(2) 4 oz (3) 6 oz (4) 8 oz (5) 9 oz 8. A school teacher is distributing 2 gallons of milk equally to her 16-student class. How many cups of milk will each student receive? (1) 2 (2) 4 (3) 8 (4) 16 (5) 32 9. Last year Sari was 4 feet 7 inches tall. This year she is 5 feet 1 inch tall. How much did she grow in the year? (1) 0 feet 3 inches (2) 0 feet 6 inches (3) 1 foot 3 inches (4) 1 foot 6 inches (5) 1 foot 9 inches 10. Sam has two pet snakes. The red snake is 2 feet 7 inches long, and the green snake is 3 feet 11 inches long. How much longer is the green snake than the red snake? (1) 1 foot 4 inches (2) 1 foot 7 inches (3) 1 foot 11 inches (4) 2 feet 3 inches
(5) 3 feet 9 inches 11. Jeff is using 20 feet of leather to make 8 belts. What will be the length, in inches, of each belt? (1) 20 (2) 30 (3) 40 (4) 50 (5) 60 12. Today, Stacy is getting a haircut. Right now her hair is 1 foot 5 inches in length. How long will her hair be, in inches, if she cuts off 7 inches? (1) 5 (2) 7 (3) 10 (4) 12 (5) 14 13. Emily leaves her house at 1:26 p.m. to drive to a friend’s house. What time will she arrive at her friend’s house if she drives for 57 minutes? (1) 12:29 a.m. (2) 2:23 a.m. (3) 12:29 p.m. (4) 2:23 p.m. (5) 3:57 p.m. 14. On four plays in a football game, a running back runs for a gain of 5 yards, a loss of 1 yard 2 feet, a loss of 2 feet, and a gain of 6 yards 1 foot. What is the distance traveled by this player?
(1) 2 yards 1 foot (2) 3 yards 2 feet (3) 5 yards 0 feet (4) 7 yards 1 foot (5) 9 yards 0 feet 15. The world’s largest cheeseburger is made up of 10 pounds of beef, 2 pounds of cheese, 5 ounces of lettuce, and 1 pound 4 ounces of bread. What is the total weight, in ounces, of this cheeseburger? (1) 217 (2) 240 (3) 271 (4) 278 (5) 313 16. Kelly is making her 6 grandchildren scarves for their birthdays. If she buys 4 yards of fabric, what will be the length, in feet, of each of the six scarves? (1) 1 (2) 2 (3) 3 (4) 4 (5) 5 17. Mona bought 2 packages of ground beef, each weighing 3 pounds 4 ounces, and a package of pork chops weighing 2 pounds 9 ounces. How many pounds of meat did she buy in total? (1) 5 pounds 13 ounces (2) 6 pounds 9 ounces
(3) 8 pounds 9 ounces (4) 9 pounds (5) 9 pounds 1 ounce 18. A child is learning to walk. She walks 2 feet on Monday, 2 feet 9 inches on Tuesday, and 3 feet 7 inches on Wednesday. How far does she walk over those three days? (1) 2 feet 5 inches (2) 6 feet 4 inches (3) 7 feet 1 inch (4) 7 feet 9 inches (5) 8 feet 4 inches 19. A man is cutting a 12-foot piece of wood into 24 equal pieces for a building project. What is the length, in inches, of each piece? (1) 3 (2) 6 (3) 9 (4) 12 (5) 15 20. A snail travels 60 inches every day. How many days will it take the snail to travel 5 yards? (1) 3 (2) 6 (3) 9 (4) 12 (5) 15
21. An ice cream machine makes 1.5 quarts of ice cream in 20 minutes. How many pints of ice cream will the machine make in 1 hour? (1) 2 (2) 5 (3) 6.5 (4) 9 (5) 15 22. George is cutting a long sheet of paper into 72 smaller sheets of equal length. The original sheet of paper is 45 feet long. What is the length, in inches, of each of the smaller sheets? (1) 7.5 (2) 12 (3) 36.75 (4) 72 (5) 432 23. The time difference between San Francisco, California, and Helsinki, Finland, is +10 hours (Helsinki is 10 hours ahead of San Francisco). If it is 7:35 a.m. in San Francisco, what is the time in Helsinki? (1) 9:35 a.m. (2) 11:35 a.m. (3) 1:35 p.m. (4) 3:35 p.m. (5) 5:35 p.m. 24. A truck driver drives an average of 6 hours and 45 minutes every day for 5 days. What is the total amount of time that the truck driver spends driving over those 5 days?
