7th graders: click on the following link for practice problems before Friday's test on Probability https://sites.google.com/site/lassmath/myforms Problem of the Week 3/23/18 Adrienne's Axel Adrienne is a figure skater. Her coach Mandy is planning Adrienne's new competition program, and has told Adrienne that the jumps in the new program will all be Axels, double Salchows, or LutzLoopLoop Combinations. When a skater does a single jump, such as a Lutz, a Loop, or a Salchow, she does one rotation in the air, jumping backward and landing backward. When a skater does a double jump, such as a double Salchow, she does two rotations in the air (all in one jump). The Axel is an unusual jump, because the skater jumps forward and lands backward, doing oneandahalf rotations in the air. A LutzLoopLoop Combination is three jumps in a row without any skating in between. In addition to ice skating, both Mandy and Adrienne enjoy mathematical puzzles. Mandy says, "Your new program will have nine jumps, eleven rotations, and two combinations. Adrienne says, "Now I know exactly what jumps I will have in my new program!"
What does Adrienne know and how does she know it? (Adrienne thinks algebraically.) 3/16/18 Cheap Sunglasses Gilbert was working on lighting for the school play and thought that he could make a cool pair of sunglasses by attaching some of the red filter film to his glasses. He did some research and learned that 20 percent of the light passing through the filter is absorbed. He also read that good sunglasses should absorb 80 percent of the incoming light, so he figured that if he put four layers of filter film on his glasses he'd be all set. 1. Calculate how much light would be absorbed by 4 layers of filter film. Is Gilbert right?
Solution 1. Gilbert is wrong, four layers would let in 40.96% of the light and 59.04% would be absorbed
2/26/18 Persistence of Numbers One property of numbers is their persistence. Take the number 723 as an illustration: if you multiple the digits 7, 2 and 3 together, the product is 42; multiply 4 and 2 together to get 8. Because this operation takes two steps to reach a singledigit number, the persistence of 723 is 2.
What are the smallest numbers that lead to persistences of 2, 3 and 4? 2/12/18 Tennis Anyone? A tennis player computes her "win ratio" by (number of matches won) divided by (total number of matches played). At the start of a weekend, her win ratio is 0.500. During the weekend, she wins three matches and loses one. At the end of the weekend her win ratio is greater than 0.503.
What is the greatest number of matches she could have won before the weekend began? Solution
The largest number of matches she could have won before the weekend began is 164. Let x/(2x) = the win ratio before the weekend began, where x is the number of games won. (x + 3)/(2x + 4) represents the win ratio after the weekend (x + 3)/(2x + 4) > 0.503 (x + 3) > 1.006x + 2.012 0.988 > 0.006x x < 164.66666 Therefore, the greatest integer less than 164.66666 is 164. 2/5/18 An "Average" Problem Doug, Dick, and Dean are triplets in Mr. Finch's firstperiod Algebra I class. Friday's exam is to be their last in that school.
On the following Monday, after the triplets have left the school, Mr. Finch announces the class averages. "Before Doug, Dick, and Dean left, the class average was 74%  not bad, really. However, removing their scores causes the class average to rise by one percentage point. Okay?"
"Furthermore," he adds, "the ratios of the triplets' scores is
6 : 5 : 3. What are the scores of our three departing friends?" Solution The scores would have been 90, 75, and 45. We compute the average by adding the scores and dividing that by the number of scores involved, i.e. m = T/n. In this case we know the mean “m” and the number of scores “n”. So for the “before departure” case we write: T = 74 and T = 74 * 15 = 1110 For the “after departure” case, we have: T = 75 and T = 75 * 12 = 900 Subtracting yields 210 points earned by the three boys. Now let x = the scale factor of the ratio, giving us that 3x + 5x + 6x = 210 14x = 210 and x = 15 This means that one boy got (6 * 15) = 90; Another got (5 * 15) = 75; and the third got (3 * 15) = 45.
A second possibility is 84, 70 and 42 if there are a total of 29 students, or 26 after the twins leave the school. Solution The scores would have been 90, 75, and 45. We compute the average by adding the scores and dividing that by the number of scores involved, i.e. m = T/n. In this case we know the mean “m” and the number of scores “n”. So for the “before departure” case we write: T = 74 and T = 74 * 15 = 1110 For the “after departure” case, we have: T = 75 and T = 75 * 12 = 900 Subtracting yields 210 points earned by the three boys. Now let x = the scale factor of the ratio, giving us that 3x + 5x + 6x = 210 14x = 210 and x = 15 This means that one boy got (6 * 15) = 90; Another got (5 * 15) = 75; and the third got (3 * 15) = 45.
