Applied Mathematics PDF Past Paper

# MARGALLO' BUCKLING REVIEW REFRESHER I ## Applied Mathematics 1. Rufus T. Flypaper drives two miles to work every morning. Very precise, he knows he must average 30 mph to arrive on time. One morning a woman driver impedes him for the first mile, cutting his average to only 15 mph. He quickly ca...

# MARGALLO' BUCKLING REVIEW REFRESHER I ## Applied Mathematics 1. Rufus T. Flypaper drives two miles to work every morning. Very precise, he knows he must average 30 mph to arrive on time. One morning a woman driver impedes him for the first mile, cutting his average to only 15 mph. He quickly calculated his proper speed for the rest of the trip to arrive on time. Assume that his car could do 120 mph. Could he arrive on time? **Ans. No, he has already used 4 minutes** 2. Station A and B are 120 miles apart on a single track railroad. At the same time a train leaves A for B at 15 mph. Just as the first train leaves A, a South American botfly flies from the front of the engine straight toward the other train at 100 mph. On meeting the second train it immediately turns back and flies straight for the first train. It continues to fly back and forth with undiminished speed until it is crushed in the eventual collision. How far had the fly flown? **Ans. 300 miles** 3. [MAY 2017] Assuming the sun rises at 6:00 a.m., sets at 6:00 p.m., and moves at a uniform rate, how can a lost boy scout determine south by means of a watch on a cloudless days? **Ans. Align the hour hand, with the sun's azimuth, and south will be midway between the hour hand and 12** 4. [MAY 2017] What is the least number of links that must be disengaged from a 23-link chain so that any number of links from 1 to 23 can be obtained by taking one or more of the pieces? **Ans. 2** 5. The set A contains the integers 0, 4, 5, 9, 11, 12, 13, 14, 19... The set B contains 1, 2, 3, 6, 7, 8, 10, 15, 16, 17, 18... Place 20 and 21 in their proper sets. **Ans. 20 is in A, 21 is in B** 6. A neat computer programmer wears a clean shirt every day. If he drops off his laundry and picks up the previous week's load every Monday night, how many shirts must he own to keep him going? **Ans. 15** 7. [MAY 2019] The passengers on an excursion bus consisted of 14 married couples, 8 of whom brought no children, and 6 of whom brought 3 children apiece. Counting the driver, the bus had 31 occupants. How is this possible? **Ans. Included among the 18 children were 8 married couples** 8. Very few people are aware of the growth pattern of Jack's beanstalk. On the first day it increased its height by 1/2, on the second day by 1/3, on the third day by 1/4, and so on. How long did it take to achieve it maximum height (100 times its original height)? **Ans. 198 days** 9. [NOV 2023] A novice librarian shelved a twelve volume set of encyclopedias in the following order from left to right. Volumes 8, 11, 5, 4, 9, 1, 7, 6, 10, 3, 12, and 2. Using her system, where will the annual supplement, Volume 13, go? **Ans. Between volumes 10 and 3** 10. With some sharp reasoning, you ought to be able to determine the last member of the sequence for which the first 20 members are: 11, 31, 71, 91, 32, 92, 13, 73, 14, 34, 74, 35, 95, 16, 76, 17, 37, 97, 38, 98, ? **Ans. 79** 11. What operation can be performed three successive times on a solid cube, so that at each stage, the surface area is reduced in the same proportion as the volume? **Ans. Turning** 12. [NOV 2021] A lighthouse shows successive one-second flashes of red, white, green, green, white, red. A second lighthouse does the same only with two-second flashes. The six-second sequence of the first lighthouse is repeated steadily, as it the twelve-second sequence of the other lighthouse. What fraction of the time do the two lights show the same color if the given sequences start at the same time? **Ans. 1/6** ## 13. Mary Ann Moore's father has a yacht and so has each of his four friends: Colonel Downing, Mr. Hall, Sir Barnacle Hood, and Dr. Parker. Each of the five also has one daughter and each has named his yacht after a daughter of the others. Sir Barnacle's yacht is the Gabrielle, Mr. Moore owns the Lorna: Mr. Hall the Rosalind. The Melissa, owned by Colonel Downing, is named after Sir Barnacle's