H.S.E. PREP COURSE @ SEC VERSION 2.0, 2018 MODULE B RATIONALS STUDENT WORKBOOK
H.S.E. PREP COURSE MODULE B: RATIONALS CONTENTS REVIEW... 3 OPERATIONS WITH INTERGERS... 3 DECIMALS... 4 BASICS... 4 ADDING AND SUBTRACTING DECIMALS... 5 MULTIPLYING AND DIVIDING DECIMALS... 5 SCIENTIFIC NOTATION... 6 FRACTIONS... 7 ADDING AND SUBTRACTING FRACTIONS... 8 MULTIPLYING AND DIVIDING FRACTIONS...10 EQUIVALENT RATIONAL NUMBERS...13 CONVERTING FROM A Decimal...13 CONVERTING FROM A Fraction...14 CONVERTING FROM A Percent...15 Extended Practice...16 DATA AND MEASUREMENTS...17 Ratios And Proportions...17 Simple Interest...20 Combinations and Permutations...22 Measurements of Central Tendencies...24 GED MATH FORMULA SHEET...27 2
H.S.E. PREP COURSE @ SEC - MODULE B: RATIONALS REVIEW OPERATIONS WITH INTERGERS 1. 51 (-11) 2. (-7)(6)(-4) 3. 33 + (-48) 4. 240 (-6) 5. -24 * -48 6. -67 1 7. -5 + -13 8. -288-24 9. -18 + -23 + 10 10. -27 - - 27 11. -6-5 -4 12. 15 - -28 63-14 3
H.S.E. PREP COURSE MODULE B: RATIONALS DECIMALS BASICS 1. Write the whole number 3 as a decimal number with one zero after the decimal point. 2. Remove the unnecessary zeros from the following numbers. 002.060 3.50 5.0150 3. Given the decimal number 7.9284 (a) Which digit is in the thousandths place? (b) Which digit is in the tenths place? (c) Which digit is in the hundreths place? (d) Which digit is in the ten-thousandths place? 4. Write the following decimal numbers: (a) three hundredths (b) two tenths (c) seven thousandths (d) forty-eight hundredths (e) two and sixty-five thousandths ROUNDING 5. to the hundredths, 0.1586 6. to the tenths, 2.14512 7. to the thousandths, 1.7493 8. to the whole number, 3.45 4
H.S.E. PREP COURSE @ SEC - MODULE B: RATIONALS ADDING AND SUBTRACTING DECIMALS 1. Add: HINT: LINE THE DECIMAL POINT UP WHEN YOU ADD & SUBTRACT DECIMALS! 2. Subtract: MULTIPLYING AND DIVIDING DECIMALS 3. Multiply. 4. Divide. 5
H.S.E. PREP COURSE MODULE B: RATIONALS SCIENTIFIC NOTATION 1. Length of one year: 2. Speed of light: 3. Mass of the sun: 4. Mass of Earth: 5. Power output of sun: 6
H.S.E. PREP COURSE @ SEC - MODULE B: RATIONALS FRACTIONS 7
H.S.E. PREP COURSE MODULE B: RATIONALS ADDING AND SUBTRACTING FRACTIONS ADDITION AND SUBTRACTION (LIKE DENOMINATORS) 8
ADDITION AND SUBTRACTION (UNLIKE DENOMIMATORS) H.S.E. PREP COURSE @ SEC - MODULE B: RATIONALS 9
H.S.E. PREP COURSE MODULE B: RATIONALS MULTIPLYING AND DIVIDING FRACTIONS 10
H.S.E. PREP COURSE @ SEC - MODULE B: RATIONALS 11
H.S.E. PREP COURSE MODULE B: RATIONALS 12
H.S.E. PREP COURSE @ SEC - MODULE B: RATIONALS EQUIVALENT RATIONAL NUMBERS CONVERTING FROM A DECIMAL DECIMAL TO PERCENT To manually change a decimal to a percent, multiply by 100 and add the percent sign (%). Example, 0.35 multiple by 100, 0.35 100 = 35 add percent sign, 35% OR move the decimal point 2 places to the right and add the percent sign (%). Example, 0.35 move decimal point 2 places, 0.35 = 35. add percent sign, 35% When using the TI-30XS Multiview, enter the decimal then the percent convert button. For example, to convert the decimal 0.35 to a percent enter the following: DECIMAL TO FRACTION To manually change a decimal to a fraction, count the number of decimal places. Drop the decimal point then place the result over 1 followed by the same number of 0 s as decimal places. Write the answer in the lowest terms (reduce). Example, 0.35 35 7 drop decimal and place as fraction, reduce to lowest terms, 100 20 *two decimal places, so two zeros When using the TI-30XS Multiview, enter the decimal then the percent convert button. For example, to convert the decimal 0.35 to a fraction enter the following: PRACTICE EXERCISES Convert each decimal to a percent. a) 0.25 = b) 0.5 = c) 0.7 = d) 0.07 = e) 0.45 = f) 0.09 = g) 0.4 = h) 0.375 = Convert each decimal to a fraction. i) 0.3 = j) 0.5 = k) 0.6 = l) 0.02 = m) 0.05 = n) 0.25 = o) 0.36 = p) 0.125 = 13
