as much as the more experienced landscaper B mowed.

Transcription

1 Final exam review: Study the 1st and nd exam and the perform the following. START NOW! 1. At Riverdale Middle School, 1 of the students are in the band. Two out of every three students in the 8 band are girls. A) The number of boys in the band is times the number of girls in the band. B) What fraction of the students who play in the band are boys? C) What fraction of the students at Riverdale are boys who play in the band? D) The number of girls in the band is times the number of students in the school. E) What is the ratio of girls who do not play in the band to the boys who do not play in the band?. Five of every six students interviewed favored a change in library hours. A) Among those interviewed, what is the ratio of those who favor a change to those who do not favor a change? B) Among those interviewed, there are times as many students who favor a change as there are students who do not favor a change. C) Among those interviewed, there are times as many students who do not favor a change as there are students who do favor a change. D) What fraction of the students interviewed favor a change in the library hours? 3. According to a U.S. News/CNN poll, three out of 10 people went away on vacation in August. A) What is the ratio of those who went away on vacation in August to those who didn't? B) What percent of people did not go away on vacation in August? C) The number of people who did not go on vacation is times the number who did. 4. Two landscapers mowed the lawn of a wealthy family. When they finished, landscaper A had mowed only 3 7 as much as the more experienced landscaper B mowed. A B A) Mark on the drawing of the lawn to show how the mowing might have been done. B) A's part is times as much as B's part. C) A's part is what part of the lawn? D) What is the ratio of A's part to B's part? E) If they are paid $10 for mowing the lawn, what would be a fair split of the $10?. Alan and Bob started mowing a rectangular lawn. After mowing 3 of the whole lawn, they got tired and 4 stopped. When they stopped, Alan had mowed as much as Bob had mowed. 7 A) Mark the diagram of the lawn below to show how much of the lawn each boy mowed and label the parts with each boy's initial to indicate clearly the parts mowed by each. B) The area of lawn that Bob mowed is times as large as the area that Alan mowed. C) Alan mowed of the total lawn area. D) The ratio of the area of lawn Bob mowed to the area Alan mowed is. E) Together they were paid $18 for the work they did. How much money did each boy get if they were paid proportionally to the amount they worked? Page 1

2 6. John drives 7 miles to campus each day, while Vaneta drives only 4 miles to campus. John drives how many times as far as Vaneta? 7. If the Browns were to save an additional $14,000, they would have1 times as much money as the amount the Joneses have in their savings account. The Joneses have $6,000. How much do the Browns currently have in their savings account? 8. Tim worked 30 hours last week, which was times as many hours as Robert worked. 3 A) How many hours did Robert work? B) Who worked more hours? C) Which type of comparison does the question in part B address, additive or multiplicative? 9. A) Cut the rectangle into two regions, A and B, so that A is 3 B) What fraction of the whole rectangle is A? as large as B of an amount is $0. How many dollars is the full amount? 11. Pat and Ron split a cake. Pat's share is /3 as large as Ron's share. A) Sketch fairly accurately on the "cake" to show Pat's and Ron's shares. Label them P and R. B) Ron's share is times as large as Pat's share. C) Pat's share is what fractional part of the whole cake? D) What is the ratio of Ron's share to Pat's share? 1. Two painters paint a wall from opposite ends. When they finish, painter P has painted only 3 as much as the more experienced painter Q painted. P Q A) Mark on the drawing of the wall to show how the painting might have been done. B) Q's part is times as much as P's part. C) Q's part is what part of the wall? D) What is the ratio of P's part to Q's part? E) If they are paid $100 for painting the wall, what would be a fair split of the $100? Page

3 13. Any driveway ramp that is 3.0 ft high is less steep than any other ramp that is 3.98 ft high. A) True B) False 14. Ruben and Ofilia are painting the walls of a large lecture hall. They mixed gallons of blue paint with gallons of white paint for a total of 7 gallons of paint. They ran out of paint. They estimated that they needed one half gallon to finish. Find the portion of the room which has been painted. A) Find the portion of the room which has been painted. Draw a neatly labeled diagram displaying you solution. B) If they mix 1/4 gallon of blue paint with 1 gallon of white paint, will the paint colors match? Exp your answer briefly. C) Suggest what amounts of blue paint and white paint would finish the room and match the color, perh with a little left over for touch-ups. 1. There are four water slides at the Six Flags Atlantis water park. The following are the measurements of the four water slides. Circle the letter of the steepest slide. A) Length: 100 ft.; height: 81 ft. B) Length: 60 ft.; height: 4 ft. C) Length: 80 ft.; height: 70 ft. D) Length: 10 ft.; height: 7 ft. 16. A) 3 bags of tea are used with 4 cups of water in one teapot, and 10 bags of tea with 1 cups of water in another teapot. Which pot will have the stronger tea, or will the two have the same strength? Explain your answer. B) scoops of coffee are used with 8 cups of water in one coffee pot, while 7 scoops of coffee are used with 10 cups of water in another coffee pot. Which pot will have the stronger coffee? Explain your answer. C) The two problems above were given to a sixth-grade class. Which one do you think was more difficult, and why? 17. Two painters on a large project want to paint different areas the same color. Painter A mixes 3 quarts of red paint with gallons of white paint, and Painter B mixes quarts of the same kind of red paint with 4 gallons of white paint. Painter A says the two mixtures will be the same color and Painter B says his mixture will be redder than Painter A's. Explain the thinking of each one. Which one, if either, is correct? Explain your decision. 18. Terry uses 1 cup of Mr. Spiffy in1 1 gallons of water to clean the kitchen floor. What percent of the cleaning solution is Mr. Spiffy? (Note: 16 cups = 1 gallon) 19. Miguel runs 00 meters in 40 seconds; Paul runs 10 meters in 1 minute. Who runs faster? A) Miguel B) Paul C) They run at the same speed. D) More information is needed. Page 3

