Topics Reading: Four out of Five Doctors Recommend... 2 Naming Fractions... 3 Converting Improper Fractions to Mixed Numbers... 3 Converting Mixed Numbers to Improper Fractions... 3 Equivalent Fractions... 4 Comparing Fractions... 4 Reading: Prime and Composite Numbers... 5 Prime Factorization... 6 Greatest Common Factor (GCF)... 6 Simplifying (Reducing) Fractions... 6 Multiplying Fractions... 7 Dividing Fractions... 7 Multiplication and Division Word Problems... 8 Least Common Multiple (LCM)... 8 Adding/Subtracting Fractions... 8 Addition and Subtraction Word Problems... 9 Exponents and Square Roots with Fractions... 10 Order of Operations with Fractions... 10 Word Problems... 11 Fraction Word Puzzle... 11 1 K r u c z i n s k i P r e - G E D M A T H R E V I E W
Reading: Four out of Five Doctors Recommend You probably seen many advertisements for medicines. The ads try to convince you to buy the medicine. They may say, for example, that four out of five doctors recommend a particular headache product. What does this really mean? Exactly how many doctors like the headache medicine? To understand the math behind advertising claims like these, you need to think about how fractions work. Let s say 500 doctors are in a survey, and 400 of them like the medicine. Then the fraction 400 shows 500 what part of the whole group recommends the product. This fraction is equal to 4. But so are the 5 fractions 40 4000 40000 and and even. So you cannot really tell how many doctors were questioned in the 50 5000 50000 survey. All you know is that four out of five said the medicine was good. So why do they not tell you how many doctors were asked? They could say, We asked 500 doctors, and 400 of them say the medicine is great. But they do not. Instead they write Four out of five doctors recommend our medicine. You are not told the actual number of doctors in the survey. Maybe they only asked 50. But by telling you four out of five, they may hope you will think that thousands of doctors were questioned. Next time you see an ad that tells you four out of five doctors like a medicine, stop and guess how many doctors that is. You might even write a letter or email to find out the actual number. Reading Check 1. Mark the following statements as the main idea, too broad, or too narrow. a. Advertisers use numbers like four out of five to mislead people. b. The fraction 40 50 is equal to 4 5. c. Fractions are sometimes confusing. 2. Another good title for this passage is a. Reading Labels on Medicine Bottles. b. Why Doctors Recommend Certain Medicines. c. Fractions Always Tell the Truth. d. How Fractions are Used in Advertising. 3. A statement like four out of five a. always means 40 out of 50 people. b. does not tell exactly how many people were counted. c. is the fairest way to explain a survey. d. should convince you to buy certain medicines. 4. The author of this passage thinks you should a. think carefully about ads you hear. b. realize that all ads are full of lies. c. get better at doing math. d. realize that advertisers do not want to mislead people. 5. The final paragraph of the passage is intended as a a. criticism of people who do not write letters. b. comparison. c. recommendation. d. joke. 6. In this passage, the word claims means a. statements that something is true. b. fractions. c. pieces of land belonging to a person. d. lies. 2 K r u c z i n s k i P r e - G E D M A T H R E V I E W
Naming Fractions Part 1: Write each fraction as a word statement and state if they are proper, improper, or a mixed number. a) 1 5 b) 9 5 c) 7 1 3 d) 8 15 e) 4 5 7 f) 11 3 g) 17 20 h) 8 3 Part 2: Write each word statement as a fraction and state if they are proper, improper, or a mixed number. Do not simplify. a) nine-seventieths b) seven-thirds c) sixteen and one-quarter d) eleven-halves e) thirty-five and ten-elevenths f) nine-thousandths g) eighty-three-quarters h) forty one-seventieths Converting Improper Fractions to Mixed Numbers Directions: Convert the following Improper Fractions to Mixed Numbers. a) 8 3 b) 10 7 c) 40 9 d) 11 5 e) 14 5 f) 32 11 g) 12 7 h) 50 33 Converting Mixed Numbers to Improper Fractions Directions: Covert the following Mixed Numbers to Improper Fractions. a) 1 3 8 b) 2 1 4 c) 1 12 17 d) 3 2 5 e) 4 1 2 f) 6 3 5 g) 1 2 15 h) 5 1 8 3 K r u c z i n s k i P r e - G E D M A T H R E V I E W