(1) 15 hours 30 minutes (2) 22 hours 15 minutes (3) 33 hours 45 minutes (4) 37 hours 0 minutes (5) 43 hours 45 minutes 25. Juanita walked in the park for 1 hour and 20 minutes every day for one full week. At the end of the week, how many hours had she spent walking in the park altogether? (1) 7 hours 20 minutes (2) 8 hours (3) 9 hours 20 minutes (4) 10 hours 10 minutes (5) Not enough information is provided. Answers 1. (3) First, convert the measurements to inches. Then, subtract Jamie’s height from Joe’s height 6 feet 1 inch = 73 inches 5 feet 4 inches = 64 inches 73 inches − 64 inches = 9 inches 2. (2) Convert the mixture measurement to ounces: 2 pounds 4 ounces = 36 ounces: 36 ounces + 12 ounces = 48 ounces = 3 pounds
3. (3) Convert 3 quarts to pints. A quart equals 2 pints, so 3 quarts = 6 pints. Then subtract 3 pints: 6 pints – 3 pints = 3 pints. 4. (3) Convert the minutes to hours. 30 minutes is of an hour, or 0.5 hours. 15 minutes is of an hour, or 0.25 hours. 0.5 hours + 1 hour + 0.25 hours = 1.75 hours. 5. (4) Subtract 45 min from 3 hr 15 min. First convert the hours to minutes. 15 minutes is of an hour, so 3 hours and 15 minutes = 3.25 hours. 45 minutes is of an hour, or 0.75 hours. Then subtract: 3.25 hours – 0.75 hours = 2.5 hours, or 2 hours 30 minutes. 6. (2) There are 2 cups to a pint. Add the number of cups and divide by 2 to get the number of pints. 3 + 1 + 2 = 6 cups ÷ 2 = 3 pints. 7. (5) Convert 4 lbs to ounces. Then divide by 8, the number of guests: 4 × 16 = 72 72 ÷ 8 = 9 8. (1) 2 gallons = 8 quarts = 16 pints = 32 cups 32 cups ÷ 16 students = 2 cups per student 9. (2) Convert Sari’s two heights to inches: 4 feet 7 inches = 55 inches 5 feet 1 inch = 61 inches Then subtract last year’s height from this year’s height: 61 inches − 55 inches = 6 inches. 10. (1) 3 feet 11 inches
11. (2) Convert the belt length to inches: 20 feet = 240 inches Then divide by the number of belts: 240 inches ÷ 8= 30 inches 12. (3) Convert feet to inches: 1 foot 5 inches = 17 inches Subtract 7 inches from 17 inches: 17 inches − 7 inches = 10 inches 13. (4) If Emily drove for a full hour, she would arrive at 2:26 p.m. However, she drives 3 minutes less than one hour. So, she arrives 3 minutes earlier: 1:26 p.m. + 57 minutes = 2:23 p.m. 14. (5) Convert yards to feet: Then add and subtract the gains and losses: 15 − 5 − 2 + 19 = 27 feet Divide by 3 to obtain the number of yards traveled: 27 feet ÷ 3 = 9 yards 15. (1) Add together the weights of the different components. Then convert to ounces: 13 pounds 9 ounces = (13 × 16) + 9 = 208 + 9 = 217 ounces
16. (2) Convert yards to feet: 4 yards = 12 feet Divide by the number of scarves: 12 feet ÷ 6 = 2 feet 17. (5) Add 3 lb 4 oz, 3 lb 4 oz, 2 lb 9 oz = 8 lb 17 oz. Convert the 17 oz to pounds (1 lb 1 oz), and add to the answer = 9 lb 1 oz. 18. (5) Add together the distances walked over the three days. 2 feet + 2 feet 9 inches + 3 feet 7 inches = 7 feet 16 inches 16 inches = 1 foot 4 inches, so 7 feet 16 inches = 8 feet 4 inches. 19. (2) Convert feet to inches. 12 feet = 144 inches Then divide by the number of pieces to find the length of each piece. 144 ÷ 24 = 6 inches 20. (1) Convert yards to inches. 5 yards = 15 feet = 180 inches Then divide by 60 to find the number of days. 180 inches ÷ 60 inches = 3 days 21. (4) There are 60 minutes in an hour. (20 minutes + 20 minutes + 20 minutes = 60 minutes.) So, in one hour, the machine will make 1.5 + 1.5 + 1.5 = 4.5 quarts of ice cream. But the question asks you how many pints the machine will make in one hour. So, you need to convert the quarts to pints: 4.5 × 2 = 9 pints. 22. (1) Convert feet to inches: 45 feet = 45 × 12 = 540 inches. Divide 540 by 72 to determine the length of each piece: 540 ÷ 72 = 7.5 inches. 23. (5) 7:35 a.m. + 10 hours = 17:35, or 5:35 p.m. 24. (3) Convert the time driven each day to minutes: 6 hours 45 minutes = 405 minutes. Multiply 405 minutes by 5 days: 405 minutes × 5 = 2025 minutes. Divide the total number of minutes by 60 to determine the number of hours
driven: 2025 ÷ 60 = 33.75 hours The figure 0.75 hours equals 45 minutes, so the driver drove for 33 hours and 45 minutes. 25. (3) Multiply 1 hour and 20 minutes by 7 days = 7 hours 140 minutes. Convert 140 minutes into hours. 140 ÷ 60 = 2 hours 20 minutes. Add the converted hours to the answer. 7 hours + 2 hours + 20 minutes = 9 hours 20 minutes.