H.S.E. PREP COURSE MODULE B: RATIONALS CONVERTING FROM A FRACTION FRACTION TO DECIMAL To manually change a fraction to a decimal, divide the numerator by the denominator. Example, 3 5 divide, 3 5 = 0.6 When using the TI-30XS Multiview, enter the fraction then the fraction/decimal convert button. For example, to convert the fraction 3/5 to a decimal enter the following: FRACTION TO PERCENT To manually change a fraction to a percent, first change the fraction to a decimal, then change the decimal to a percent. Example, 3 5 change to decimal by dividing, 3 5 = 0.6 multiple by 100, 0.6 100 = 60. add percent sign, 60% When using the TI-30XS Multiview, enter the decimal then the percent convert button. For example, to convert the fraction 3/5 to a percent enter the following: PRACTICE EXERCISES Convert each fraction to a decimal. a) 10 7 = b) 5 1 = c) 5 2 = d) 4 3 = e) 7 = f) 2 = g) 9 = h) 7 8 3 20 25 = Convert each fraction to a percent. i) 10 1 = j) 5 1 = k) 10 9 = l) 4 3 = m) 4 5 = n) 17 20 = o) 1 3 = p) 2 3 = 14
H.S.E. PREP COURSE @ SEC - MODULE B: RATIONALS CONVERTING FROM A PERCENT PERCENT TO DECIMAL To manually change a percent to a decimal, drop the percent sign (%) and multiply by 0.01 Example, 1.2% drop the percent sign (%), 1.2% = 1.2 multiply by 0.01, 1.2 0.01 = 0.012 OR move the decimal point 2 places to the left and drop the percent sign (%). Example, 1.2% move decimal point 2 places, 1.2% = 0.012% drop percent sign, 0.012 When using the TI-30XS Multiview, enter the decimal then the percent convert button. For example, to convert the percent 1.2% to a decimal enter the following: PERCENT TO FRACTION To manually change a percent to a fraction, change the percent to decimal, then the decimal to a fraction. Example, 1.2% drop the percent sign (%), 1.2% = 1.2 multiply by 0.01, 1.2 0.01 = 0.012 12 drop decimal and place as fraction, 1000 *three decimal places, so three zeros 3 reduce to lowest terms, 250 When using the TI-30XS Multiview, enter the decimal then the percent convert button. For example, to convert the percent 1.2% to a fraction enter the following: PRACTICE EXERCISES Convert each percent to a decimal. a) 3% = b) 30% = c) 25% = d) 80% = e) 8% = f) 12% = g) 67% = h) 17.5% = Convert each percent to a fraction. i) 20% = j) 75% = k) 5% = l) 30% = m) 40% = n) 15% = o) 24% = p) 35% = 15
H.S.E. PREP COURSE MODULE B: RATIONALS EXTENDED PRACTICE EQUIVALENT RATIONAL TABLE Complete the following table by finding the equivalent rational numbers to the ones given. DECIMAL 0.35 1.75 FRACTION 1 5 5 6 PERCENT 12% 37.5% TRUE OR FALSE Determine if the following statements are true or false. Write out the word true or false in the blank. a) 6.35 > 6.7 b) 835% < 0.95 c) 0.32 < 0.5 d) 60% = 0.6 e) The following is in order from least to greatest: 40%, 4 9, 0.52 f) A walk from our classroom to the front office is about 7.48 feet. MATCHING Match the following rational numbers with their equivalent counterpart(s). a) 30% 0.44444 1.25 2/5 b) 2.35 3% 17.5% 30.0 Number Bank c) 12.5% 40% 3/10 1/9 d) 0.65 0.125 23.5% 13/20 e) 4 9 65% 1/8 0.40 f) 2 5 44.44% 6.5% 235% g) list all rational numbers not used 16
H.S.E. PREP COURSE @ SEC - MODULE B: RATIONALS DATA AND MEASUREMENTS RATIOS AND PROPORTIONS WHAT IS A RATIO? A RATIO is a comparison between two quantities. We use ratios everyday; one Pepsi costs 50 cents describes a ratio. On a map, the legend might tell us one inch is equivalent to 50 miles or we might notice one hand has five fingers. Those are all examples of comparisons ratios. A ratio can be written three different ways. If we wanted to show the comparison of one inch representing 50 miles on a map, we could write that as; 1 to 50 or using a colon 1:50 or using a fraction 1 50 WHAT IS A PROPORTION? A PROPORTION is a statement of equality between 2 ratios. To solve problems, most people use either equivalent fractions or cross multiplying to solve proportions. EXAMPLE 5 10 = n 50 If a turtle travels 5 inches every 10 seconds, how far will it travel in 50 seconds? To find n, you must cross multiply. 10n = 5 50 Simplify. 10n = 250 Divide by 10, both sides. The turtle will travel 25 inches in 50 seconds n = 25 STORY PROBLEMS Solve these problems by setting up a proportion. a) If there were 7 males for every 12 females at the dance, how many females were there if there were 21 males at the dance? b) David read 40 pages of a book in 5 minutes. How many pages will he read in 80 minutes if he reads at a constant rate? c) On a map, one inch represents 150 miles. If Las Vegas and Reno are five inches apart on the map, what is the actual distance between them? 17