Equivalent Fractions Part 1: Find the equivalent fraction of the following with the given numerator or denominator. a) 1 5 =? 45 b) 49? = 7 6 c)? 32 = 3 8 d) 9 = 27 16? e)? 48 = 6 12 f) 11 15 = 44? g) 5 9 =? 63 h) 24? = 8 13 i) 1 4 =? 28 j) 3 7 = 27? k) 6 5 =? 25 l) 6 12 = 12? Part 2: Find the equivalent fraction of the following with the given numerator or denominator. 1. Write three equivalent fractions for 2 that have 10, 25, and 40 as denominators. 5 2. Write three equivalent fractions for 4 that have 8, 12, and 20 as numerators. 9 3. Barnaby had 30 math questions to do for homework. If he was able to only complete 5 of his 6 homework assignment for math, how many math questions did he complete? 4. In Mr. Kruczinski s math class 3 of his students finished their homework. If there are 25 students 5 in the class, how many students completed their homework? Comparing Fractions Part 1: Compare the following numbers, by placing =, >, or < between the two numbers. a) 1 6 1 12 b) 3 7 3 9 c) 4 9 1 2 d) 7 15 28 60 e) 3 7 3 14 f) 5 6 7 12 g) 1 9 3 27 h) 5 4 16 20 Part 2: Answer the following questions. 1. State which of the following fractions is larger. 1 or 1 9 6 2. State which of the following fractions is smaller. 4 or 1 25 5 3. Order the following fractions from greatest to least. 1, 7, 1 and 4 60 20 6 15 4. Order the following fractions from least to greatest. 1/8, 2/5, 5/8, 5/6, 3/49, 3/56, 4/7, 3/10 Part 3: Answer the following question with the diagram below. Sally thought she discovered that sometimes ¼ really is bigger than ½. She draws the following diagrams to show what she found. Is Sally s reasoning correct? If not, what is her misunderstanding? 4 K r u c z i n s k i P r e - G E D M A T H R E V I E W
Reading: Prime and Composite Numbers One important way of classifying a whole number is by whether it is prime or composite. A prime number is a whole number greater than 1. It can have only two factors, or numbers that can be multiplied to produce it. These are the number itself and the number 1. The number 6 is not prime because it has the factors 1, 2, 3, and 6. If a number is not prime, like the example number 6, then it is composite it is composed of many factors. To identify all the prime numbers less than 100, start with a grid. Shade in all the multiples of twos, such as 4, 6, as so on (as shown to the right). Shade in the multiples of 3, 5, and 7. The unshaded numbers are prime numbers. You will see from your diagram that a prime number greater than 2 has to end in the digit 1,3,7, or 9. Other than that, there is not any obvious pattern to the frequency of the primes. In fact, many renowned mathematicians have been fascinated with prime numbers, wondering if there is a complex pattern to their occurrence. Prime numbers have recently been put to an important use. Let s say a very large number, one with 120 digits, has only four factors: 1, itself, and two roughly equal prime numbers. (Those prime numbers might have 40 or 50 digits themselves.) Mathematicians believe that figuring out what those factors are can take years, even on the fastest computers. So these sorts of numbers are used in creating a procedure called oblivious transfer, a method of providing secure internet transactions. Reading Check 1. Mark the following statements as the main idea, too broad, or too narrow. a. The only even prime number is 2. b. Prime numbers are part of mathematics. c. Mathematicians consider prime numbers both interesting and useful. 2. The passage is mainly concerned with a. prime numbers and their use. b. prime numbers between 1 and 200. c. whether there is a pattern in how often prime numbers occur. d. composite numbers and their use. 3. The number 6 is not a prime number because it a. is less than 100. b. is an even number. c. has other factors beside itself and 1. d. is not a whole number. 4. This passage leads the reader to conclude that a. there are no prime numbers over 100. b. prime numbers can be very large. c. prime numbers cannot end with a digit of 3. d. mathematicians have identified every prime number that exists. 5. The diagram helps show that prime number a. are only even numbers. b. are greater than 2. c. require multiplication skills. d. Have no obvious repeating pattern. 6. In this passage, the word secure means a. unlikely. b. simple. c. safe. d. inexpensive. 5 K r u c z i n s k i P r e - G E D M A T H R E V I E W