GEOMETRY When you take the GED, you will be provided with a page of formulas that you will need to complete the geometry and algebra questions on the test. You do not need to memorize the formulas since they will be given to you, but you do need to be familiar with the process for using formulas.
Geometric Concepts Angles Angles are indicated by the symbol . They are measured in degrees (°). The letter “m” is used to indicate the measure of an angle. The lines that form an angle are called rays. The rays of an angle meet at a point called the vertex. Angles are named by letters. Look at the angle on the following page. The name of this angle is ABC. This angle is called a right angle and it measures 90°. The small square drawn in the angle indicates that it is a right angle. The two lines that meet to form a right angle are perpendicular to each other. An angle that measures less than 90° is called an acute angle. VWX below is an acute angle. An angle that measures more than 90° is called an obtuse angle. EFG below is an obtuse angle.
A straight angle measures 180°. XYZ is a straight angle. Two angles whose measurements add up to 180° are called supplementary angles. The diagram below shows how DEF and FEG are supplementary angles. Two angles are called complementary angles when their measurement adds up to 90°. In the diagram below, m ABC = 90°. ABE and CBE are complementary because they add up to 90°. In geometry, when a set of points makes up a flat surface, it is referred to as a plane. When two lines in the same plane never meet, no matter how far they are extended, they are considered to be parallel lines. Because parallel lines never meet, they cannot form angles. If those two lines are intersected by a third line, however, eight angles are formed. The line that intersects two parallel lines is called the transversal. Look at the diagram below.
In the diagram, you can see that eight angles have been formed. The four acute angles ( 2, 3, 6, and 7) are equal in measurement, and the four obtuse angles ( 1, 4, 5, and 8) are also equal in measurement. The sum of any acute angle plus any obtuse angle, in this picture, equals 180°. If the transversal intersects two parallel lines perpendicularly, all eight angles that are formed are right angles. Triangles A triangle is a shape that is made up of three angles. The sum of the three angles that make up a triangle is 180° no matter how large or what the shape the triangle is. The diagram below shows the three types of triangles with which you will be working. The isosceles triangle has two sides of equal length. B and C have the same measurement. and are the same length. In the equilateral triangle, all three angles have the same measurement, and all three sides are the same length. In the
right triangle, one angle is a right angle, and the other two angles are acute angles. The longest side of a right triangle (in this case, ) is called the hypotenuse. The Pythagorean Theorem The Pythagorean Theorem is a very important concept in working with right triangles, and you will no doubt be asked to use it in solving problems on the GED. The formula for the Pythagorean Theorem is c 2 = a 2 + b 2 . You do not have to memorize this formula because it will be on the formula page you will be given with the GED test, but you should be familiar with how to use it. Look at the example below. In our diagram, we are given the length of line a and line b, but we need to find the length of line c. To do this, we will use the Pythagorean Theorem. If you are working with a problem where you are given the length of the hypotenuse and are asked to find the length of one of the other lines, you can do this by adjusting the formula as in the example below.
