LearningExpress Skill Builders • CHAPTER 4 69 RATIO AND PROPORTION 7. If it takes 27 nails to build 3 boxes, how many nails will it take to build 7 boxes? a. 64 b. 72 c. 56 d. 63 8. In Mrs. Sam’s first grade class, the ratio of boys to girls is 3 to 4. There are 28 students total. How many are girls? a. 12 b. 20 c. 16 d. 4 PERCENTS 9. Change 35% into a decimal. a. 3.5 b. .35 c. 35.0 d. .035 10. 75 people were invited to the Frazzettas’wedding. All but 9 were able to attend. What percent couldn’t come? a. 8.33% b. 7.5% c. 12% d. 9% ABSOLUTE VALUE 11. What is | 47 Ϫ 64 |? a. 17 b. Ϫ17 c. 111 d. 47 12. Find | Ϫ ᎏ 2 3 ᎏ |. a. Ϫ ᎏ 2 3 ᎏ b. ᎏ 3 2 ᎏ c. 1 ᎏ 1 2 ᎏ d. ᎏ 2 3 ᎏ EXPONENTS 13. Calculate 43 2 ϫ 4. a. 172 b. 129 c. 7,396 d. 1,849 14. Calculate (Ϫ ᎏ 1 5 ᎏ ) 3 . a. ᎏ 1 1 25 ᎏ b. ᎏ 1 5 ᎏ c. – ᎏ 1 1 25 ᎏ d. – ᎏ 1 3 5 ᎏ SCIENTIFIC NOTATION 15. What is the correct way to write 3,600,000 in sci- entific notation? a. 3,600 ϫ 100 b. 3.6 ϫ 10 6 c. 3.6 ϫ 10 Ϫ6 d. 36 ϫ 10 6 16. 7.359 multiplied by 10 Ϫ6 is equal to a. 0.0007359 b. 0.00007359 c. 0.000007359 d. 0.0000007359 –ESSENTIAL PRACTICE WITH MATH– CHAPTER 4 • LearningExpress Skill Builders 70 SQUARE ROOTS 17. Which of the following equations is correct? a. ͙36 ෆ + ͙64 ෆ = ͙100 ෆ b. ͙25 ෆ + ͙16 ෆ = ͙41 ෆ c. ͙9 ෆ + ͙25 ෆ = ͙64 ෆ d. There is no correct equation. 18. What is another way to write 5͙12 ෆ ? a. 12͙5 ෆ b. 10͙3 ෆ c. 6͙3 ෆ d. 12 CALCULATING MEAN, MEDIAN, AND MODE For questions 19 and 20, memorize these definitions: Mean: When you are calculating the mean of a series of numbers, you are simply finding the average. Median: The median is the number in the mid- dle of a series. If there are two middle numbers in a set, the median is the average of the two. Mode: The mode is the number that appears most frequently in a series. 19. Calculate the mean of the following test scores: 92, 89, 96, 93, 93, and 83. a. 93 b. 91 c. 92.5 d. 91.5 20. Find the mode of the following series of num- bers: 2 3 7 7 9 9 9 9 14 a. 2 b. 7 c. 9 d. 14 GEOMETRY Measurement 1. What is the area of the shaded region in the fig- ure below? a. 42 Ϫ 4.5π b. 42 Ϫ 9π c. 24 d. 42 This question requires the knowledge of 2 area formulas: ■ Area of rectangle = length ϫ width ■ Area of circle = πr 2 This question also requires some reasoning. Exactly how much of the whole figure is shaded? How can you use these area formulas to help? Well, you might’ve noticed that the shaded region is just the area of the rectangle minus the area of ᎏ 1 2 ᎏ the circle. You can write a formula for yourself: Area shaded = Area of Rectangle Ϫ ᎏ 1 2 ᎏ Area of Circle Let’s get all the pieces we need by marking up the fig- ure a different way: 6 7 4 3 6 4 3 –BASIC SKILLS FOR COLLEGE– LearningExpress Skill Builders • CHAPTER 4 71 Notice that by drawing a new radius, we know that the length of the rectangle is 7. We already knew that the width was 6, so the area of the rectangle is just length ϫ width ϭ 7 ϫ 6 ϭ 42. Now, we will figure out the area of the circle using A ϭ πr 2 , which becomes π(3) 2 ϭ 9π. If the area of the whole circle is 9π, then the area of half the cir- cle will be ᎏ 1 2 ᎏ ϫ 9 π ϭ 4.5π. Thus, the area of the shaded region is Area shaded = Area of Rectangle Ϫ ᎏ 1 2 ᎏ Area of Circle Area shaded = 42 Ϫ 4.5π The correct