Prime Factorization Directions: Find the prime factorization of the following numbers without a calculator. If a number is prime, state that the numer is prime. a. 8 b. 50 c. 72 d. 89 e. 180 f. 126 g. 100 h. 64 i. 12 j. 27 k. 77 l. 17 Greatest Common Factor (GCF) Directions: Find the Greatest Common Factor of the following pairs of numbers. a. 6 and 8 b. 15 and 55 c. 9 and 18 d. 5 and 7 e. 12 and 20 f. 18 and 42 g. 44 and 55 h. 25 and 225 i. 2 and 7 j. 9 and 81 k. 4, 7, and 9 l. 12, 27, and 36 Simplifying (Reducing) Fractions Part 1: Find the GCF of the numerator and denominator and then simplify the fraction. You MAY NOT use a calculator. a. 9 12 b. 15 18 c. 20 42 d. 25 50 e. 17 49 f. 15 20 g. 21 49 h. 63 70 i. 6 12 j. 12 18 k. 20 32 l. 225 550 m. 21 63 n. 30 45 o. 21 48 p. 26 69 Part 2: Solve the following word problems. Write your answer as an integer or a simplified fraction. 1. There are 5280 feet in a mile. What fraction of a mile is represented by 160 feet? 2. A wall is 20 inches wide. Six inches of the wall is concrete, 8 inches is brick, and 6 inches is limestone. What fraction of the wall is concrete? 3. A work shift for an employee at a restaurant consists of 10 hours. What fraction of the employee s work shift is represented by 6 hours? 4. A company operates stores under multiple banners in 16 states in the United States of America. a. How many states do not have a store operated by the company? b. What fraction of the states do not have a store operated by the company? 6 K r u c z i n s k i P r e - G E D M A T H R E V I E W
Part 3: Read each question. Then use the drag and drop options to compete each answer. The final answer must be simplified. 1. 2. 3. 4. Multiplying Fractions Directions: Perform the indicated operation. Write your answer below the question. Simplify the result if possible. You MAY NOT use a calculator for the problems below. a. 2 3 7 9 b. 7 10 15 17 c. 7 9 9 10 d. 1 5 1 7 e. ( 3 5 ) (2 4 ) f. 8 8 9 g. 8 15 4 5 h. 3 22 10 11 i. 3 1 2 1 3 j. 3 4 2 1 3 k. (1 2 3 ) (2 2 9 ) l. 2 2 7 3 1 2 Dividing Fractions Directions: Perform the indicated operation. Write your answer below the question. Simplify the result if possible. You MAY NOT use a calculator for the problems below. a. 3 4 7 9 b. 2 5 11 12 c. 6 7 7 10 d. 1 4 1 9 e. 5 6 4 7 f. 12 3 5 g. 7 14 4 9 h. 3 55 10 11 i. 7 13 11 26 j. 6 8 4 k. 3 5 9 l. 1 1 3 8 9 7 K r u c z i n s k i P r e - G E D M A T H R E V I E W
Multiplication and Division Word Problems Directions: State the operation that is required and then solve the problem. Type your answer as an integer or a simplified fraction. You MAY use your calculator with these questions. 1) What is two-eighths of five-sixths? 2) Peter's truck gets him 10 miles per gallon. Suppose Peter's tank is empty and he puts 5 gallons in, how 6 far can Peter go with the truck? 3) Babette must cut pipes into lengths of 5/6 of a foot. How many pipes can she make from a pipe that is 20 feet long? 4) At a certain factory, products are made in vats that have the capacity to hold 200 quarts. If each bottle of a product contains 2/25 of a quart, how many bottle can be made from each vat? Least Common Multiple (LCM) Directions: Find the Least Common Multiple (LCM) of the following pairs of numbers. a. 9 and 12 b. 15 and 60 c. 7 and 12 d. 5 and 8 e. 14 and 20 f. 10 and 15 g. 4 and 16 h. 6 and 14 i. 3 and 7 j. 9 and 81 k. 3, 5, and 8 l. 12, 27, and 36 Challenge Problems Directions: State if the following statements are true or false. 1. Five is a multiple of ten. TRUE or FALSE 2. Twelve is a multiple of two. TRUE or FALSE 3. Eight is a factor and a multiple of eight. TRUE or FALSE Adding/Subtracting Fractions Part 1: Perform the indicated operation. Write your answer below the question. Simplify the result if possible. You MAY NOT use a calculator for the problems below. a. 1 5 + 3 5 b. 2 3 + 4 11 c. 5 8 + 7 10 d. 1 7 + 1 14 e. 5 12 + 2 4 f. 5 9 + 7 18 g. 1 6 + 5 24 h. 5 1 2 + 10 11 i. 7 16 3 16 j. 5 8 5 12 k. 9 12 3 4 l. 12 55 2 11 m. 3 4 1 8 n. 1 2 7 16 o. 23 24 5 8 p. 6 1 3 8 K r u c z i n s k i P r e - G E D M A T H R E V I E W
Part 2: Answer the following questions. Jack tries to add ⅙ +⅙ using the following model and says: These both represent ⅙ since one out of size parts is shaded. Now, if we put these pictures together we have two out of twelve squares shaded, so ⅙ +⅙ =2/12. What is Jack s misconception and how do you help him to understand? Addition and Subtraction Word Problems Part 1: State the operation that is required and then solve the problem. Type your answer as an integer or a simplified fraction. You MAY use your calculator with these questions. 1) What is the difference of two-eighths and fivesixths? 2) Joshua went out to eat and ate four-fifths of his meal How much of his meal does Joshua have left? 3) A person bought 3/5 pound of peanut butter fudge and 24/25 pound of fudge with nuts. How many pounds of fudge did the person buy? 4) Mercedes walked 4/5 of a mile before lunch and 2/3 of a mile after lunch. How many miles did she walk in all? Part 2: Use the table below to answer the following questions. Read each question. Then use the drag and drop options to compete each answer. 9 K r u c z i n s k i P r e - G E D M A T H R E V I E W