The order in which the formula is set up is changed slightly to solve this problem. The unknown line (in this case, line b) starts the formula and, instead of adding the two squared numbers, you must subtract them. Congruent and Similar Triangles Two geometric figures that have the same size and shape are said to be congruent. The symbol for congruent figures is ≅. Two figures that have the same shape are called similar. Look at the figure below.
In order to find the length of the corresponding side of the larger triangle, you must set up a proportion. Because the triangles are similar, their sides are in proportion. To solve the proportion, cross multiply. 2x = 120 Now, to find x, divide 120 by 2. 120 ÷ 2 = 60 The unknown side is 60 feet long. Perimeter and Area of Triangles The perimeter of a triangle is the sum of the lengths of the three sides of the triangle. The formula for the perimeter of a triangle is P = a + b + c. The area of a triangle means the space inside the figure. It is equal to one-half the length of any side times the height. The formula for the area of a triangle is A = bh. This means that you would multiply the base times the height, then divide that number by two.
Practice 1. If two angles of a triangle measure 72° and 60°, what is the measure of the third angle? 2. Find the area of the triangle below. 3. A right angle measures how many degrees? 4. What is the measurement of A in the diagram below? 5. What is the perimeter of a triangle whose sides measure 8 feet, 6 feet, and 5 feet? 6. Two angles whose sum is 180° are called what? 7. Find the length of the unknown side of the triangle below.
8. One acute angle of a right triangle measures 60°. What does the other acute angle measure? 9. If the vertex angle of an isosceles triangle measures 98°, what is the measure of each base angle? 10. If the two triangles below are similar, what is the length of the unknown side? D Answers 1. 48° 2. 12 sq in 3. 90° 4. 120°
5. 19 ft 6. supplementary 7. 20 in 8. 30° 9. 41° 10. 3
Circles There are several terms relating to circles that you will need to know to solve some of the geometry problems on the GED. The diameter of a circle is the distance through the center of the circle. The radius is one-half of the diameter. The circumference is the distance around a circle, and the area is the amount of space inside the circle. The diameter and radius are shown in the diagram below. If the diameter of our example circle is 12 inches, then the radius is 6 inches. To find the circumference of our circle, we must follow a formula. The formula for finding the circumference is C = πd. C stands for the circumference, π stands for the Greek letter pi, and d stands for the diameter. Unless you are told to do otherwise, use the number 3.14 as the value of π. Below is the solution to finding the circumference of our example circle. C = πd C = 3.14 × 12 C = 37.68 The formula for finding the area of a circle is A = πr 2 . This means that you will multiply π times the radius squared. A = πr 2 A = 3.14 × 6 × 6 A = 113.04 Other Geometric Shapes Besides triangles and circles, you should be familiar with rectangles and squares. A
rectangle is a figure whose length and width are not the same size, while a square has all sides the same. To find the perimeter of a rectangle, multiply the length by 2, multiply the width by 2, then add the figures together. Because a square has all four sides equal, you simply have to multiply one side times four. Remember that the perimeter is the distance around the outside of the figure. Look at the diagram below. To find the perimeter of this figure, use the formula P = 2l + 2w P = 2l + 2w P = (2 × 30) + (2 × 10) P = 60 + 20 P = 80 ft The area of a square or a rectangle is the space inside the perimeter of a figure. To find the area, simply multiply the length times the width. The formula for finding the area is A = l × w. The area of the figure above is 10 × 30 = 300 sq. feet. Volume is the amount of space a solid or three dimensional figure occupies. Volume is measured in cubic measurement. Look at the following diagram of a rectangular solid. To find the volume, or how much space the rectangular solid occupies, the formula is V = lwh. This means that we must multiply the length times the width times the height. V = lwh
V = 6 × 3 × 4 V = 72 cubic inches To find the volume of a cube, because the length, width, and height are equal, the formula is V = s 3 . Since s is the side, we multiply that figure by itself three times. To find the area of the surface of the solid rectangle, the formula is A = 2lw + 2wh + 2lh. The area of our example can be expressed as follows: A = 2lw + 2wh + 2lh A = 2 × 18 + 2 × 12 + 2 × 24 A = 36 + 24 + 48 A = 108 in A cylinder is another geometric figure you might be asked to work with. In a cylinder, the top and bottom bases are circles that lie in parallel planes. Look at the diagram below. To find the volume of a cylinder, you must multiply the area of one base by the height. The formula for finding the volume of a cylinder is V = πr 2h. V = πr 2h V = 3.14 × 82 × 12 V = 3.14 × 64 × 12 V = 2411.52 cu in