daughter. Gabrielle's father owns the yacht which is named after Dr. Parker's daughter. Who is Lorna's father? **Ans. Colonel Downing** 14. Five suspects were rounded up in connection with the famous "Cock Robin Murder." Their statements were as follows: A: "C and D are lying." B: "A and E are lying." C: "B and D are lying." D: "C and E are lying." E: "A and B are lying." Who is lying? **Ans. A** 15. Six Grocers in a town each sell a different brand of tea in four ounce packets at 25 cents per packet. Once of the grocers gives short weight, each packet of his brand weighing on 3% ounces. If I can use a balance for only one weighing, what is the minimum amount I must spend to be sure offinding the grocer who gives short weight? **Ans. 3.7 dollars** 16. Between Kroflite and Beeline are five other towns. The seven towns are an integral number of miles from each other along a straight road. The towns are so spaced that if one knows the number of miles a person has traveled between any two towns he can determined the particular towns uniquely. What is the minimum distance between Kroflite and Beeline to make this possible? **Ans. 25 miles** 17. [APR 2023] [NOV 2016] In Bristol 90% of the citizen drink tea; 80% drink coffee; 70% drink whiskey; and 60% drink gin. No one drinks all four beverages. What percent of Bristol's citizens drinks liquor. **Ans. 100%** 18. A teenager wants to go out 2 consecutive nights out of a 3-day weekend. Permission for each night is obtained (or denied) by asking either Father or Mother. Father is known to be more likely to grant permission. However, if the same parent is asked on 2 consecutive days the answer are never the same 2 days running. Whom should he ask first? **Ans. Mother** 19. [MAY 2017] You and a friend spot a loose $20 bill simultaneously and agree to an auction in which you write down your bids and compare them. High bidder gets the $20 and pays the other the amount of the higher bid. Tie bidders split the $20. How much do you bid? **Ans. $9.99** 20. Hard Knox College is a member of a six-school basketball league in which every pair of schools plays each other twice. The other five schools ended the season with respective league records of 200, 300, 500, 600, and .800. How did Hard Knox make out? **Ans. 0.600** 21. A prisoner is given 10 white balls, 10 black and two boxes. He is told that an executioner will draw one ball from one of the two boxes. If it is white, the prisoner will go free; if it is black, he will die. How should the prisoner arrange balls in the boxes to give himself the best chance for survival. **Ans. 73.7%** 22. [NOV 2016] Smith and jones, both 50% marksmen, decide to fight to fight a duel in which they exchange alternate shot until one is hit. What are the odds in favor of the man who shoots first? **Ans. 213** 23. On a certain day, our parking lot contains 999 cars, no two of which have the same 3 digit license number. After 5:00 p.m. what is the probability that the license numbers of the first 4 cars to leave the parking lot are in increasing order of magnitude? **Ans. 24** ## 24. [NOV 2016] Three marksmen simultaneously shoot at and hit a rapidly spinning spherical target. What is the probability that the three points of impact are on the same hemisphere? **Ans. 1** 25. A hospital nursery contains only two baby boys; the girls have not yet been counted. At 2:00 p.m. a new baby is added to the nursery. A baby is then selected at random to be the first to have its footprint taken. It turns out to be a boy. What is the probability that the last addition to the nursery was a girl? **Ans. 2/5** 26. Assume that a depth charge has a probability of 1/2 of sinking a submarine, 1/4 of damage and 1/4 missing. Assume also that two damaging explosions sink the sub. What is that probability that 4 depth charge will sink the sub? **Ans. 25/11256** 27. If 2 marbles are removed at random from a bag containing black and white marbles, the chance that they are both white is 1/3. If 3 are removed at random, the chance that they all are white is 1/6. How many marbles are there of each color? **Ans. 6 whites, 4 blacks** 28. [NOV 2018] A coin is so unbalance that you are likely to get two heads in two successive throws as you are to get tails in one. What is