answer is a. 2. Using the formula V = ᎏ 1 3 ᎏ πr 2 h, what is the vol- ume of the cone below? a. ᎏ 4 3 5 ᎏ π b. 45π c. 75π d. 125π Looking at the figure, we see that the radius, r, is 5 and the height, h, is 9. We plug the values r ϭ 5 and h ϭ 9 into the volume formula V ϭ ᎏ 1 3 ᎏ πr 2 h. The formula becomes V ϭ ᎏ 1 3 ᎏ π(5) 2 (9) ϭ V ϭ ᎏ 1 3 ᎏ π(25)(9).At this point, you may be inclined to multiply 25 by 9. But remem- ber what we told you in the beginning of the book about questions working out nicely? Does it seem nice to mul- tiply 25 by 9 and then take a third of that number? No. How about this: Take ᎏ 1 3 ᎏ of the 9 instead. V ϭ ᎏ 1 3 ᎏ (9)π(25) ϭ 3π(25) ϭ 75π Thus, the answer is c. Quadrilaterals 1. What is the area of the trapezoid shown below? a. 260 b. 210 c. 160 d. 130 The area of a trapezoid is A ϭ ᎏ 1 2 ᎏ (base 1 ϩ base 2) ϫ height. In this case, the formula becomes A ϭ ᎏ 1 2 ᎏ (10 ϩ 16) ϫ 10 ϭ ᎏ 1 2 ᎏ (26) ϫ 10 ϭ 13 ϫ 10 ϭ 130. Thus, choice d is correct. 2. What is the area of the parallelogram shown below? a. 64 b. 32 c. 16 d. It cannot be determined by the informa- tion given. 4 8 10 10 16 9 5 –ESSENTIAL PRACTICE WITH MATH– CHAPTER 4 • LearningExpress Skill Builders 72 The area of a parallelogram is A ϭ base ϫ height. Look- ing at the diagram, we see that the base is 8 and the height is 4. The area, A ϭ 8 ϫ 4 ϭ 32. Thus, choice b is correct. Triangles 1. In the right triangle below, A ෆ B ෆ = 4 and A ෆ C ෆ = 5. What is the value of B ෆ C ෆ ? a. 3 b. between 6 and 7 c. 7 d. between 7 and 8 To solve this question,we will use the Pythagorean the- orem, a 2 ϩ b 2 ϭ c 2 ,where a and b represent 2 legs of the right triangle, and c represents the hypotenuse of the right triangle. The hypotenuse is the longest side of a right triangle and it is always opposite the 90° angle (the right angle). Let’s fill in the information that we know: a 2 ϩ b 2 ϭ c 2 (4) 2 ϩ (5) 2 ϭ c 2 16 ϩ 25 ϭ c 2 41 ϭ c 2 c ϭ ͙41 ෆ Because we know 6 2 ϭ 36 and 7 2 ϭ 49, we know that ͙41 ෆ will be between 6 and 7, choice b. 2. In the figure shown below, what is the value of x ? a. 16 b. 13 c. 9 d. 6 The figure is comprised of 2 triangles. These triangles happen to be similar triangles. Triangles are similar when they have all three angles in common. The sides of similar triangles are in proportion. We know that these 2 triangles are similar because they both have right angles, and the angles marked below are equal as well. It follows that the third angles must also be equal because all triangles have 180°. (90° ϩ marked angle ϩ 3rd angle ϭ 180° for both triangles.) In order to figure out the proportion, you just look at the sides opposite the equal angles. SAMPGEO_4 SAMPGEO_5 SAMPGEO_6 SAMPGEO_7 SAMPGEO_8 3 7 8 14 x Equal angles SAMPGEO_4 SAMPGEO_5 SAMPGEO_6 SAMPGEO_7 3 7 8 14 x SAMPGEO_8 SAMPGEO_4 SAMPGEO_5 SAMPGEO_6 SAMPGEO_7 B A C SAMPGEO_8 –BASIC SKILLS FOR COLLEGE– LearningExpress Skill Builders • CHAPTER 4 73 The top triangle has a side of 7,and the bottom triangle has a side of 14, so we know that the sides of the bottom triangle are double the sides of the top triangle. Since x is opposite the 3rd angle, we look at the top triangle to see that 3 is opposite the 3rd angle. We double the 3 to get x ϭ 6. Thus, the answer is d. Parallel