Practice 1. Find the area of the floor of a room that is 6 ft long and 8 ft wide. 2. If the diameter of a circle is 46 inches, what is its radius? 3. What is the area of a circle that has a diameter of 6 feet? 4. A rectangular solid measures 8 in wide by 6 in long by 12 in high. What is its surface area? 5. What is the volume of the rectangular solid described above? 6. What is the volume of a cylinder that measures 18 inches high and has a diameter of 9 inches? 7. What is the volume of a cube with sides that measure 4 feet? 8. If a circle has a diameter of 16 inches, what is its circumference? 9. What is the perimeter of a rectangle whose length is twice its width and whose width is 8 feet? 10. If the circumference of a circle is 25.12 feet, what is its diameter? Answers 1. 48 sq ft 2. 23 in 3. 28.26 sq ft 4. 432 sq in 5. 576 cu in 6. 1144.53 cu in
7. 64 cu ft 8. 50.24 in 9. 48 ft 10. 8 ft
COORDINATE GEOMETRY (DISTANCE AND SLOPE) Finding points on a plane is the study of coordinate geometry. A grid is commonly used to do this. Grids are divided into four sections, and each section is called a quadrant. The two number lines that divide the grid into quadrants are called the xaxis and the y-axis. The points that are drawn on the grid are called ordered pairs. The order in which the pairs of numbers are written will determine where on the grid the numbers are placed. The x, or horizontal coordinate, is always written first. This number is separated from the y coordinate by a comma. Both numbers in a pair are written in parentheses. Look at the grid below. http://bit.ly/hippo_alg30 The ordered pair for point A is (4,5). Starting at the zero in the middle of the grid, count over four squares to the right on the x-axis. This gives you the coordinate for the first number of the pair. Now, count up 5 squares on the y-axis. Look at point C on the grid. What is the ordered pair to express point C’s location? Because you must count two squares to the left of the zero and four squares below the zero, the ordered pair is (–2,–4).
Distance Finding the distance between two points that are directly horizontal or vertical from each other is simply a matter of counting the number of squares that separate the points or subtracting the lesser number from the greater number. If you are asked to find the distance between two points, you use the Pythagorean theorem. Look at the grid below. The distance between points A and B is 5. The distance between points D and C is also 5. To find the distance between points E and F, first determine the distance between E and G. Then figure the distance between F and G. The distance between E and G is 4, and the distance between F and G is 3. When you set up the formula for the Pythagorean theorem, c will be the distance between E and F. The solution for finding the distance between E and F is shown below.
The distance between points E and F is 5.
Slope The slope of a line is the distance that the line rises between two points on the line. The formula for determining the slope of a line is http://bit.ly/hippo_alg26 The formula indicates that the slope (m) is the change in the y coordinates divided by the change in the x coordinates of the two points. Look at the grid below. http://bit.ly/hippo_alg27 The coordinates of point A are (2,1), the coordinates of point B are (5,1), and the coordinates of point C are (5,8). The solution to finding the slope of is below.
Practice Using the grid below, answer the following questions. 1. What are the coordinates for point A? 2. The coordinates for point B are? 3. What is the distance between points A and B? 4. What are the coordinates for point C? 5. (–3,4) are the coordinates for what point? 6. (2,4) are the coordinates for what point? 7. (2,–8) are the coordinates for what point? 8. What is the distance between points D and F? 9. What is the slope of ?
10. What is the distance between points G and D? Answers 1. (–5,–3) 2. (–5,3) 3. 6 4. (2,–5) 5. D 6. E 7. F 8. 13 9. 1 10.
Test Yourself 1. A new rectangular swimming pool at the community center measures 12 feet by 20 feet. How many square feet does the pool take up? (1) 24 sq ft (2) 32 sq ft (3) 64 sq ft (4) 120 sq ft (5) 240 sq ft 2. If the pool measures 12 feet by 20 feet, and a decorative border of tiles runs around the inside of the pool’s edge, how many feet long is the decorative border? (1) 24 ft (2) 32 ft (3) 48 ft (4) 64 ft (5) 120 ft Items 3–6 refer to the following diagram.