the probability of getting heads in a single throw? **Ans. 0.618** 29. Three dart players threw simultaneously at a tic-tac-toe board, each hitting a different square. What is the probability that the three hits constituted a win at tic-tac-toe? **Ans. 2/21** 30. All the members of a fraternity play basketball white all but one play ice hockey; yet the number of possible basketball teams (5 members) is the same as the number of possible ice hockey teams (6 members). Assuming there are enough members to for either type of team, how many are in the fraternity? **Ans. 15** 31. There are three families, each with two sons and two daughters. In how many ways can all these young people be married? **Ans. 80** 32. How many three digit telephone area codes are possible given that: (a) the first digit must not be zero or one; (b) the second digit must be zero of one; (c) the third digit must not be zero; (d) the third digit may be one only if the second digit is zero **Ans. 136** 33. Max and his wife Min each toss a pair of dice to determine where they will spend their vacation. If either of Min's dice displays the same number of spots as either of Max's she wins and they go to Bermuda. Otherwise, they go to Yellowstone. What is the chance they'll see "Old Faithful" this year? **Ans. Bermuda, 0.514** 34. There are four volumes of an encyclopedia on a shelf, each volume containing 300 pages, (that is, numbered 1 to 600), but there have been placed on the shelf in random order. A bookworm starts at the first page of Vol. 1 and eats his way through to last page of Vol. 4. What is the expected number of pages (excluding cover) he has eaten through? **Ans. 500** 35. [MAY 2019] The local weather forecaster says "no rain" and his record is 2/3 accuracy of prediction. But the Federal Meteorological Service predicts rain and their record is 3/4. With no other data available, what is the chance of rain? **Ans. 3/15** 36. [MAY 2019] To stimulate his son in the pursuit of partial differential equations, math professor offered to pay him $8 for every equation correctly solved and to fine him $5 for every incorrect solution. At the end of 26 problems, neither owned any money to the other. How many did the boy solve correctly? **Ans. 10** ## 37. An expert on transformer design relaxed one Saturday by going to the races. At the end of the race, he had doubled his money. He bet $30 on the second race and tripled his money. He bet $54 on the third race and quadrupled his money. He bet $72 on the fourth race and lost it, but still had $48 left. With how much money did he start? **Ans. 29** 38. Two men are walking towards each other at the side of a railway. A freight train overtakes one of them in 20 seconds and exactly ten minutes later meets the other man coming in the opposite direction. The train passes this man in 18 seconds. How long after the train has passed the second man will the two men meet? (Constant speeds are to be assumed throughout.) **Ans. 5562** 39. [NOV 2023] Using the French Tricolor as a model, how many flags are possible with five available colors if two adjacent rows must not be colored the same? **Ans. 50 flags** 40. A cubic box with sides 'a' feet long is placed flat against a wall. A ladder 'p' feet long is placed in such a way that it touches the wall as well as the free horizontal edge of the box. If a = 1 and p = m, calculate at what height the ladder touches the wall, using quadratics only. **Ans. 3.62 or 1.38** 41. [NOV 2017] Dr. Irving Weiman, who is always in a hurry, walks up an up-going escalator at the rate of one step per second. Twenty steps bring him to the top. Next day he goes up to two steps per second, reaching the top in 32 steps. How many steps are there in the escalator? **Ans. 80** 42. [APR 2024], [NOV 2016] There are nine cities which served by two competing airlines. One or the other airline (but not both) has a flight between every pair of cities. What is the minimum number of triangular flights (i.e., trips from A to B to C and back to A on the same airline)? **Ans. 12** 43. Two snail start from the same point in opposite directions toward two bits of food. Each reaches his destination in one hour. If each snail had gone in the direction the other took, the first snail would