copyright 005 by dvanced Placement Strategies 8

27 Example : If it takes 5 people 9 hours to do a certain job, how many hours would it take people to do / of the job? (a) 5 hours (b) /5 hours (c) / hours (d) /5 hours (e) hours This problem is fairly easy if you will remember one thing. Start by figuring out how long it will take one person to do the job. You do this by figuring the total hours spent on the job. In this problem, five people each are working for 9 hours. That would mean a total of 5 (5 x 9) hours would be needed to finish the job. So, if one person were doing it all by himself, he would need 5 hours. From there you figure how many hours it will take two people to do the job. This would be / hours. Since these two people are just doing / of the job, it will take them / of / hours, which is 5 hours. Example is similar to Example. Example : If it takes 9 people 0 hours to do a certain job, how long will it take 6 people to do /5 of the same job? (a) hours (b) hours (c) / hour (d) /5 hours (e) 5 hours Example : certain basketball, after being dropped, will bounce / of its previous height. If the height after the th bounce is / feet, what was the height after the st bounce? (a) 5 / feet (b) feet (c) 7/6 feet (d) feet (e) /6 feet Example 5: The sum of three consecutive even integers is M. In terms of M, what is the sum of the three consecutive integers that follow the greatest of the original even integers? (a) M + 8 (b) M + (c) M + 6 (d) M + (e) M + 5 Example 6: If and and if + + C = 00, then what is? = C = 5 (a) 6 (b) 7 (c) 0 (d) (e) 6 / Copyright 005 by dvanced Placement Strategies 9

28 9N Example 7: If N not be equal to? is an odd integer, then which of the following could (a) - (b) (c) 0 (d) 9 (e) Example 8: The ratio of Joe's weight to ill's is to. The ratio of Joe's weight to Pete's is to. What is the ratio of ill's weight to Pete's weight? (a) 9 to 8 (b) 8 to 9 (c) to (d) to (e) to n easy way to do #8 (a difficult problem) is to assign actual weights that fit the ratios. Start by calling Joe's weight 0 and then find ill's weight and Pete's weight by using the ratios. fter that, compare ill's weight to Pete's weight. Example 9: is equal to which of the following? (a) - 0 (b) 0 5 (c) (d) (e) 0 Start #9, and any like it, by factoring. 0 (? -?) Example 0: If x = + 8, what is the value of x? (a) 60 (b) 5 (c) 96 (d) 0 (e) 08 Example : Given the set of integers that are greater than 0 and less than 00, how many of the integers are multiples of both and 5? (a) 9 (b) 0 (c) 9 (d) 8 (e) 58 nswers. c. d. c. c 5. d 6. e 7. e 8. e 9. c 0. d. e. d. d. d 5. c 6. c 7. d 8. d 9. e 0. b. d. a. b. a 5.d 6. a 7. e 8. a 9. d 0. e. c Copyright 005 by dvanced Placement Strategies 0

29 Math concept #8 Difficult geometry problems If you are not convinced that geometry teaches you how to think, or if you do not think it helps you in figuring out how much paint or fertilizer to buy, or if you feel that the only reason for geometry to exist is to give poor geometry teachers a job, think again. The difficult geometry problems are what separate the superstars from the others on the PST and ST. ngle measure Example : What is the sum of the degree measures of the numbered angles? (a) 60 (b) 80 (c) 50 (d) 70 (e) Example : In the given figure, what is y in terms of x? y o (a) y = x - 90 (b) y = x (c) y = 90 - x (d) y = x (e) y = 80 - x x o Example : Which of the following is always equal to 90 - q? (a) x + z (b) y + q (c) z + q (d) q (e) q q x y z Example : The given circle has a center Q. If the radius of the circle is and m QD = 5, what is the total area of the shaded region? (a) π (b) π (c) 6π (d) 8π (e) π 5 Q D C Copyright 005 by dvanced Placement Strategies

