LearningExpress Skill Builders • CHAPTER 6 115 C • H • A • P • T • E • R SUMMARY Now you can apply the math skills that you have learned in this book. You can use this practice test to help you identify your strengths and weaknesses and see where you might need some more practice to get your math skills in shape for college. PRACTICE TESTS IN ARITHMETIC, ALGEBRA, AND GEOMETRY 6 6 CHAPTER 6 • LearningExpress Skill Builders 116 P RACTICE A RITHMETIC T EST Directions: Circle the correct answer to the following problems.You can check your answers at the end of the chapter 1. After she started exercising, Patty started losing weight at a steady rate. She lost 24 pounds in one year. How much weight did Patty lose per month? a. 2 pounds b. 1 pound c. 3 pounds d. 12 pounds 2. At her party, Mackenzie put out a bowl con- taining 360 jellybeans. Marina came by and ate ᎏ 1 1 2 ᎏ of the jellybeans, Christina ate ᎏ 1 4 ᎏ , Athena ate ᎏ 1 5 ᎏ , and Jade ate ᎏ 1 8 ᎏ . How many jellybeans were left? a. 120 b. 240 c. 237 d. 123 3. At DeCavallas Home Improvements, industrial cable sells for $4.98 per yard. If Gina needs to purchase 108 feet of cable, how much will this cost her? a. $179.28 b. $537.84 c. $18.02 d. $268.92 4. If it takes 5 workers to build 3 sheds, how many would it take to build 18? a. 90 b. 18 c. 15 d. 30 5. Change 0.525 to a percent. a. 525% b. 5.25% c. 0.525% d. 52.5% 6. What is |Ϫ423| ϩ |423| equal to? a. 0 b. Ϫ|423| c. 846 d. 423 7. What is (4 ϩ 2) 3 ? a. 196 b. 72 c. 216 d. 18 8. What does 8.2 ϫ 10 9 equal? a. 8,200,000,000 b. 820,000,000 c. 820,000 d. 820,000,000,000 9. Calculate ͙(97 Ϫ ෆ 16) ෆ multiplied by ͙(48 Ϭ ෆ 3) ෆ . a. ͙36 ෆ b. 9 c. 6 2 d. 4 10. What is the median of the following group of numbers? 10 20 30 40 50 60 a. 30 b. 35 c. 60 d. 40 –BASIC SKILLS FOR COLLEGE– LearningExpress Skill Builders • CHAPTER 6 117 11. The average weight of a male Proboscis monkey, Nasalis larvatus, is 20,370 grams. The average weight of a male Douc langer, Pygathria nemaeus, is 10,900 grams. How much bigger is the average male Proboscis monkey than the average male Douc langer? a. 10,430 grams b. 8,980 grams c. 9,470 grams d. 10,470 grams P RACTICE A LGEBRA AND G EOMETRY T EST Directions: Circle the correct answer to the following problems.You can check your answers at the end of the chapter. 12. If V = πr 2 h, what is h equal to? a. ᎏ π r V 2 ᎏ b. Vπr 2 c. ᎏ π V r 2 ᎏ d. ᎏ V r 2 h ᎏ 13. Factoring 2pq 2 Ϫ 4p 2 q 3 yields which of the fol- lowing expressions? a. 2pq(q Ϫ 2pq 2 ) b. 2pq(q Ϫ 2pq) c. 2p 2 q(q Ϫ 2pq 2 ) d. 2pq(q Ϫ 4pq 2 ) 14. (3x 4 y 2 )(5xy 3 ) is equivalent to a. 15x 4 y 5 b. 15x 5 y 4 c. 15x 5 y 5 d. 15x 4 y 4 15. If x is a positive integer, solve for x:3x ϩ x 2 ϭ 28 a. 4 b. Ϫ4 c. Ϫ7 d. a and c 16. What is the value of 3x 2 Ϫ 2xy 3 when x ϭ 1 and y ϭϪ2? a. Ϫ19 b. Ϫ5 c. 13 d. 19 17. ᎏ 4 x ᎏ Ϫ ᎏ 2 3 x ᎏ ϩ ᎏ 5 6 x ᎏ ϭ a. ᎏ 1 2 2 x ᎏ b. ᎏ 1 5 2 x ᎏ c. ᎏ 4 6 x ᎏ d. ᎏ 2 6 2x ᎏ 18. If BC is parallel to DE, and DB = 6, what is the value of AE ? a. 4 b. 6 c. 8 d. 10 A BC DE 32 6 –PRACTICE TESTS IN ARITHMETIC, ALGEBRA, AND GEOMETRY– CHAPTER 6 • LearningExpress Skill Builders 118 19. Circle O has a diameter of 8 cm.What is