Number Systems and Fractions
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1 Number Systems and Fractions Section 1: Fractions A. Adding and Subtracting Fractions The rule: When adding or subtracting fractions find a common denominator in order to add or subtract the fractions. Then add the numerators and keep the denominators. Reduce the fraction if necessary. ( ) ( )
2 B. Multiplying and Dividing Fractions Multiplication of fractions rule: When multiplying fractions, multiply the numerators and the denominators straight across. Reduce the fraction if necessary. Division of fractions rule: When dividing fractions follow an acronym called keep, change, flip. This means that you keep the first fraction, change the division sign to multiplication, and flip the second fraction. Then follow the multiplication rule
3 Section 2: Statistics and Probability A. Mean: To find the mean, add up the sum of the numbers and then divide by how many numbers there are. Example: Find the mean of: 5,7,8,5,5 Answer: 6. The sum of the numbers is equal to 30. Now divide by 5 since there are 5 numbers. 1. Find the mean: 4, 2, 5, 6 = 2. Find the mean: 110, 202, 5, 7, 3 = 3. Find the mean: 4, 2, 7, 3, 10, 2 = 4. Find the mean: 1000, 2, 5200, 4 = 5. Find the mean: 12, 10, 5, 11, 3, 4, 8 = B. Median: To find the median, line the data up from least to greatest. The median is the number that appears in the middle of the data set. If there is no number in the middle of the dataset because there is an even amount of numbers, add the two numbers closest to the middle and then divide by two. Example 1: Find the median of: 10, 2, 3, 5, 1 Answer: 3. Reorder to 1, 2, 3, 5, falls directly in the middle of the dataset. Example 2: Find the median of: 8, 12, 6, 4, Answer: 7. Reorder to 4, 6, 8, 12. There is an even amount of numbers in the set. 6 and 8 are the two numbers closest to the middle of the data set. Now take the sum of and divide by two. Your answer will be Find the median: 4, 7, 1, 3, 9 2. Find the median: 8, 5, 0, 2, 10, 10, 3 3. Find the median: 9, 12, 5, 2, 1 4. Find the median: 3, 4, 5, 6, 5, 9, 5, Find the median: 1000, 1002, 1092, 2000, 5000 C. Mode: The mode is the number that appears the most in the dataset. There may be more than one mode or no mode if each number appears only once. Example: Find the mode of: 4, 6, 7, 8, 7, 2, 3 Answer: 7; The number 7 appears twice in the data set. 1. Find the mode: 2, 4, 6, 6, 8, 8, 8, 10, Find the mode: 1, 2, 3, 9, 2000, 9, 2, 9, 1 3. Find the mode: 18, 9, 4, 204, 1, 6, 4, 2, 4 4. Find the mode: 3, 5, 9, Find the mode: 7, 8, 20, 10, 20, 20, 7
4 Number Systems and Fractions Answers to select problems Section 1: Fractions A. Adding and Subtracting Fractions B. Multiplying and Dividing Fraction Section 2: Statistics and Probability A. Mean B. Median C. Mode No mode
5 Ratios, Rates & Proportions 1. a. Write a ratio for the shaded squares to the unshaded squares. b. Write a ratio for the unshaded squares to the total number of squares. 2. a. Which shapes are in the ratio of 3:2? b. 5:15 is the ratio for to total shapes. 3. a. Two trucks can carry 6 boxes. Write this ratio using the to form. b. There are 30 birds to 3 cages. Write this ratio using the : form. c. Each truck can carry 3 boxes. Write this ratio as a fraction. 4. a. In a deck of 52 cards, one out of every 4 cards is a heart. How many cards are hearts? b. In a deck of cards, there are 3 face cards (Jack, Queen, and King) in each of the 4 suits. c. Express in fraction form the ratio of face cards that are diamonds to the total number of diamonds in the deck. d. Express in fraction form the ratio of the total number of face cards in the deck to the total number of cards in the deck. 5. Mrs. Brown buys packages of pencils with 5 pencils per package. Fill in the chart below to show how many pencils she will get, depending on the number of packages that she buys. Packages Pencils
6 6. The perimeter of a square can be found by multiplying the side S x 4. Fill in the chart below. Graph the side length (x axis) compared to the perimeter (y axis). Label the graph and axes. Side Length (S) Perimeter (P) 7. Using the graph that you made for Problem 6, answer the following questions: A. What is the perimeter of a square with a side of 3.5? B. If the perimeter is 21 units, what would be the length of one side? C. Extend the line on your graph. If the perimeter is 36 inches, what will be the length of the side? D. Mark and label your answers A, B, and C on your graph. 8. Use a tape diagram to answer this question: One recipe makes a batch of 12 cookies. If you need to make 84 cookies, how many batches of cookies do you need to make? 9. Use a double line diagram to answer this question:the speed of a runner is 5 meters every 2 seconds. How many meters will the runner go in 9 seconds? Meters 5 2 Seconds
7 Equivalent Ratios: 10. Which two teachers received equivalent rations of apples from their total number of students? Teachers Received Apples Number of Students Brown 3 27 Thompson 3 48 Smith 2 25 Jones Which two types of cars have equivalent ratios of miles traveled to gallons of gas used? Cars Miles Traveled Gallons of Gas Used Dodge 64 4 Honda 48 3 Fiat 27 3 Toyota Which two recipes have equivalent ratios of cups of flour needed to the number of cookies Recipes Cups of Flour Number of Cookies Sugar 4 32 Macadamia Nut 3 17 Peanut Butter 4 34 Chocolate Chip 5 40 Ratios and Proportions Answers to select problems 1. a. 9: 3 b. 3:12 or 1:4 3. a. 30:3 b. 1/3 c. 2 to 6 5. Packages Pencils a. 14 b or 5 c. 9 units d. Mark your answers and label A, B, C or Dodge and Honda
8 Expressions and Equations Evaluate each expression. 1. t a 6, when t = 4 and a = 4 2., when u = 39 3., when g = a y, when a = 10 and y = 1 Write the phrase as an expression more than x 6. y to the fourth power 7. one-third of a 8. u tripled Solve the one-step equation Simplify the following expressions: 14. 5a + 6a = 15. 3a + a = 16. 8a 3a = a 2a = 18. 9a + 4a = a 7a = 20. 4b + 3b = b 6b = 22. 5b + 9b =
9 Complete the following expressions: Expressions and Equations Answers to select problems /6 5. x /3a or a/3 9. a = m = p = a 17. 8a 19. 4a 21. 6b
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