30 Example 5: In the given figure, what is the value of a + b + c + d + e? a o b o c o (a) 80 (b) 70 (c) 60 (d) 00 (e) 50 d o e o Example 6: Which of the following cannot be the measure of exterior angle of a regular polygon? (a) (b) 0 (c) 5 (d) 5 /7 (e) 0 Example 7: In the given figure, what is the sum of the measures of a, b, c and d? (a) 90 (b) 0 (c) 80 (d) 60 (e) It cannot be determined from the given information. a b c d Example 8: In the given figure, what is the sum of the measures of a, b, c, d, e and f? (a) 80 (b) 70 (c) 00 (d) 60 (e) It cannot be determined from the given information. a f c e b d Example 9: In the given figure, what is the sum of the measures of a, b, c, d, e, f and g? (a) 50 - s - t (b) 900 (c) 50 (d) 60 (e) It cannot be determined from the given information. s f a b c e d g t Copyright 005 by dvanced Placement Strategies

31 Example 0: In the given figure, what is the value of x? (5y + 8) (a) 0 (y + ) (b) Note: Figure not (c) 6 drawn to scale. (d) 0 (e) It cannot be determined from the given information. 6x (y + b) Example : If the m = 90, what is the average (arithmetic mean) of the degree measures of angles through 0? 5 (a) 7 (b) 50 (c) 5 (d) 60 (e) Example : In the given figure, what is the sum of the degree measures of,,,, 5, 6, 7, 8, 9, 0,,, and? (a) 60 (b) 70 0 (c) 0 (d) 800 (e) It cannot be determined from the given information Example : In the given figure, if m XE = 6 and m CXD = 5, what is the degree measure of DXE? (a) 77 (b) 78 (c) 79 (d) 80 (e) 8 D E X C Copyright 005 by dvanced Placement Strategies

32 Circles Example : In circle Q, points,, C, D, E and F divide the circle into 6 equal arcs. If the area of the circle is 6π, what is the length of arc D? F Q (a) π (b) π (c) π (d) 6π (e) π E D C Example 5: Circle Q has a radius of 6. If the total area of the shaded region is 6π, what is the measure of QD? Q D (a) 0 (b) 5 (c) (d) 50 (e) 65 C Example 6: The centers of two circles are 7 cm apart. The area of one circle is 00π, and the circumference of the other circle is 6π. t how many points do the circles intersect? (a) 0 (b) (c) (d) infinite (e) It cannot be determined from the given information. Example 7: In the given figure, M is the center of the large circle, and TM is a diameter of the small circle. What is the ratio of the area of the small circle to the area of the shaded region? (a) : (b) : (c) : (d) : T M Example 8: In the given figure, regular octagon CDEFGH is inscribed in circle Q. If circle Q has a radius of r, which of the following represents the length of arc GH? (a) πr (b) πr 8 (c) πr (d) πr (e) πr 8 G H F Q D C E Copyright 005 by dvanced Placement Strategies

33 Example 9: In the given figure, circles, and C each have a radius of. What is the radius of circle D? (a) (b) (c) - (d) - (e) C D Example 0: In the given figure, circle Y has a radius of 0. If the area of the shaded region is 90π, what is the measurement of? (a) 0 (b) 6 (c) 5 (d) 60 (e) Y Example : C is equilateral and is inscribed in circle W, which has a radius of 6. What is the perimeter of the shaded region? (a) 6 + π (b) 6 + π (c) 6 + π (d) 6 + π (e) 6 + π W C Example : In circle Q, m = 6 and m D =. What is the m QD + m QC? 6 (a) 7 (b) 98 (c) Q (d) 96 (e) 06 D C Copyright 005 by dvanced Placement Strategies 5