H.S.E. PREP COURSE MODULE B: RATIONALS d) Bob had 21 problems correct on a math test that had a total of 25 questions, what percent grade did he earn? (In other words, how many questions would we expect him to get correct if there were 100 questions on the test?) e) If there should be three calculators for every 4 students in an elementary school, how many calculators should be in a classroom that has 44 students? If a new school is scheduled to open with 600 students, how many calculators should be ordered? f) If your car can go 350 miles on 20 gallons of gas, at that rate, how much gas would you have to purchase to take a cross country trip that was 3000 miles long? PRACTICE EXERCISES a) Express each ratio as a fraction. 3 to 4 8 to 5 9:13 15:7 b) Express each ratio in simplest form. 12 to 10 24:36 15 18 c) A certain math test has 50 questions. The first 10 are true false and the rest are matching. Find: The ratio of true-false questions to matching questions. The ratio of true-false questions to the total number of questions. The ration of the total number of questions to matching questions. d) A 30 pound moonling weighs 180 pounds on the earth. How much does a 300 pound Earthling weigh on the moon? 18
H.S.E. PREP COURSE @ SEC - MODULE B: RATIONALS e) The ratio of the sides of a certain triangle is 2:7:8. If the longest side of the triangle is 40 cm, how long are the other two sides? f) If a quarterback completes 20 out of 45 passes in his first game, how many passes do you expect him to complete in his second game if he only throws 18 passes? g) On a trip across the country Joe used 20 gallons of gas to go 300 miles. At this rate, how much gas must he use to go 3500 miles? h) On a map 3 inches represents 10 miles. How many miles do 16 inches represent? i) If our class is representative of the university and there are 2 males for every 12 females. How many men attend the university if the female population totals 15,000? j) The ratio of length to width of a rectangle is 8:3. Find the dimensions of the rectangle if the perimeter is 88dm. 19
H.S.E. PREP COURSE MODULE B: RATIONALS SIMPLE INTEREST FORMULA Simple interest is a quick method of calculating the interest charge on a loan. Simple interest is determined by multiplying the daily interest rate by the principal by the number of days that elapse between payments. Using the TI-30XS Multiview, you can enter the Interest Rate as a percent. Just reminder to use the percent symbol! EXAMPLE: Solving for Interest Colin has $60 in a savings account that earns 10% interest per year. The interest is not compounded. How much interest will he earn in 1 year? STEP ONE: Step up the formula. I = P R T $60 = P 10% = R 1yr = T I = $60 10% 1yr STEP TWO: Evaluate using TI-30XS Multiview I = $60 10% 1yr I = $6 Colin will earn $6 in interest. EXAMPLE: Solving for Principal, Rate, or Time Colin has $1100 in a savings account that earned $110 in interest over two years. The interest is not compounded. What is the interest rate for the saving account? STEP ONE: Step up the formula. I = P R T $1100 = P $110 = I 2yrs = T $110 = $1100 R 2yrs STEP TWO: Simplify numerical values by multiplying $1100 and 2yrs. $110 = $2200 R STEP THREE: Divide each side by $2200. $110 $2200 = R 0.05 = R The savings account interest rate is 5% (0.05). 20
H.S.E. PREP COURSE @ SEC - MODULE B: RATIONALS PRACTICE EXERCISES Evaluate for the missing information. a) A bank is offering 2.5% simple interest on a savings account. If you deposit $5000, how much interest will you earn in one year? b) To buy a car, Jessica borrowed $15,000 for 3 years at an annual simple interest rate of 9%. How much interest will she pay if she pays the entire loan off at the end of the third year? What is the total amount that she will repay? [This has two questions!!] c) Nancy invested $6000 in a bond at yearly rate of 3%. She earned $450 in interest. How long was the money invested? d) Mr. Johnson borrowed $8000 for 4 years to make home improvements. If he repaid a total of 10, 