3. Find the length of in the above triangle. (1) 5 (2) 6 (3) 8 (4) 12 (5) 20 4. Find the area of ACE. (1) 30 (2) 60 (3) 120 (4) 144 (5) 240 5. Find the area of BDE. (1) 30 (2) 48 (3) 90 (4) 140 (5) 260 6. Find the length of . (1) 13 (2) 17 (3) 18 (4) 20
(5) 26 Items 7–11 refer to the following diagram. 7. What is the volume of the can shown above? (1) 24 cu in (2) 48 cu in (3) 192.90 cu in (4) 204.24 cu in (5) 301.44 cu in 8. What is the area of one of the circular ends of the can? (1) 4π sq in (2) 8π sq in (3) 16π sq in (4) 24π sq in (5) 36π sq in 9. What is the circumference of the can? (1) 2π in (2) 8π in
(3) 12π in (4) 36π in (5) 48π in 10. If the can has a rectangular label that wraps all the way around it, what is the area of that label? (Hint: The height of the label is the height of the can, and the width of the label is the circumference of the can.) (1) 16π sq in (2) 24π sq in (3) 36π sq in (4) 48π sq in (5) 96π sq in 11. What is the total surface area of the can? (Hint: The total surface area is twice the area of one of the circular ends, plus the area of the label.) (1) 48π sq in (2) 64π sq in (3) 80π sq in (4) 112π sq in (5) 128π sq in 12. What is the surface area of a cube that measures 2 inches on each side? (1) 4 sq in (2) 8 sq in (3) 12 sq in (4) 24 sq in (5) 48 sq in 13. The diameter of a circular playground is 60 feet. If Anthony walks around
the playground twice, how many feet will he have walked? (1) 120 ft (2) 188.4 ft (3) 300.8 ft (4) 376.8 ft (5) 396.7 ft Items 14–17 refer to the diagram below. Lines a and b are parallel. 14. If 5 measures 140°, what is the measurement of 6? (1) 25° (2) 40° (3) 45° (4) 60° (5) 90° 15. Which angle is equal to 1, 4, and 5? (1) 2 (2) 3
(3) 6 (4) 7 (5) 8 16. What is the sum of 6 and 8? (1) 120° (2) 140° (3) 180° (4) 260° (5) 360° 17. If 5 measures 140°, then 1 + 3 + 4 = (1) 160° (2) 180° (3) 240° (4) 260° (5) 320° Items 18–20 refer to the following diagram. 18. The room shown above will be carpeted with wall-to-wall carpeting. In square yards, how much carpet will be needed?
(1) 12 (2) 18 (3) 21 (4) 29 (5) 32 19. Two adjacent walls in the room are to be wallpapered. What is the approximate area, in square yards, of the two walls? (1) 2.4 (2) 10.8 (3) 14.6 (4) 18.9 (5) Not enough information is provided. 20. A strip of molding will be placed all along the edges of the ceiling. How many yards of molding will be needed? (1) 7 (2) 12 (3) 14 (4) 21 (5) 42 Items 21 and 22 refer to the following diagram.
21. What is the measurement of AFD? (1) 30° (2) 45° (3) 60° (4) 90° (5) 180° 22. Which angle is complementary to EFC? (1) BFA (2) DFA (3) AFC (4) BFC (5) DFC Items 23–25 refer to the following grid.
23. What are the coordinates of point B? (1) (4,2) (2) (2,4) (3) (–2,4) (4) (–2,–4) (5) (–4,–2) 24. What is the distance between points A and C? (1) 4 (2) 5 (3) 6 (4) 7 (5) 8
25. What is the distance between points B and C? (1) 8 (2) (3) (4) 16 (5) Not enough information is provided. Answers 1. (5) Multiply the length by the width: 12 × 20 = 240 2. (4) Use the perimeter formula: 2l + 2w = 2(20) + 2(12) = 64 3. (1) Set up a proportion for similar triangles = 4. (3) bh = (24)(10) = 120 5. (1) bh = (12)(5) = 30 6. (1) Use the Pythagorean Theorem:
7. (5) Use the formula V = πr 2h V = 3.14 × 4 2 × 6 V = 3.14 × 16 × 6 V = 301.44 cu in 8. (3) Use the formula A = πr 2 A = π4 2 A = 16π 9. (2) Use the formula C = 2πr C = 2π(4) C = 8π 10. (4) The circumference is 8π and the height is 6. 6 × 8π = 48π 11. (3) The area of one of the ends is 16π; the area of the label is 48π. (16π × 2) + 48π = 80π 12. (4) The surface area formula is SA = 2lw + 2wh + 2lh, and because all the sides are the same in a cube, l = w = h = 2. SA = 2lw + 2wh + 2lh SA = 2(2 2 ) + 2(2 2 ) + 2(2 2 ) SA = 24 13. (4) Use the formula C = πd. C = 3.14 × 60 C = 188.4 Because he walked around the playground twice, you must multiply 188.4 by 2 = 376.8.