the area of Circle O? a. 64π cm 2 b. 32π cm 2 c. 16π cm 2 d. 8π cm 2 20. What is the perimeter of the rectangle shown below? a. (2 Ϫ a) 2 b. (2 Ϫ a)(a) 2 c. 4 ϩ 8a d. 4 21. If a 10 ft ladder is leaning against a building as shown in the diagram below, how many feet above the ground, h, is the top of the ladder? a. 8 b. 10 c. ͙8 ෆ d. ͙10 ෆ 22. The graph of y ϭ 3x Ϫ 12 crosses the x-axis at which of the following coordinates? a. (Ϫ4, 0) b. (4, 0) c. (0, 4) d. (0, Ϫ4) 23. If the side of the cube below is doubled, what happens to its volume? a. It is doubled. b. It is tripled. c. It is quadrupled. d. It is multiplied by eight. 2 10 ft 6 ft h 2 - a a O –BASIC SKILLS FOR COLLEGE– LearningExpress Skill Builders • CHAPTER 6 119 24. How much greater is the area of Circle B than the area of Circle A? a. 5π cm 2 b. 12π cm 2 c. 20π cm 2 d. 36π cm 2 25. If r ϭ 5 cm and the water is 4 cm high, what is the volume of water in the right cylinder below? a. 20π cm 3 b. 80π cm 3 c. 100π cm 3 d. 800π cm 3 26. If line segment A ෆ B ෆ is parallel to line segment C ෆ D ෆ , what is the value of x? a. 58° b. 62° c. 56° d. 60° 27. A line has a slope ϭ ᎏ 3 2 ᎏ and passes through the points (1, q) and (Ϫ5, Ϫ6). What is the value of q? a. 3 b. 1 c. Ϫ3 d. Ϫ1 28. How many times does the graph x 2 Ϫ 81 ϭ y cross the x-axis? a. not at all b. once c. twice d. three times 29. Which inequality below is equivalent to 8x Ϫ 3 Ͼ 29? a. Ϫ4 Ͻ x Ͼ 4 b. x Ͻ 4 c. x Ͼ 4 d. Ϫ4 Ͻ x AB CD x - 6 2x r = 5 cm 8 cm 2 3 A B –PRACTICE TESTS IN ARITHMETIC, ALGEBRA, AND GEOMETRY– CHAPTER 6 • LearningExpress Skill Builders 120 30. Alan is 5 years less than twice Helena’s age. If Alan is 27, then which equation can be used to solve for Helena’s age? a. 22 ϭ 2H b. 27 ϭ H Ϫ 5 c. 22 ϭ 2H ϩ 5 d. 27 ϭ 2H Ϫ 5 ANSWERS PRACTICE ARITHMETIC TEST 1. a. There are 12 months in one year, so 24 Ϭ 12 ϭ 2 pounds per month. 2. d. Marina ate ᎏ 1 1 2 ᎏ of 360: ᎏ 1 1 2 ᎏ ϫ ᎏ 36 1 0 ᎏ = ᎏ 3 1 6 2 0 ᎏ , which is equal to 30. Christina ate ᎏ 1 4 ᎏ of 360: ᎏ 1 4 ᎏ ϫ ᎏ 36 1 0 ᎏ ϭ ᎏ 36 4 0 ᎏ , which is equal to 90. Athena ate ᎏ 1 5 ᎏ of 360: ᎏ 1 5 ᎏ ϫ ᎏ 36 1 0 ᎏ ϭ ᎏ 36 5 0 ᎏ , which is equal to 72. Finally, Jade ate ᎏ 1 8 ᎏ of 360: ᎏ 1 8 ᎏ ϫ ᎏ 36 1 0 ᎏ ϭ ᎏ 36 8 0 ᎏ which is equal to 45. Add them all up: 30 ϩ 90 ϩ 72 ϩ 45 ϭ 237. Then subtract that from the original amount: 360 Ϫ 237 ϭ 123. 3. a. First convert 108 feet into yards. Since there are 3 feet in one yard, divide 108 by 3: 108 Ϭ 3 ϭ 36. Then multiply your answer by $4.98 to get $179.28. If you chose answer b, you forgot to convert the feet into yards. 4. d. First set up a proportion: ᎏ 5 3 ᎏ ϭ ᎏ 1 x 8 ᎏ .Then cross multiply: 3x ϭ 18 ϫ 5. Then solve for your answer: 3x ϭ 90, so x ϭ 30. 5. d. First, move the decimal point two digits to the right: .525 becomes 52.5. Next, add a percent sign: 52.5%. 6. c.|Ϫ423| ϭ 423, |423| ϭ 423, so add the two numbers together to get 846. 7. c. Calculate what is in the parentheses first: 4 ϩ 2 ϭ 6, and then find the value of 6 3 , which is 216. 8. a. Count nine spaces to the right of the deci- mal, so it becomes 8,200,000,000. 