have reached his food 35 minutes after the second. How do their speeds compare? **Ans. The first snail traveled at three-fourths the speed of the second** 44. A necklace consists of pearls which increase uniformly from a weight of 1 carat for the end pearl to a weight of 100 carats for the middle pearl. If the necklace weighs altogether 1650 carats and the clasp and string together as much (in carat) as the total number of pearls, how many pearls does the necklace contain? **Ans. 33** 45. Mr. field, a speeder, travelers on a busy highway having the same rate of traffic flow in each direction. Except for Mr. Field, the traffic is moving at the legal speed limit. Mr. Field passes one car for every nine which he meets from the opposite direction. By what percentage is he exceeding the speed limit? **Ans. 25%** 46. The teacher marked the quiz on the following basis: one point for each correct answer, one point off for each question left black and two points off for each question answered incorrectly. Pat made four times as many errors as Mike, But Mike left nine more questions Black. If they both got the same score, how many errors did each make? **Ans. Pat: 8, Mike: 2** 47. A student, just beginning the study of logarithms, was required to evaluate an expression of the form A^1/10. He proceeded to... **Ans. 12** ## 48. [NOV 2022], [MAY 2017] There are four towns at the corners of a square. Four motorists set ... **Ans. A = 914 and B = 2718** 49. [APR 2023] A pencil, eraser and notebook together cost $1.00. A notebook cost more than two pencil, and three pencils cost more than four erasers. If three erasers cost more than a notebook, how much does each cost? **Ans. P= 26, E = 19, N=55** 50. The faces of a solid figure are all triangles. The figure has nine vertices. At each of six of these vertices, four faces meet, and at each of the other three vertices, six faces meet. How many faces does the figure have? **Ans. 14** 51. [NOV 2017] A new kind of atom smasher is to be composed of two tangents and a circular arc which is concave towards the point of intersection of the two tangents. Each tangent and the arc of the circle is I mile long. What is the radius of the circle? **Ans. 1437.45 ft** 52. A spider and a fly are located at opposite vertices of a room of dimensions 1,2 and 3 units. Assuming that the fly is too terrified to move, find the minimum distance the spider must crawl to reach the fly **Ans. √18** 53. [NOV 2017] In a room 40 feet long, 20 feet wide, and 20 feet high, a bug sits on an end wall at a point one foot from the floor, midway between the sidewalls. He decides to go on a journey to a point on the other end wall which is one foot from the ceiling midway between the sidewalls. Having no wings, the bug must make this trip by sticking to the surfaces of the room. What is the shortest route that the bug can take? **Ans. 58 ft** 54. [NOV 2023] [NOV 2018] A farmer owned a square field measuring exactly 2261 yards on each side. 1898 yards from one corner and 1009 yards from an adjacent corner stood a beech tree. A neighbor offered to purchase a triangular portion of the field stipulating that a fence should be erected in a straight line from one side of the field to an adjacent side so that the beech tree was part of the fence. The farmer accepted the offer but made sure that the triangular portion was of minimum area. What was area of the field the neighbor received, and how long was the fence? **Ans. A = 919, 120 and L = 2018** 55. A man leaves from the point where the prime meridian crosses the equator and moves forty five degrees northeast by geographic compass which always points toward the north geographic pole. He constantly corrects his route. Assuming that he walks with equal facility on land and sea, where does he end up and how far will he have travelled when he gets there? **Ans. √2 x 10 meters** 56. [NOV 2021] Near the town of Lunch, Nebraska there is a large triangular plot of land bounded by three straight roads which are 855, 870, and 975 yards long respectively. The owner of the land, a friend of mine, told me that he had decided to sell half the plot to a neighbor, but that the buyer had stipulated that the seller of the land should erect the fence which was to be a