BARC Final Exam: Solution and Q&A

Sum of Areas of Equilateral Triangles Inscribed In a Circle - BARC Finals

An equilateral triangle is inscribed within a circle whose diameter is 12 cm. In this triangle a circle is inscribed; and in this circle, another equilateral triangle is inscribed; and so on, indefinitely. Find the sum of the areas of all the triangles.

Solution	Problem
	A cat takes 4 leaps to a dog's 3; but 2 of the dog's leaps are equivalent to 3 of the cat’s. The cat has a start of 50 leaps. How many leaps must the dog take to catch the cat?
	In a certain municipality, all seven-digit telephone numbers begin with 321. How many telephone numbers maybe assigned to that municipality if the last four digits should not begin or end in zero?
	A drawer contains red socks and black socks. When two socks are drawn at random, the probability that both are red is 1/2. How small can the number of socks in the drawer be?
	A truth serum given to a suspect is known to be 90% reliable when the person is guilty and 99% reliable when the person is innocent. If the suspect was selected from a group of suspects of which only 5% have ever committed a crime, and the serum indicates that he is guilty, what is the probability that he is innocent?
	M. Poirot and Miss Marple walk into a turned-off escalator to realize that he is able to walk 3 steps during the time she needs to walk only 2. When the escalator is turned-on and it is already working at constant speed, M. Poirot enters again the escalator to realize he needs to take 25 steps to go through it, while Miss Marple realizes she needs only 20 steps to reach its end (at the very same conditions). How many steps are there on the escalator?
	In a certain town there are 1000 married couples. Two-thirds of the husbands who are taller than their wives are also heavier and three-quarters of the husbands who are heavier than their wives are also taller. If there are 120 wives who are taller and heavier than their husbands, how many husbands are taller and heavier than their wives?
	A substance goes into a solution at the rate of 25% of the undissolved portion each minute. If the initial amount was 64/27 grams, what amount (in grams) is undissolved after 6 minutes?
	I was walking along the road at 6kph when a car dashed past me and only missed me by a few inches. From the moment it passed me to its disappearance round a corner I took thirty-one steps and walking on reached that corner with four hundred and thirty-four steps more. Assuming that I walked and the car ran at uniform speeds, what was the speed of the car?
	A multiple-choice examination has 200 questions, each with 4 possible answers, of which only one is correct. What is the probability that sheer guess work yields from 25 to 30 correct answers for 80 of the 200 problems about which the student has no knowledge?
	What is the probability that a waitress will refuse to serve alcoholic beverages to only 2 minors if she randomly selects the I.D.’s of 6 students from among 9 students of which 4 are not of legal age?
	Studies have shown that 72.5% of apprehended snatchers in Metro Manila are drug users. In a random sample of 16 apprehended snatchers, what is the probability that exactly 10 of them are drug users?
	There are three candidates for president of Silk Residences Condominium Unit Owners Association. The probability that Albert will be elected is 0.3, the probability that Bryan will be elected is 0.5 and the probability that Charlie will be elected is 0.2. Should Albert be elected the probability for an increase in monthly dues is 0.8. Should Bryan or Charlie be elected, the probabilities for increase in monthly dues are 0.1 and 0.4, respectively. What is the probability that there is an increase in monthly dues?
	The probability that both stages of a two-stage missile will function correctly is 0.95. The probability that the first stage will function correctly is 0.98. What is the probability that the second stage will function correctly given that the first one does?
	A man arrives at the railroad station nearest his country home an hour and 5 minutes before the time at which he had ordered his driver to meet him with his car. He starts to walk at the rate of 4mph and meets his car when it has travelled 8 miles. It turns back and takes him the remainder of the way home, where he arrives just 10 minutes ahead of the time he had originally planned to get there. Find the distance from the station to his home. Assume no waiting time on the part of his driver.
	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 a minimum area. How long was the fence?
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	A coin is so unbalanced that it may come both heads in 2 tosses as it may come tails in a single toss. What is the probability of getting a head in a single toss?
	Two casks A and B were filled with two kinds of sherry. In cask A, the ratio of the two kinds of sherry is 2:7 while in B, it is 1:5. How many gallons will be taken from each cask to form a mixture which shall consist of 2 gal of one kind and 9 gal of the other?
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	Three people toss a coin, and the odd man pays for the coffee. If the coins all turn up the same, they are tossed again. Find the probability that fewer than 4 tosses are needed.
	A man passed one-sixth of his life in childhood, one-twelfth in youth, and one-seventh more as a bachelor. Five years after his marriage, a son was born who died four years before his father at half his father’s final age. What was the man’s final age?
	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?
	A man and his wife invite 8 of their friends to a dinner. They are to be seated around a round table. How many arrangements can be made if the man and his wife are to sit side by side?
	Find the values of $b$ for which the line $y = 3x + b$ and the parabola $y^2 = 5x$ will have no common point.
	A contractor has 50 men of the same capacity at work on a job. They can complete the job in 30 days, but the contract expired in 20 days. He decides to put 20 additional men. If all the men get P600/day for a full or part day and if the liquidated damages are P50,000 for every full or part of the day he requires over his contract time. How much money he saves or loss by putting 20 additional men?
	What time between 2 and 3 o’clock will the angle between the hands of the clock be bisected by the line connecting the center of the clock and the 3 o’clock mark?
	The king’s minter boxes his coins 100 to a box. In each box he puts 1 false coin. The king suspects the minter and from each of 100 boxes draws a random coin and has it tested. What is the chance the minter’s peculations go undetected?
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	Stations A and B are 120 miles apart on a single-track railroad. At the same time that a train leaves A for B at 25 mph, a train leaves B for A at 15 mph. Just as the train leaves A, a 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 continuous to fly back and forth with undiminished speed until it is crushed in the eventual collision. How far had the fly flown?
	To encourage Elmer’s promising tennis career, his father offers him a prize if he wins (at least) two tennis sets in a row in a three-set series to be played with his father and the club champion alternately: Series A: FATHER-CHAMPION-FATHER or Series B: CHAMPION-FATHER-CHAMPION according to Elmer’s choice. The champion is a better player than his father. Which series should Elmer choose?
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	Situation. The number of flaws in a meter length of carbon fiber material is known to follow a Poisson distribution with parameter 0.4. Find the probabilities that… There are no flaws in a 1-meter length. There is 1 flaw in 3-meter length. There is 1 flaw in a piece of material which is half a meter long.