Lines 1. If A ෆ B ෆ is parallel to C ෆ D ෆ , what is the value of x ? a. 77° b. 87° c. 103° d. 113° We know that A ෆ B ෆ and C ෆ D ෆ are parallel, so any line that intersects them will create the same angles as it crosses each line. Notice how we can write 103° in the figure below: Also, we know that 103° ϩ x ϭ 180°, because there are 180° in a straight line: We can solve for x by subtracting 103° from both sides of the equation 103° ϩ x ϭ 180°. Thus, x ϭ 77°,which is choice a. 2. Given that l ෆ m ෆ and n ෆ o ෆ are parallel, use the figure below to determine the value of a ϩ b ϩ c ϩ d. a. 120° b. 180° c. 270° d. 360° Just by knowing that straight lines are 180°, we can fill in all the values for a, b, c, and d: Now we just add up the values: a ϩ b ϩ c ϩ d ϭ 70° ϩ 110° ϩ 70° ϩ 110° ϭ 360°, choice d. SAMPGEO_4 SAMPGEO_5 SAMPGEO_6 SAMPGEO_7 SAMPGEO_8 lm n o 70 110 70 110 110110 7070 oo oo o o o o SAMPGEO_1 SAMPGEO_2a SAMPGEO_2b SAMPGEO_3 lm n o a b c d 110110 7070 oo oo SAMPGEO_1 SAMPGEO_2a SAMPGEO_2b SAMPGEO_3 103 103 x AB CD o o o SAMPGEO_1 SAMPGEO_2a SAMPGEO_2b SAMPGEO_3 103 103 x AB CD o o o SAMPGEO_1 SAMPGEO_2a SAMPGEO_2b SAMPGEO_3 103 x AB CD o o SAMPGEO_4 SAMPGEO_5 SAMPGEO_6 SAMPGEO_7 SAMPGEO_8 3 7 8 14 x –ESSENTIAL PRACTICE WITH MATH– CHAPTER 4 • LearningExpress Skill Builders 74 Coordinate Geometry 1. Which line below has no slope? a. Line A b. Line B c. Line C d. Line D Let’s review how to tell the slope of a line by looking at each graph: Thus, choice d is correct. This line has no slope because slope ϭ ᎏ c c h h a a n n g g e e i i n n x y ᎏ There is no change in x for Line D. No change ϭ zero, which means we would have a zero in the denom- inator of our slope formula. Zeroes and denominators do not mix! (Actually dividing by zero is technically termed undefined, as in you can’t do it!) Therefore, there is no slope! Line C is interesting to look at as well. Here there is a zero slope because there is a zero in the numerator of the slope formula. There is a zero in the numerator of the slope formula because there is no change in y. 2. Line A ෆ B ෆ below contains the points (2, 3) and (Ϫ3, Ϫ2). What is the equation of line AB? a. y ϭ x Ϫ 1 b. y ϭϪ3x ϩ 2 c. y ϭ x ϩ 1 d. y ϭ 2x ϩ 3 The equation of a line is y ϭ mx ϩ b, where m is the slope of the line ( ᎏ Δ Δ x y ᎏ ) and b is the y intercept. We are given 2 points to work with, so first we will determine the slope. m = ᎏ Δ Δ x y ᎏ = ᎏ x y2 2 Ϫ Ϫ y x 1 1 ᎏ x y 1234567 1 2 3 4 5 6 7 -1 -2 -3 -4 -5 -6 -7 -1-2-3-4-5-6 -7 (-3,-2) (2,3) C D X Y X Y zero slope no slope A B X Y X Y positive slope negative slope C D X Y X Y A B X Y X Y –BASIC SKILLS FOR COLLEGE– . NOTATION 15. What is the correct way to write 3 ,60 0,000 in sci- entific notation? a. 3 ,60 0 ϫ 100 b. 3 .6 ϫ 10 6 c. 3 .6 ϫ 10 6 d. 36 ϫ 10 6 16. 7.359 multiplied by 10 6 is equal to a. 0.0007359 b. 0.00007359 c correct? a. ͙ 36 ෆ + 64 ෆ = ͙100 ෆ b. ͙25 ෆ + ͙ 16 ෆ = ͙41 ෆ c. ͙9 ෆ + ͙25 ෆ = 64 ෆ d. There is no correct equation. 18. What is another way to write 5͙12 ෆ ? a. 12͙5 ෆ b. 10͙3 ෆ c. 6 3 ෆ d. 12 CALCULATING. c 2 16 ϩ 25 ϭ c 2 41 ϭ c 2 c ϭ ͙41 ෆ Because we know 6 2 ϭ 36 and 7 2 ϭ 49, we know that ͙41 ෆ will be between 6 and 7, choice b. 2. In the figure shown below, what is the value of x ? a. 16 b.