14. (2) 180° – 140° = 40° 15. (5) Vertical angles are equal. In this figure, 8 = 5 = 4 = 1. 16. (3) The two angles lie on a straight line, so their sum is 180°. 17. (5) 1 + 3 + 4 = 140° + 40° + 140° = 320° 18. (1) First convert the feet to yards: 9 ft = 3 yds (width) 12 ft = 4 yds (length) Use the formula A = l × w 3 × 4 = 12 square yards of carpet will be needed. 19. (4) Convert the feet to yards h ≈ 2.7 yd l = 4 yd w = 3 yd Use A = l × w to find the area of each wall 2.7 × 4 = 10.8 2.7 × 3 = 8.1 10.8 + 8.1 = 18.9 So, the approximate area of the two walls is 18.9 square yards. 20. (3) Convert the feet to yards l = 4 yd w = 3 yd
Use P = 2l + 2w to find the perimeter of the ceiling: 2(4) + 2(3) = 14 So 14 yards of molding will be needed. 21. (4) A right triangle measures 90°. 22. (3) AFC. The sum of complementary angles is 90°. 23. (5) Point B is at (–4,–2). 24. (4) The distance between points A and C is 7. 25. (3) Use theg> Use the formula c 2 = a 2 + b 2 . c 2 = 6 2 + 7 2 (the distance between A and B is 6 and the distance between A and C is 7) c 2 = 36 + 49 c 2 = 85 c =
ALGEBRA In algebra, numbers are sometimes replaced by letters. These letters are called variables. You have already worked with variables in some of the problems you solved in previous sections when you were working with basic math. For example, when you were asked to solve the problem 6 +? = 13, the variable was the question mark. To write that problem as an algebraic expression, you would write 6 + x = 13. In the expression 5x, the x is the variable and the 5 is called the coefficient. Look at the term below: 4a + 6a + 2b + 2a − x http://bit.ly/hippo-alg15 The first thing to do is to simplify the problem. There are three a combinations, one b, and one x. To simplify, you combine all the same letter terms. After simplifying, the problem looks like this: 12a + 2b − x http://bit.ly/hippo_alg16 A division problem in algebra is usually expressed using the fraction symbol. For example, instead of writing 12 ÷ x = 6, the problem would be written = 6. The multiplication symbol is not used in algebra. Instead, a multiplication problem would be written in one of three ways: • The numbers and variables are written with nothing between them: 12xy • The numbers are written with a raised dot between them: 12 · x · y • Each number, variable, or group of numbers and letters are written in parentheses: (12)(x)(y) Parentheses are also used in algebraic expressions to indicate the order of the procedure to follow to solve the problem. Look at the problem below: 5(6 + 10) In this problem, you would first add the numbers that are in parentheses, then
multiply the sum by 5. 5(16) = 80 To combine like terms when there are exponents and you are multiplying, the exponents are added together. In the term (3x 2 )(4x 2 ), the solution would be 12x 4 . To combine like terms where there are exponents and you are dividing, the exponents are subtracted. In the term , the solution would be 5x. 15 ÷ 3 = 5 and when x 3 is divided by x 2 , the answer would be x 1 , which is simply written as x. y 2 ÷ y2 = 1, so that variable is eliminated from the equation. Factoring is the process by which you divide common elements out of an expression and rewrite the expression in a simplified way. Look at the expression below: 3a 2b – 2a 2c In this expression, the common element on either side of the minus sign is a 2 , so that is simplified by only writing it one time as follows: a 2 (3b − 2c)
Practice Factor the following expressions 1. 6a + 6b 2. 4x 2 + 4y 2 3. x 2 − xy 4. 2xy + 6ab Simplify the following terms 5. 4a − 2a + −3a + 3b 6. 4x 3 − x + 3b 7. 8. (6x 3 )(3x 2 ) Answers 1. 6(a + b) 2. 4(x 2 + y 2 ) 3. x(x − y) 4. 2(xy + 3ab) 5. −a + 3b 6. 4x 3 − x + 3b 7. 3a
8. 18x 5