34 Example : In circle Q, C and D are diameters of length 0. If m QD = 7, what is the length of arc DTC? () π () π (C) π (D) 6π (E) 8π D Q T C Example : If the circumference of circle X is π cm longer than the circumference of circle Y, then how many cm longer is the radius of circle X than the radius of circle Y? (a) π (b) π (c) (d) (e) It cannot be determined from the given information. Example 5: wheel comes off a race car and rolls down the track 00 feet. If the radius of the wheel is feet, which of the following is the closest approximation of the number of revolutions that the wheel makes while traveling this 00 feet? (a) 0 (b) 60 (c) 00 (d) 00 (e) 00 Example 6: The circumference of circle Q is 0π cm. The sum of the length and the width of rectangle RSTQ is cm. What is the perimeter of quadrilateral RT in centimeters? (a) (b) (c) + 0 R Q T S (d) (e) Example 7: This circle has an area of 65π sq cm and is divided into five equal regions. What is the perimeter of the shaded region in centimeters? (a) 5π + 50 (b) 0π + 50 (c) 5π + 50 (d) 0π + 0 (e) 5π + 50 Copyright 005 by dvanced Placement Strategies 6

35 Example 8: The given figure is half of a rectangular solid. Find the surface area of the figure in square centimeters. (a) (b) (c) (d) (e) rea and volume 6 cm cm cm Example 9: Find the ratio of the area of a rectangle whose length is three times its width to the area of an isosceles right triangle whose legs are each equal to the width of the rectangle. 6 (a) (b) (c) (d) (e) 6 9 Example 0: Find the ratio of the area of a rectangle whose length is three times its width to the area of an isosceles right triangle whose hypotenuse is equal to the width of the rectangle. 6 (a) (b) (c) (d) (e) 6 Example : The volume of a cube is 6 cubic centimeters. What is the total surface area in centimeters? (a) 6 (b) 6 (c) 96 (d) 56 (e) 8 Example : The lengths of the edges of a rectangular solid are all whole numbers. If the volume of the solid is cubic centimeters, what is the total surface area in square centimeters? (a) 8 (b) 5 (c) 5 (d) 78 (e) 69 Example : If the areas of these two figures are equal and (p)(q) = 00, what is the value of (x)(y)? (a) 0 (b) 0 (c) 50 (d) 00 (e) 00 p q x y Copyright 005 by dvanced Placement Strategies 7

36 Example : tank that has a height of 8 feet and a base that is feet by feet contains 0 cubic feet of water. If cubic feet of water are added to the tank, how many inches will the water in the tank rise? (a) (b) 6 (c) (d) (e) / Example 5: rectangular floor, 0 feet by feet, is going to be tiled with square tiles that each have a perimeter of one foot. What is the least number of tiles needed to cover the floor? (a) (b) 0 (c) 80 (d) 00 (e) 90 Example 6: The volume of a cube with an edge of is how many times the volume of a cube with an edge? (a) (b) (c) (d) (e) 9 Example 7: The height of triangle C is 80% of the width of rectangle RSTQ. The base of triangle C is 75% of the length of rectangle RSTQ. If the area of triangle C is, what is the area of rectangle RSTQ? (a) 9 (b) 0 (c) 0 (d) 0 (e) 80 Example 8: square with each side equal to is divided into regions. One is a square with an area of 9; another is a triangle named "" and another is a polygon named "." What is the area of polygon ""? 9 9 (a) 6 (b) 50 (c) 56 (d) 86 (e) It cannot be determined from the given information. Example 9: Five cubic blocks are stacked on top of each other. The block on the bottom has an edge of / feet. Each additional block that is stacked has an edge that is / foot less than the preceding block. What is the ratio of the volume of the block on top to the volume of the block on the bottom? (a) (b) (c) (d) (e) Copyright 005 by dvanced Placement Strategies 8

37 Example 0: What is the volume of a cube with a surface area of 96s? (a) 6s (b) 6s (c) 56s (d) 0s (e) 96s Example : If the area of a square is equal to the area of an isosceles right triangle, what is the ratio of a side of the square to the hypotenuse of the right triangle? (a) (b) (c) (d) (e) Example : What is the area of an equilateral triangle in which each side is equal to /? 6 6 (a) (b) (c) (d) (e) Example : What is the ratio of an edge of a cube with a volume of 5 to an edge of a cube with a volume of 0? (a) : (b) : (c) 7:8 (d) : (e) 8:7 Example : The given diagram is a square with two semicircles. What is the ratio of the shaded area to the area of the square? π (a) π (b) (c) (d) (e) π 8 6 π π Example 5: On a map, inch represents 6 miles. circle on the map with a circumference of π inches represents a circular region with what area? (a) 8π square miles (b) π square miles (c) π square miles (d) 88 square miles (e) 576 square miles Example 6: Four semicircles are drawn on the number line. What is the total shaded area? (a) π (b) π (c) π (d) π (e) π Copyright 005 by dvanced Placement Strategies 9