Exponents and Square Roots with Fractions Directions: Perform the indicated operation. Write your answer below the question. Simplify the result if possible. You MAY NOT use a calculator for the problems below. a. ( 1 5 ) 3 b. ( 3 7 ) 2 c. ( 6 5 ) 2 d. ( 2 3 ) 3 e. ( 1 5 ) 6 f. ( 3 2 ) 5 g. 81 144 h. 16 25 i. 8 32 j. 27 12 k. 80 125 l. 81 16 Order of Operations with Fractions Directions: Perform the indicated operation. Write your answer below the question. Simplify the result if possible. You MAY NOT use a calculator for the problems below. a. ( 1 5 1 10 ) + 3 5 b. 5 7 (3 5 + 1 5 ) c. 5 8 (16 25 ) (2 3 ) d. ( 1 7 + 1 14 ) 1 2 e. ( 1 12 1 6 ) 2 f. 5 4 3 2 1 2 + 3 4 g. 3 8 3 4 2 7 h. 23 + 8 5 8 4 Challenge Problems Directions: Examine the information. Then read each question, and use the drag and drop options to complete each answer. The distributive law of addition and subtraction is shown by: [(3)(5) + (3)(7)] = (3)(5 + 7). 10 K r u c z i n s k i P r e - G E D M A T H R E V I E W
Word Problems Directions: State the operation that is required and then solve the problem. Type your answer as an integer or a simplified fraction. You MAY use your calculator with these questions. 1) What is one-fourth of a half? 2) How many one-thirds are there in twenty-sevenninths? 3) My recipe calls for 2 cups of white flour and ¼ cups 3 of whole wheat flour. How much flour do I need in total for my recipe? 4) A mother and her teenage son are painting their home. In one day, the mother completes 1/4 of the job and the son completes 1/5 of the job. How much more of the job did the mother complete than her son? 5) An airplane covers 50 miles in one-fifth of an hour. What is the airplanes average speed in miles per hour? 6) Business people have determined that 2 of the 3 items on a mailing list will change in one year. A business has a mailing list of 2121 people. After one year, how many addresses on that list will be incorrect? 7) A sidewalk is built 10 bricks wide by laying each brick side by side. How many inches wide is the sidewalk if each brick measures 2 3 inches wide? 8 8) A recipe for banana oat muffins calls for ¾ cup of old fashioned oats. You are making ½ of the recipe. How much oats should be used? 9) An Italian sausage is 8 inches long. How many pieces of sausage can be cut if each piece is to be two-thirds of an inch? 10) A recipe needs one-fourth tablespoon salt. How much salt does 8 such recipes need? 11) In a certain country, 1/5 of college freshmen major in chemistry. A community college in a region of the country has a freshman enrollment of approximately 900 students. How many of these freshmen might we project are majoring in chemistry? 12) After building a house the homeowners discover the land is sinking three-eighths of an inch every year. How many years will it take the house to sink 24 inches? 13) Find the quotient of 1 2 and 2 3. 14) Sally walked 3/4 of a mile before lunch and 1/2 of a mile after lunch. How many miles did she walk in all? 15) Find one-half of 90. 16) Jim bought a candy bar and ate 2/3 of it. How much of the candy bar does Jim have left? 17) A chewy fruit bar recipe calls for 15/16 cup of brown sugar and 5/8 cup of granulated sugar. How many more cups of brown sugar than granulated sugar are in the recipe? Fraction Word Puzzle Directions: Read each clue and follow the instructions to form words. You will end up with a question that needs to be answered. 1. Write the first 2/3 of who and the first 2/9 of attention. 2. Write the last ½ of this. 3. Write 3/3 of the. 4. Write the first ¼ of tomorrow and the second ¼ of open. 5. Write the first 2/5 of offer. 6. Write the third 1/7 of teacher. 7. Write the first 3/8 of fragment, the fourth 1/7 of discuss, and the last 4/10 of population. 8. Write the first 1/5 of calculator, the last 2/3 of all, and the last ¼ of compiled. 11 K r u c z i n s k i P r e - G E D M A T H R E V I E W