straight one. The cost of fences being high, my friend naturally wanted the fence to be as short as possible. What is the minimum length the fence can be? **Ans. 600 yards** 57. Three hares are standing in a triangular field which is exactly 100 yards on each side. One hare stands at each corner; and simultaneously all three set off running. Each hare runs after the hare in the adjacent corner on his left, thus following a curved course which terminates in the middle of the field, all three hares arriving there together. The hares obviously ran at the same speed, but just how far did they run? **Ans. 100 yards** ## 58. A one-acre field in the shape of a right triangle has a post at the midpoint of each side. A sheep is tethered to each of the side posts and a goat to the post on the hypotenuse. The ropes are just long enough to let each animal reach the two adjacent vertices. What is the total area the two sheep have to themselves, i.e", the area the goat cannot reach? **Ans. 1 acre** 59. A divided highway goes under a number of bridges, the arch over each lane being in the form of a semi-ellipse with the height equal to the width. A truck is 6 ft. wide and 12 ft. high. What is the lowest bridge under which it can pass? **Ans. 13 ft 5 in** 60. A cowboy is five miles south of a stream which flows due east. He is also 8 miles west and 6 miles north of his cabin. He wishes to water his horse at the stream and return home. What is the shortest distance he can travel and accomplish this? **Ans. 17.9 miles** 61. A pirate buried his treasure on an island, a conspicuous landmark of which were three palm trees, each one 100 feet from the other two. Two of these trees were in a N-S line. The directions for finding the treasure read: "Proceed from southernmost tree 15 feet due north, then 26 feet due west." Is the treasure buried within the triangle formed by the trees? **Ans. the treasure will lie outside the triangle** 62. The Ben Azouli are camped at an oasis 45 miles west of Taqaba. They decide to dynamite the Trans-Hadramaut railroad joining Taqaba to Maqaba, 60 miles north of the oasis. If the Azouli can cover 18 miles a day, how long will it take them to reach the railroad? **Ans. 2 days** 63. [NOV 2017] A cross section through the center of a football is a circle x inches in circumference. The football is x-8 inches long from tip to tip and each seam is an arc of a circle 1/2 x inches in diameter. Find x. **Ans. 20.69** 64. [NOV 2022], [NOV 2016] A coffee pot with a circular bottom tapers uniformly to a circular top with radius half that of the base. A mark halfway up the side says "2 cups." Where should the "3 cups" mark go? **Ans. about 2/5 of the way down from the top of the pot** 65. [MAY 2018] An icicle forming from a dripping gutter is in the shape of a cone five times as long as it is wide (at the top). A few hours later it has doubled in length and the generating angle has also doubled. How does its present weight compare with its previous weight? **Ans. 33 times** 66. A student beginning the study of trigonometry came across an expression of the form sin (X + Y). He evaluated this as sin X + sin Y. Surprisingly he was correct. The values of X and y differed by 10°; what were these values, assuming that 0° < X < Y < 360°? **Ans. 175° and 185°** 67. What is the longest 6' wide shuffle board court which will fit in a 20'x 30' rectangular room **Ans. 30 7/18 ft** 68. [NOV 2022] Three farmers, Adams, Brown and Clark all have farms containing the same number of acres. Adams' farm is most nearly square, the length being only 8 miles longer than the width. Clark has the most oblong farm, the length being 34 miles longer than the width. Brown's farm is intermediate between these two, the length being 28 miles longer than the width. If all the dimensions are in exact miles, what is the size of each farm? **Ans. 40x48, 32x60, and 30x64** 69. 