4 0. A machine can make 700 bolts in 40 minutes. At that rate, how many bolts can the machine make in one hour? A) 900 B) 100 C) 800 D) 8,000 E) None of A D 1. The pollster noticed that for every 40 men who were in favor of X, there were 8 women who were in favor of X. According to these figures, if 80 men were in favor of X, how many women were in favor of X? A) 400 B) 196 C) 168 D) 40 E) None of A D. A group of 37 persons goes to a holiday camp for 3 days. They need to buy enough sugar for the trip. They read that the average consumption of sugar is. kg per week for 10 persons. How much sugar do they need? SHOW YOUR WORK CLEARLY AND CIRCLE YOUR ANSWER. 3. A child says that the two situations below would give the same chocolatey-ness, since Each way has one more spoonful of chocolate sprinkles. Situation 1 Situation 3 spoonfuls of chocolate sprinkles on scoops of vanilla ice cream 4 spoonfuls of chocolate sprinkles on 3 scoops of vanilla ice cream Give a drawing (not just calculations) that should help the child see that the chocolatey-ness in the two situations would be different. (Use words as needed.) 4. Clarissa runs 40 meters in 4 seconds; Doña runs 160 meters in 1 A) Clarissa B) Doña C) They run at the same speed. D) More information is needed. minute. Who runs faster?. Kathy owed her dad $80 and then paid off $4 from her tip earnings. A) What percent of the original debt had she paid? B) What percent was still owed? C) What was her new debt after the payment? The following evening she paid him an additional $0 from her tip earnings. D) What percent of her new debt was paid off? E) What was her debt after the last payment F) What percent of her original debt is now paid off? Page 4

5 6. If this box represents 7% of something, modify the box so that it represents 1% of the same thing. 7. The median cost of housing rose % in one city in a particular year. The median price was then $360,000. What was the median cost of housing at the end of the previous year? 8. Gala apples are on sale today, at $1.30 per pound. This is a discount of 30%. What was the cost before the sale? 9. The Nasdaq closed at 190 today, off by 0.09%. What the Nasdaq at yesterday's closing? 30. Complete the following. For each subtraction problem, first rewrite it as an addition problem. A) = B) 3 + = C) = D) 1 + = E) 17 3 = F) = G) 13 9 = H) = I) J) 11 ( 17) K).6 4. L) Reorder these numbers from smallest to largest: 3. Complete this fact family table: 33. Explain how one could use yellow (positive) and red (negative) chips to model the following: A) 4 + ( 6) B) ( ) 34. Which properties does each of the following involve? A) ( + 3) + = (3 + ) + B) ( + 3) + = + (3 + ) C) ( + 3) + = +( 3 + ) 3. Complete the following: A) 7 8 = B) 3 = C) 16 = D) 1 = E) 18 3 = F) = G) 18 9 = H) 4 16 = I) 4 1 J) ( ) K) L) 1..4 Page

6 36. Match the operations and the names of the properties by placing the correct number to the left of the letters A E. (Not all properties on the right will necessarily be used; some may be used more than once.) A) 3 ( + ) = 3 ( + ) 1. Associative property of multiplication B) 3 ( + ) = (3 ) + (3 ). Additive identity property C) 4 + ( 3 + 1) + = (4 + 3) +( 1 + ) 3. Multiplicative inverse property D) + ( 8 + 0) = Additive inverse property E) -4 1 = -4. Commutative property of addition F) 6 + (4 + 4) = Associative property of addition G) Distributive property of over + H) 3 ( 0) = ( 0) 3 8. Multiplicative identity property I) 3 + ( + ) = (3 + ) + 9. Commutative property of multiplication 37. Show why 4 x - = Say the same thing as the following sentence, but use the word multiple. 360 is a factor of N. 39. Determine whether m and n are primes. Write only enough to make your decisions clear. A) m = 3 9 (= 667) B) n = Is 4 a prime number? Explain. 41. Give the prime factorization It is correct that = Give the prime factorization of (notice the extra two zeros). 43. Tell the difference between (a) give a prime factor of 30 versus give a prime factorization of 30 and the difference between (b) give a number that has an odd factor versus give a number that has an odd number of factors. 44. What is the largest prime number that you need to test, in checking for the primeness of A) 173 B) Circle the numbers that are prime. If a number is not prime, list at least three factors below the number Circle the numbers below that divide 110., 3, 4,, 6, 8, 9, 10, 1, 1, 18, A) What is the least common multiple of 148 and 79 (in factored form)? B) Write two other common multiple of 148 and 79. Page 6