9. c. First calculate ͙(97 Ϫ ෆ 16) ෆ ϭ ͙81 ෆ ϭ 9. Next, figure out ͙(48 Ϭ ෆ 3) ෆ ϭ ͙16 ෆ ϭ 4. Lastly, you multiply: 9 ϫ 4 ϭ 36. Since 6 2 ϭ 36, the answer is c. 10. b. Since there are two middle numbers in this set—30 and 40—the median is the average of the two, or 35. 11. c. This is a subtraction problem: 20,370 Ϫ 10,900 ϭ 9,470. PRACTICE ALGEBRA AND GEOMETRY TEST 12. c. You need to rearrange the equation V ϭ πr 2 h, into an equation that has h equal to something. In order to isolate the h, you need to get rid of the πr 2 on the right side of the equation. You can do this by dividing both sides by πr 2 . Thus, the equation becomes ᎏ π V r 2 ᎏ ϭ h. 13. a. In order to factor the original expression, first note what the two terms have in common:You can pull out a 2, a p,and a q 2 . You get: 2pq(q Ϫ 2pq 2 ). To check this, you can distribute the 2pq to yield the orig- inal expression, 2pq 2 Ϫ 4p 2 q 3 . 14. c. (3x 4 y 2 )(5xy 3 ) can first be changed to 15x 4 y 2 xy 3 . If you have the same base, when multiplying exponents, you just add the powers. Since x is the same as x 1 , when you add the powers of the x terms you get 4 ϩ 1, or x 5 . For the y terms, you add 2 ϩ 3 to get y 5 . Thus, the final answer is 15x 5 y 5 . 15. a. First, subtract the 28 from both sides of 3x ϩ x 2 ϭ 28 to yield 3x ϩ x 2 Ϫ 28 ϭ 0. We rearrange this to x 2 ϩ 3x Ϫ 28 ϭ 0. Next, you need to pick out two numbers that add to 3 (the coefficient of x) and mul- tiply to Ϫ28 (the last term). The numbers that work are Ϫ4 and 7. These go inside the parentheses as follows: (x Ϫ 4)(x ϩ 7) ϭ 0. Now you solve two equations: x Ϫ 4 ϭ 0 and x ϩ 7 ϭ 0. The solutions to these equations are x ϭ 4 and x ϭϪ7. But be careful! The question tells us that x is a positive integer. This means that x ϭ 4 ONLY. –BASIC SKILLS FOR COLLEGE– LearningExpress Skill Builders • CHAPTER 6 121 16. d. Look at the equation 3x 2 Ϫ 2xy 3 and put a 1 wherever you see an x and a Ϫ2 wherever you see a y. The equation becomes 3(1) 2 Ϫ 2(1)(Ϫ2) 3 ϭ 3(1) Ϫ 2(1)(Ϫ8) ϭ 3 Ϫ (Ϫ16) ϭ 3 ϩ 16 ϭ 19. There are two tricky parts to this question. First, notice that (Ϫ2) 3 ϭ Ϫ8.Also, notice that when subtracting a negative num- ber, you are really just adding a positive number: 3 Ϫ (Ϫ16) ϭ 3 ϩ 16 ϭ 19. 17. b. First, we need to find the least common denominator. The denominators are 4, 3, and 6, so 12 will be the least common denominator. Next, we con- vert all three terms into something over 12: ᎏ 1 3 2 x ᎏ – ᎏ 1 8 2 x ᎏ ϩ ᎏ 1 1 0 2 x ᎏ ϭ ᎏ Ϫ 1 5 2 x ᎏ ϩ ᎏ 1 1 0 2 x ᎏ ϭ ᎏ 1 5 2 x ᎏ 18. b. Triangle ABC and triangle DAE are simi- lar. This means that their sides will be in proportion. Side AB will be in proportion with side AD. On the fig- ure we can see that AB ϭ 3. We are given that DB ϭ 6, so we know that AD ϭ 9. Thus the triangles are in a 3:9 ratio, which reduces to a 1:3 ratio. This helps us because if AC ϭ 2, then AE will be three times as long, or 6. 19. c. Use the area formula for a circle, A ϭ πr 2 . If d ϭ 8, then r ϭ 4. A ϭ πr 2 becomes A ϭ π(4) 2 ϭ π(16) ϭ 16π cm 2 . 20. d. The perimeter formula for a rectangle is P ϭ 2l ϩ 2w. Here the length is 2 Ϫ a, and the width is a. Putting these values into our formula we get L ϭ 2(2Ϫa) ϩ 2(a) ϭ 4 – 4a ϩ 4a ϭ 4. 