1960 and 1961 were bad years for ice cream sales but 1962 was very good. An accountant was looking at the tonnage sold in each year and noticed that the digital sum of the tonnage sold in 1962 was three times as much as the digital sum of the tonnage sold in 1961. Moreover, if the amount sold in 1960 (346 tons), was added to the 1961 tonnage, this total was less than the total tonnage sold in 1962 by the digital sum of the tonnage sold in that same year. Just how many more tons of ice cream were sold in 1962 than in the previous year? **Ans. 361 tons** ## 70. Three rectangles of integer sides have identical areas. The first rectangle is 278 feet longer than wide. The second rectangle is 96 feet longer than wide. The third rectangle is 542 feet longer than wide. Find the area of the rectangle. **Ans. 1,466,640** 71. Every year an engineering consultant pays a bonus of $300 to his most industrious assistant, and $75 each to the rest of his staff. After how many years would his outlay be exactly $6,000 if all but two of his staff had merited the $300 bonus, but none of them more than twice? **Ans. 8 years** 72. Two wheels in the same plane are mounted on shafts 13 in. apart. A belt goes around both wheels to transmit power from one to the other. The radii of the two wheels and the length of the belt not in contact with the wheels at any moment are all integers. How much larger is one wheel than the other? **Ans. 5 inches** 73. Dad and his son have the same birthday. On the last one, Dad was twice as old as Junior. Uncle observed that this was the ninth occasion on which Dad's birthday age had been an integer multiple of Junior's. How old is Junior? **Ans. 36** 74. [NOV 2019] The undergraduates of a School of Engineering wished to form ranks for a parade. In ranks of 3 abreast, 2 men were left over; in ranks of 5, 4 over; in 7's, 6 over; and 11's, 10 over. What is the least number of marchers there must have been? **Ans. 1154** 75. The sum of the digits on the odometer in my car (which reads up to 99999.9 miles) has never been higher than it is now, but it was the same 900 miles ago. How many miles must I drive before it is higher than it is now **Ans. 100 miles** 76. A rectangular picture, each of whose dimensions is an integral number of inches, has an ordinary rectangular frame 1 inch wide. Find the dimensions of the picture if the area of the picture and the area of the frame are equal. **Ans. 3x10 or 4x6** 77. My house is on a road where the numbers run 1, 2,3,4 ... consecutively. My number is a three digit one and, by a curious coincidence, the sum of all house numbers less than mine is the same as the sum of all house numbers greater than mine. What is my number and how many houses are there on my road? **Ans. #204 and 288 houses** 78. [NOV 2022] On what days of the week can the first day of a century fall? (The first day of the twentieth century was Jan. 1, 1901) **Ans. Monday** 79. No factorial can end in five zeros. What is the next smallest number of zeros in which a factorial can not end? **Ans. 11** 80. [NOV 2023] [MAY 2017] A ball is dropped from a height of 10 feet. It rebounds one-half the distance on each bounce. What is the total distance it travels? **Ans. 30 feet** 81. Mr. X veers to the right when he walks. The curvature of his path is proportional to his latitude. He starts walking North from point A on the equator, in the area of a large level plain, and finds he is proceeding East when he is one mile north of the equator. He continues walking and arrives back at the equator at point B. What is the straight line distance from A to B? **Ans. 1.2 miles** 82. A boat owner agrees to take a group on an outing at $4.50 a piece if the number of passengers is equal to or less than his break even point. For each person above this he reduces the fare for all passengers 3 cents per person. If he has on board now the number of passengers that maximizes the total collected, what is the boat owner's profit? **Ans. O** ## 83. The price per cubic inch for platinum trays is the same as that per square inch for platinum sheets. A metal supply house has a square of platinum which will yield the same amount whether sold as a sheet, or fashioned into a tray of maximum volume with the four cut-out corners sold as sheets. How big is the square? **Ans. 1 ft** ### LITTON TREND | Month | Year | Count | |--------|------|-------| | NOV | 2016 | 5 | | MAY | 2017 | 5 | | NOV | 2017 | 4 | | MAY | 2018 | 1 | | NOV | 2018 | 2 | | MAY | 2019 | 3 | | NOV | 2019 | 1 | | NOV | 2021 | 2 | | MAY | 2022 | 0 | | NOV | 2022 | 4 | | APR | 2023 | 2 | | NOV | 2023 | 4 | | APR | 2024 | 1 |