7 48. Suppose K = , L = M = 9 and N = Name the least common multiple of each of the following (in factored form). A) K and L B) M and N C) K and M D) K, L, and N 49. Suppose K = , L = M = 9 and N = Name the greatest common factor of each of the following (in factored form). A) K and N B) K and L C) M and N D) K, L, and M 0. Use the GCF to write these numbers in simplest form: A) 6 6 B) C) A) Use the LCM of 64, and 49 to find the least common multiple of the two numbers. B) Compute the following:. (Leave the answer in factored form.) Two neighboring satellites send out signals at regular intervals. One sends a signal every 180 seconds, and the other sends a signal every 80 seconds. If both satellites send out a signal at 1:00 midnight on January 1, when will be the next time that they both send out a signal at the same time? 3. Hamburger patties come in packages of 16, and hamburger buns come in bags of 1. How many of each do you need to buy so that you have the same number of buns as you do of hamburgers? 4. As a charitable service, your class undertakes a project where they fill backpacks with donated school supplies for underprivileged children. The donations include 13 notebooks, 16 pencils, and 81 pens. You want to use all the donations and include the same number of each item in each backpack. What is the largest number of backpacks you can fill and how many items will be in each backpack?. Two football players are working out by running around a track. The first can run the track in 3 minutes, and the second one can run the track in 4 minutes. If they begin at the starting point at the same time and run in the same direction at the same rates, when will the both be at the starting point again? 6. The band has been invited to march at the Rose Parade and need to make money to cover the expenses. They divide up into three teams and shovel snow from long driveways for four days before Christmas. The first team made $31, the second $40, and the third $10. If they charged the same whole-dollar rate for each driveway, what was that rate? 7. What does the sales tax rate is 7 % mean? Page 7

8 Answer Key 1. (A diagram may be helpful for parts C, D, and E.) A) 1/ B) 1/3 C) 1/4 D) 1/1 (or /4) E) Impossible to determine. A) :1 B) C) 1/ D) /6 3. A) 3:7 B) 70% C) 7/3 4. A) The region should be marked into 10 equal pieces, with 3 labeled for A and the others for B. B) 3/7 C) 3/10 D) 3:7 E) $4 for A; $10 for B ($10 for the 10 pieces is a rate of $1 per piece). A) 3/4 of the lawn should be shaded. Then that shaded portion should be cut into nine equal parts since the A:B ratio :7 says two parts for Alan for every seven parts for Bob. So two of those parts should be labeled A and seven should be labeled B. B) 3 1/ (either from the drawing, or from B:A = 7:) C) 1/6 (more marks show 6:36) D) 7: E) Alan: $4, Bob: $14 (from $18 for 9 pieces is a rate of $ per piece) 6. 7: 4 translates into 7/4 times as far / times as much as $6,000 = $91,000. So the Browns must have $91,000 $14,000 = $77,000 in their savings account. 8. A) 18. (A drawing should make clear the :3 comparison; Tim's 30 hours means each of his five pieces could be thought of as 6 hours, so Robert's time is 18 hours.) B) Tim (without part A, just from the given sentence) C) Either. If a follow-up question had been How many more? that would clearly have been an additive comparison. 9. A) Cut the rectangle into five equal pieces: A is three of the pieces, B is two of the pieces. B) A is 3/ of the entire rectangle. 10. $ 11. A) Five pieces, with two for P and three for R. B) 1 ½ C) / D) 3: 1. A) equal pieces, for P and 3 for Q B) 1 1/ C) 3/ D) :3 E) P $40; Q $60 (each of the five pieces would be worth $0) 13. B) False. The steepness also depends on another dimension, usually the horizontal length of the driveway. 14. A) The diagram should show the walls (probably one rectangle), marked into 1 equal pieces, since the 7 gallons is 14 half-gallons. They have painted 14/1 of the room with the 7 gallons. B) No. The original rate was : which is the same as /:1. But if 1 gallon of white paint is used with ¼ gallon of blue paint, the ratio is 1/4:1, or 1:4, which is not equivalent to /:1. C) We need a ratio a:b, where a:b is equivalent to : and where a + b = 1/. ¼ gallon blue paint and /8 gallon of white paint would provide the correct ratio and the sum is slightly more than ½, but close enough for this situation. 1. C. Steepness can be measured by the height:length ratio. Using the fraction forms and number sense, 70:80 is the largest of the four ratios. 16. A) The 10 bags in 1 cups will be stronger since the first mixture has ¾ bag per cup of water and the Page 8