320, at what interest rate did he borrow the money? e) John s parents deposited $1000 into a savings account as a college fund when he was born. How much will John have in this account after 18 years at a yearly simple interest rate of 3.25%? f) To buy a laptop computer, Elaine borrowed $2000 for 3 years at an annual simple interest rate of 5%. How much interest will she pay if she pays the entire loan off at the end of the third year? What is the total amount that she will repay? [Make sure you answer both questions!!] g) TJ invested $4000 in a bond at a yearly rate of 2%. He earned $200 in interest. How long was the money invested? h) Mr. Mogi borrowed $9000 for 10 years to make home improvements. If he repaid a total of $20,000 at what interest rate did he borrow the money? i) Bertha deposited $1000 into a retirement account when she was 18. How much will Bertha have in this account after 50 years at a yearly simple interest rate of 7.5%? 21
H.S.E. PREP COURSE MODULE B: RATIONALS COMBINATIONS AND PERMUTATIONS WHAT IS THE DIFFERENCE? In English, we use the word "combination" loosely, without thinking if the order of things is important. EXAMPLE QUOTE: "My fruit salad is a combination of apples, grapes and bananas" We don't care what order the fruits are in, they could also be "bananas, grapes and apples" or "grapes, apples and bananas", its the same fruit salad. EXAMPLE QUOTE: "The combination to the safe is 472". Now we do care about the order. "724" won't work, nor will "247". It must be exactly 4-7-2. So, in Mathematics we use more precise language: When the order doesn't matter, it is a Combination. When the order does matter it is a Permutation. In other words: A Permutation is an ordered Combination. So, we should really call this a "Permutation Lock"! USING THE CALCULATOR PRACTICE EXERCISES Directions: Determine if the problem is a Combination or a Permutation, then evaluate. a) Suppose that 7 people enter a swim meet. If there are no ties, in how many ways could the gold, silver, and bronze medals be awarded? b) How many different committees of 3 people can be chosen to work on a special project from a group of 9 people? c) A coach must choose how to line up his five starters from a team of 12 players. How many ways can the coach choose the starters? 22
H.S.E. PREP COURSE @ SEC - MODULE B: RATIONALS d) John bought a machine to make fresh juice. He has five different fruits: strawberries, oranges, apples, pineapples, and lemons. If he only uses two fruits, how many different juice drinks can John make? e) How many different four-letter passwords can be created for a software access if no letter can be used more than once? f) How many ways can you elect a Chairman and Co-Chairman of a committee if you have 10 people to choose from. g) There are 25 people who work in an office together. Five of these people are selected to go together to the same conference in Orlando, Florida. How many ways can they choose this team of five people to go to the conference? h) There are 25 people who work in an office together. Five of these people are selected to attend five different conferences. The first person selected will go to a conference in Hawaii, the second will go to New York, the third will go to San Diego, the fourth will go to Atlanta, and the fifth will go to Nashville. How many such selections are possible? i) John couldn t recall the Serial number on his expensive bicycle. He remembered that there were 6 different digits, none used more than once, but couldn t remember what digits were used. He decided to write down all the possible 6 digit numbers. How many different possibilities will he have to create? j) How many different 7-card hands can be chosen from a standard 52-card deck? k) One hundred twelve people bought raffle tickets to enter a random drawing for three prizes. How many ways can three names be drawn for first prize, second prize, and third prize? l) A disc jockey must choose three songs for the last few minutes of his evening show. If there are nine songs that he feels are appropriate for that time slot, then how many ways can he choose and arrange to play three of those nine songs? 23