38 The coordinate plane Example 7: CD is a square. What is the area of CD? y D(0,5) C(,6) (a) 9 (b) 0 (c) (d) (e) 6 (,) (,) x Example 8: What is the area of a triangular region that is bounded by the x-axis, the y-axis and the graph of line x + y = 5? (a) 5 (b) 0 (c) (d) 5 (e) 5 Example 9: If is a diameter parallel to the x-axis and quadrilateral CD is a rectangle, what is the perimeter of the shaded region? (a) π + 8 (b) π + 8 (c) π + 0 y 5 D C (d) π + (e) π x nswers. d. a. c. b 5. c 6. c 7. c 8. d 9. c 0. b. e. c. e. c 5. d 6. b 7. b 8. a 9. c 0. c. e. e. b. c 5. a 6. d 7. b 8. d 9. a 0. a. c. c. c. b 5. e 6. c 7. d 8. c 9. e 0. b. b. b. b. c 5. c 6. b 7. b 8. c 9. a Copyright 005 by dvanced Placement Strategies 0

39 Math concept #9 hodgepodge of tough ones Example : Joe's car goes 5/6 as fast as Mike's car. Sue's car goes /5 as fast as Joe's car. Mike's car goes how many times as fast as the average of the other two cars' speeds? (a) (b) (c) (d) (e) 9 If you need a hint, look at the bottom of this page Example : t our cafeteria, P pints of milk are needed per month for each student. t this rate, T pints of milk will supply S students for how many months? T (a) PTS (b) (c) (d) (e) PS PT S PS T S PT Example : If a + b = c + d, what is the average of a, b, c and d in terms of c and d? c + d (a) c + d (b) (c) (d) (e) c + d c + d 8 cd Example : The sum of two consecutive odd integers is S. In terms of S, what is the value of the larger of these integers? S S (a) (b) + (c) + (d) (e) S S + S + Example 5: The stock market went up 0% Monday and from that point decreased 5% on Tuesday. What percent of the closing Tuesday price will the market have to rise on Wednesday in order to be exactly where it started on Monday? 9 (a) 5% (b) 9% (c) 9 % (d) 0% (e) % nswers are at the bottom of the last page. 9 Hint for Example : Call Mike's speed and go from there. Joe's speed will be 0, and Sue's will be 8. Copyright 005 by dvanced Placement Strategies

40 Example 6: If xy = - and (x + y) = 5, what is the value of x + y? (a) 6 (b) (c) 5 (d) (e) 5 I hope that you remember the rule. Whenever you see any of these six expressions, you immediately write the expression that it is equal to. Do this even before you try to figure what the problem is asking for. (x + y) (x - y) (x + y)(x - y) x + xy + y x - xy + y x - y So, if you see (x + y), you write x + xy + y. If you see x - y, you write (x + y)(x - y). etc. Example 7: The height of a triangle is 50% of the width of a rectangle. The base of the triangle is 0% of the length of the rectangle. If the area of the rectangle is 80, what is the area of the triangle? (a) 6 (b) (c) (d) 0 (e) 80 Example 8: Thirty students took the ST. s you know, it is scored from 00 to 800. Exactly 0 of the students made a score greater than or equal to 500. What is the lowest possible average of all thirty scores? (a) 00 (b) 00 (c) 00 (d) 500 (e) none of the above Example 9: Forty students took an algebra test, and the average score was Q. Scores on the test ranged from 0 to 9 inclusive. If the average of the first 0 tests that were graded was 70, what is the difference between the greatest possible value of Q and the least possible value of Q? (a) (b) (c) 7 (d) 5 (e) 76 Copyright 005 by dvanced Placement Strategies