21. a. The diagram shows a right triangle with a hypotenuse of 10 ft and one leg equal to 6 ft. If you know how to spot a 6-8-10 right triangle you are in luck, and you know that the other leg, h, is 8 ft. Otherwise, use the Pythagorean theorem: a 2 ϩ b 2 ϭ c 2 . This formula becomes 6 2 ϩ h 2 ϭ 10 2 , or 36 ϩ h 2 ϭ 100, or h 2 ϭ 64. Thus h ϭ 8. 22. b. The line will cross the x-axis when y ϭ 0. So we take the equation y ϭ 3x Ϫ 12 and stick 0 in for y. Thus, the equation becomes 0 ϭ 3x Ϫ 12. We add 12 to both sides to yield 12 ϭ 3x. Divide both sides by 3 to get x ϭ 4. So the line crosses at the (x, y) coordinates (4, 0). 23. d. The side of the original cube is 2, so its vol- ume is V ϭ side 3 ϭ (2) 3 ϭ 8 units 3 . When we double its side, the side ϭ 2 ϫ 2 ϭ 4. The new volume is V ϭ (4) 3 ϭ 64 units 3 . When you compare the two volumes, you see that you multiply the old volume (8) by eight to get the new volume (64). 24. a. The area of a circle is A ϭ πr 2 . The area of Circle B is π(3) 2 , or 9π. The area of Circle A is π(2) 2 ,or 4π. The difference in areas is 9π Ϫ 4π, or 5π. 25. c. The volume formula for a cylinder is V ϭ πr 2 h.We will substitute in 5 for r, and 4 for h.Make sure that you don’t use 8 as the height. We want the volume of the water, not the volume of the cylinder! The equa- tion V ϭ πr 2 h becomes V ϭ π(5) 2 (4) ϭ π(25)(4) ϭ 100π cm 3 . 26. b. The line that crosses both parallel lines will create the same angles about both lines. There is an angle marked “x Ϫ 6”under line segment AB,so we can mark an angle “x Ϫ 6” under line segment CD. Now, notice that 2x and x Ϫ 6 combine to make a straight line. Since a straight line is 180 degrees, we can write: 2x ϩ (x Ϫ 6) ϭ 180, or 3x Ϫ 6 ϭ 180, or 3x ϭ 186, or x ϭ 62°, answer choice b. 27. a. Here we need to use the slope formula, and put in the values of the given coordinates: m ϭ ᎏ Δ Δ x y ᎏ ϭ ᎏ x y 2 2 Ϫ Ϫ y x 1 1 ᎏ ϭ ᎏ 1 q Ϫ Ϫ ( ( Ϫ Ϫ 6 5 ) ) ᎏ ϭ ᎏ 1 q ϩ ϩ 6 5 ᎏ ϭ ᎏ q ϩ 6 6 ᎏ We know that m ϭ ᎏ 3 2 ᎏ , so ᎏ 3 2 ᎏ ϭ ᎏ q ϩ 6 6 ᎏ We cross multiply to get: 2(qϩ6) ϭ 18. Divide both sides by 2 to get: q ϩ 6 ϭ 9. Thus, q ϭ 3. –PRACTICE TESTS IN ARITHMETIC, ALGEBRA, AND GEOMETRY– CHAPTER 6 • LearningExpress Skill Builders 122 28. c. The graph will cross at an (x,y) coordinate that has y ϭ 0. This means we should take the equa- tion x 2 Ϫ 81 ϭ y and set y equal to 0. The equation becomes x 2 Ϫ 81 ϭ 0. Moving the 81 over, we get x 2 ϭ 81. Thus x ϭ 9 and Ϫ9. This means there are 2 points that are on the x-axis, namely (9, 0) and (Ϫ9, 0). Thus the graph of x 2 Ϫ 81 ϭ y crosses the x-axis twice. 29. c. First we will add 3 to both sides: 8x Ϫ 3 Ͼ 29 ϩ3 ϩ3 8x Ͼ 32 Next, we divide both sides by 8 to yield x Ͼ 4. 30. d. “Alan is 5 years less than twice Helena’s age” can be written mathematically as A ϭ 2H Ϫ 5. Because we are also told that Alan is 27, we know that 27 ϭ 2H Ϫ 5. –BASIC SKILLS FOR COLLEGE– . weaknesses and see where you might need some more practice to get your math skills in shape for college. PRACTICE TESTS IN ARITHMETIC, ALGEBRA, AND GEOMETRY 6. is quadrupled. d. It is multiplied by eight. 2 10 ft 6 ft h 2 - a a O BASIC SKILLS FOR COLLEGE LearningExpress Skill Builders • CHAPTER 6 119 24. How