A second possibility is 84, 70 and 42 if there are a total of 29 students, or 26 after the twins leave the school. 1/29/18 The Box Supper Back in the olden days, community groups in rural areas often held a fundraising event called a box supper. It worked something like this: Young ladies would cook a meal at home, put it in a box, and decorate the box. Once at the event, young men would bid on the boxes in auctions. Each box would go to the one who offered the most money, and he would have the privilege of eating the food with the girl who had prepared it. Of course, certain girls wanted to eat with certain boys, so word of how girls would decorate their boxes usually got around, so that boys could be sure which one to bid on. Sally Jo prepared a roast beef sandwich with homemade bread, apple salad, and pink lemonade. Her boyfriend, Joseph, paid a modest for that box. (It cost him an odd number of dollars.) Mary Lou's specialty was juicy slices of Virginia ham with a pineapple sauce, scalloped potatoes, and peppermint tea. Her boyfriend, Leroy, won that box for 25 fewer dollars than three times what Joseph spent. But it was Judy Ann's specialty of Southern Fried Chicken that really stole the show. She topped the meal off with creamy mashed potatoes and gravy, and that new soft drink sensation everybody was talking about, Dr. Coker. Her boyfriend Jay paid top dollar for that scrumptious meal. He paid five times the positive difference between what Joseph and Leroy paid. We want to know just how much Jay paid, but you need to know two more facts: · The total amount of money raised was twelve times Joseph's payment, and · The sum of the money paid by these three gentlemen was $4 more than the amount paid by everybody else who bought boxes. Solution Jay paid $65 for his meal. 1/22/18 What Time Is It? At 3:00, the angle between the hands of a clock is 90 degrees. As the hands move, the angle between them changes. Note that at 3:30 the hands will not be 90 degrees apart since the hour hand will have moved halfway between the 3 and the 4. Not only do both hands move, they move at different rates. In one hour the minute hand goes completely around the clock face while the hour hand moves only to the next number. Keeping these ideas in mind, answer the following questions: Find the angle between the hands at 3:12. Also find the angle at 3:39.
Find the time to the nearest minute between 3:00 and 4:00 when the angle between the hands will be 90 degrees.
Solution 1/15/18 Counterfeit Coins
Five counterfeit coins are mixed with nine authentic coins. If two coins are drawn at random, find the probability that one coin is authentic and one is counterfeit. Solution 45/91. If the counterfeit coin is chosen first, the probability is 5/14 • 9/13. If the authentic coin is chosen first, the probability is 9/14 • 5/13. The total probability is 5/14 • 9/13 + 9/14 • 5/13 = 45/91.
1/8/18 Tires tires tires
A threewheeled vehicle travels 100 km. Two spare tires are available. Each of the five tires is used for the same distance during the trip. For how many kilometers is each tire used? Solution
60. Since only three of the five tires are in use at any time, the total distance traveled by all the tires is 3(100) = 300 km. However, each of the five tires is used for the same distance during the trip. Thus, each tire is used for 300 ÷ 5 = 60 km. 12/18/17 Workshop
Toni makes stools and tables in her workshop. At the end of a long day, she has made 35 legs. If each stool has 3 legs and each table has 4 legs, and the legs are all identical, what are the possible combinations of stools and tables that can she now assemble? Solution
1 stool and 8 tables; or 5 stools and 5 tables; or 9 stools and 2 tables. Since each stool requires 3 legs and each table requires 4 legs, the scenario can be described as all possible integral solutions for 3x + 4y = 35. This has 3 possible solutions: (1, 8), (5, 5), (9, 2). Note that in terms of legs, 4 stools are equivalent to 3 tables. 12/11/17 Palindromic Speeding The odometer of a family car shows 15,951 miles. The driver noticed that this number is palindromic: it reads the same backward as forward. Surprised, the driver saw his third palindromic odometer reading (not counting 15,951) exactly five hours later. How many miles per hour was the car traveling in those 5 hours (assuming speed was constant)?
Solution
62. Realistically, the first digit of 15,951 could not change in 5 hours. Therefore, 1 is the first and last digit of the new number. Clearly, the second and fourth digits changed to 6. So the next three resulting palindromes are 16,061, 16,161, and 16,261. Thus, the car traveled 310 miles in 5 hours and must have been traveling 62 miles per hour. 12/4/17 Equal Row Sums Using the diagram, place the numbers 1 to 10 in the circles so that the sums in the rows of three circles are the same and the sums in the rows of four circles are the same. Solution One possible arrangement is shown. The sum of the rows of four is 23, and the sum of the rows of three is 16. 11/27/17 Engine Failure
An aircraft is equipped with three engines that operate independently. The probability of an engine failure is .01. What is the probability of a successful flight if only one engine is needed for the successful operation of the aircraft? Solution 0.999999. Let P(S) be the probability of a successful flight, P(S') the probability of an unsuccessful flight, and P(F_{n}) the probability of n engines failing. Since the flight is unsuccessful only when all three engines fail, then the probability of unsuccessful flight is P(S') = P(F1 ∩ (F2 ∩ F3)) =(.01)(.01)(.01) =(.01)^{3}. But P(S) = 1 − P(S') = 1 − (.01)^{3} = 1 − .000001 = 0.999999.