9 second 10/1, or /6, bag per cup of water, and /6 > ¾. B) The 7 scoops, 10 cups of water coffee will be stronger because 7/10 scoop per cup of water gives more coffee taste than /8 scoop per cup of water will. C) Part B almost certainly would be more difficult for sixth graders because the additive comparisons for the two recipes are the same, a difference of 3. And, there is not an easy relationship between the quantities in part B, as there is in part A: 1 cups is 3 times as much water as 4 cups, so repeating the first recipe 3 times would mean 9 bags for 1 cups, a weaker tea than with 10 bags for 1 cups. 17. Painter A is using an additive comparison; painter B is looking at the larger amount of red paint in his mixture. Neither reasoning is correct since it is the ratio of the number of quarts of red to the number of gallons of white that is important. For A, R:W = 3: = (1 1/):1, for B, R:W = :4 = (1 1/4):1, so A's mixture will be redder. 18. The 1 1/ gallons, or 4 cups, of water. He is also adding a cup of Mr. Spiffy to the mixture for a total of cups. This gives us a 1/ or 4 %. 19. C 0. B 1. B. One way: The 3 days is weeks, so they will need. = 11 kg for every 10 persons for the whole period. Since there are 37 persons, they will need = 40.7 kg of sugar for the camp. Alternatively, gives 1.1 kg per person, so 37 people will need = 40.7 kg of sugar. 3. Draw to show that in Situation 1, each scoop gets 1 1/ spoonfuls, and in Situation, each scoop gets 1 1/3 spoonfuls. 4. C. A) 30% B) 70% C) $6 D) 3.7% E) $36 F) % 6. Divide the box into three equal parts, then add an additional two parts of the same size. 7. About $88, About $ About (the main idea is to catch incorrect thinking or poor number sense) 30. A) 1 B) C) D) 3 E) 17 3 = = 0 F) = + = 3 G) 13 9 = = H) = = 30 I) 3 J) 0 K) = 7.1 L) = % A) Page 9

10 Subtracting the red chips allows you to flip the reds to make yellows. 34. A and B) Commutativity of addition C) Associativity of addition 3. A) 6 B) 1 C) 3 D) 4 E) 6 F) G) H) 1 4 I) 1 J) 1 K) L) A) B) 7 C) 6 D) E) 8 F) 4 G) 3 H) 9 I) X - means 4 negative fives, so = -0. Or you could use a 4 X array of red chips. 38. N is a multiple of A) Not a prime (3 or 9 is a third factor) B) Square root of 133 is 11.3, so prime because none of, 3,, 7, 11 is a factor. At least one of the primes up to 11 would need to be a factor of 133 for 133 to be a composite number because there is only one distinct set of prime factors for every number. 40. No, is a third factor (Students may overlook the 1 in the original factorization; despite the hint, some may not recognize that the target number is just 100 times the original one.) 43. A) Give a prime factor of 30 means to identify only one of the prime factors of 30, whereas give a prime factorization of 30 means to give a product of primes equal to 30. B) Give a number that has an odd factor means to find a number that has a factor that is 3,, 7, 9, etc., but give a number that have an odd number of factors means to find a number such that when you find all of its factors, there are an odd number of them. 44. A) 13 because = 169, which is the largest square smaller than 173 B) 31 because = 961, which is the largest square less than Only 43 is a prime. 11 = and the sum of the digits of 3111 add up to a multiple of All except 8, 9, and 18 divide A) B) 3760 and 3640 are two possible answers. Answers will vary. 48. A) B) C) D) A) 11 B) 7 11 C) 9 D) 0. A) B) 7 9 C) A) LCM is B) :4 AM. The next simultaneous occurrence will happen when a multiple of 180 seconds next Page 10

11 coincides with a multiple of 80 seconds, that is, at the least common multiple of 180 and 80. This will first happen 0 seconds, or 4 minutes, later. 3. The LCM of 16 and 1 is 48, so buy three packages of patties and 4 dozen buns. 4. notebooks, 8 pencils, and 3 pens in 7 backpacks: GCF is the number of backpacks.. 1 minutes 6. GCF (31, 40, 10) = 1, so $1. ($, $3, and $1 would be possible but unlikely answers.) 7. The tax will be $7 for every $100 of goods (or 7 3/4 for every 1 dollar of goods). Page 11