H.S.E. PREP COURSE MODULE B: RATIONALS MEASUREMENTS OF CENTRAL TENDENCIES MEAN GED Math Formula Sheet Definition: mean is equal to the total of the values of a data set, divided by the number of elements in the data set. The mean is the average of the numbers: a calculated "central" value of a set of numbers. To calculate: Just add up all the numbers, then divide by how many numbers there are. EXAMPLE: What is the mean of 2, 7 and 9? Add the numbers: 2 + 7 + 9 = 18 Divide by how many numbers (i.e. we added 3 numbers): 18 3 = 6 So the Mean is 6 MEDIAN GED Math Formula Sheet Definition: median is the middle value in an odd number of ordered values of a data set, or the mean of the two middle values in an even number of ordered values in a data set The median is the middle number (in a sorted list of numbers). To find the Median, place the numbers you are given in value order and find the middle number. EXAMPLE: Find the Median of {13, 23, 11, 16, 15, 10, 26}. Put them in order: {10, 11, 13, 15, 16, 23, 26} The middle number is 15, so the median is 15. EXAMPLE: Find the Median of {13, 23, 11, 16, 15, 10, 26, 14}. Put them in order: {10, 11, 13, 14, 15, 16, 23, 26} The middle numbers are 14 and 15, so the median is 14.5. *When you have two middle numbers, add them up and divide by 2! MODE The mode is the number which appears most often in a set of numbers. EXAMPLE 1: in {6, 3, 9, 6, 6, 5, 9, 3} the Mode is 6 (it occurs most often). EXAMPLE 2: in {6, 3, 9, 6, 6, 5, 9, 3, 9} the Mode is 6 and 9 (both occur 3 times). EXAMPLE 3: in {1, 1, 2, 2, 3, 3, 4, 4, 5, 5} the Mode is No Mode. All numbers occur at the same frequency. 24
H.S.E. PREP COURSE @ SEC - MODULE B: RATIONALS RANGE The range is the difference between the lowest and highest values. Your answer will always be positive! EXAMPLE 1: In {4, 6, 9, 3, 7} the lowest value is 3, and the highest is 9, so the range is 9 3 = 6. EXAMPLE 2: In {4, 6, 9, 3, 7, -2} the lowest value is -2, and the highest is 9, so the range is 9-2 = 11. USING THE CALCULATOR This section to be added later. PRACTICE EXERCISES a) Which of the following sets of data has a median of 17.5? A) {10.0, 17.5, 14.0, 16.0, 27.5} B) {12.5, 26.0, 17.5, 11.5, 10.5} C) {13.0, 17.5, 15.0, 15.5, 17.5} D) {14.5, 19.5, 16.0, 17.5, 24.0} b) The number of hours Nadia spent painting each day during a one-week period are shown below. {1.5, 4.25, 1.0, 3.75, 6.0, 0.75, 0.25} What is the mean number of hours per day that Nadia spent painting for this week? c) CHALLENGE: The mean of four numbers is 70. When a fifth number is included, the mean of the five numbers is 80. What is the fifth number? A) 40 B) 90 C) 120 D) 250 25
H.S.E. PREP COURSE MODULE B: RATIONALS d) At Oliver's Pizza Palace in the 6 hours they were open they sold the following number of pizzas: 55 pepperoni, 57 sausage, 50 cheese, 51 mushroom, 61 anchovies and 50 pineapple. Determine the mean (rounded to the nearest tenth), median, mode and range of the number of pizzas sold. e) Jerry was counting the money he received for his birthday. From his aunt he received $9. From his uncle he received $9. His best friends gave him $22, $23 and $22 and $22. And his sister gave him $7. Determine the mean (rounded to the nearest tenth), median, mode and range of the money he received. f) Dave counted the number of times people sharpened their pencils in class for a week. He counted: 4, 13, 4, 1, 14 and 11. Determine the mean (rounded to the nearest tenth), median, mode and range of the numbers. g) Victor was selling chocolate for a school fund raiser. On the first week he sold 75. On the second week he sold 67. On the third week he sold 75. On the fourth week he sold 70 and on the last week he sold 68. Determine the mean (rounded to the nearest tenth), median, mode and range of the chocolate bars he sold. h) During the first 6 hours of the fair there were the following number of customers: 58, 58, 62, 55, 49 and 48. Determine the mean (rounded to the nearest tenth), median, mode and range of the number of customers. 26
H.S.E. PREP COURSE @ SEC - MODULE B: RATIONALS GED MATH FORMULA SHEET *Perimeter, Area, Volume and Surface Area formulas not listed in this packet. 27