Course:MATH110/Archive/2010-2011/003/Groups/Group 06/Homework 3

1) A bus traveled from the terminal to the airport at an average speed of 30 mi/hr and the trip took an hour and 20 min. The bus then traveled from the airport back to the terminal and again averaged 30 mi/hr. However, the return trip required 80 min. Explain. -->Write down all known data given in the equation and draw a picture if possible.

   Trip to terminal via bus

•Velocity = 30miles/hr

    *	Change to m/s – 30miles/hr x 1hr/3600sec x 1600miles/metre = 13.333m/s

•Time = 1 hr and 20 min.

    * Change to seconds - 1 hr = 60 min. 60 min + 20 min = 80 min x 60sec/1min = 4800seconds.
  Trip returning from terminal via bus

•Velocity = 30 miles/hr

    *	Like above change to m/s = 13.3m/s

•Time = 80 min

    * Change to seconds – 80min x 60seconds/1min = 4800seconds.

→ Look at your data and ensure all units are similar. If they are not, change to similar units. This ensures your answer in the end will not end up with units you are not familiar with or have to change throughout the equation multiple times. Write down the units each time you use them in a calculation, allowing you to go back and check where and if any mistakes have been made when calculating or cancelling out units. The questions asks why one bus took an hour and 20min, while the other only took 80 min. When you change them to similar units (seconds) you see they are the exact same time with different units. This is an easy way to make mistakes and ending up with the wrong number in the end or making the problem more difficult than need be. You ask yourself what am I looking for? The time of each bus trip. If theory if they are traveling at the same speed along the same route they should have the same time for travel – ignoring outside variances. Change units and this confirms your thought.

2) A lady did not have her liscence when she failed to stop at a stop sign and proceeded 3 blocks down a one way in the wrong direction. A policeman did see her but did not stop her. Why? Information known: She ran a stop sign and drove down a one way for 3 blocks in the wrong direction. Question – why wouldn’t a policeman stop here. Look at your assumptions – she was driving, maybe the policeman was lazy and let her off, was she driving recklessly. Reread your question with those assumptions in mind, are they confirmed anywhere ? It states she “failed” to stop, not that she ran. She “went” 3 blocks, not drove. Always think of the assumptions your are making and see if they make sense/any data to back them up. → The only thing to make sense for the policeman not stopping her would be because she is NOT driving. Walking/running or biking maybe. Reread the question/statement with the thought of her walking in mind, does it make sense? YES.

3) One of three boxes contains apples, another box contains oranges, and another box contains a mixture of apples and oranges. The boxes are labeled APPLES, ORANGES and APPLES AND ORANGES, but each label is incorrect. Can you select one fruit from only one box and determine the correct labels? Explain. Draw a picture of each box and label each of the boxes with oranges, apples and apples and oranges. You want to know the correct labels and how to find out. Which box should you draw from first? Does it make a difference? If you draw from the box labeled oranges and apples first and get an apple. It goes to say that this box must be “only apples”, as it is the only label that works other than the one it is already labeled with, but the question/statement clearly says they are incorrectly labeled, so by elimination it goes to say it is “Apples only”. You have two boxes left. One labeled “oranges” and the other “apples”. You do not even need to continue selecting fruit, you should be able to decipher by elimination now. If this label is incorrect, the only other label it can possibly be is just oranges. You can ask why can’t it be “oranges and apples”? Because if it were than the 3rd box labeled oranges would have to be just that, once again the labels are incorrect, so it cannot. This third box must be “apples and oranges”. Check → You can draw from that both boxes, from your box originally labeled “apples” you will draw an orange confirming your guess. The box labeled “oranges”, you will select either an apple or an orange, what you select doesn’t tell you much but that makes sense because it could be either.

4) I am the brother of the blind fiddler, but brothers I have none. How can this be? Questions to ask? Who is the fiddler, male or female? Remember assumptions are not always right and may affect how you interpret the question. If the fiddler were a male, would this statement make sense? If the fiddler were a female would this statement make sense? Reread the statement with each of these thoughts in mind, there is nothing saying the fiddler is a male, and it would only make sense if the blind fiddler were a female. Draw a picture and describe relationships to each other if need be and you can see how the fiddler being a female works with the statement.

5) Two quarters rest next to each other on a table. One coin is held fixed while the second coin is rolled around the edge of the first coin with no slipping. When the moving coin returns to its original position, how many times has it revolved? What information is known? What are you trying to find out? You can see you are dealing with two circular objects of the same size, one lying flat and the other being rolled around the edge of the one lying flat. The “same size” clue gives you all the information needed. Diameter of circular objects = the same, therefore the circumferences = same. If you are rolling the object on its edge around the edges of the other “same sized” object while its lying flat on the counter, you can only assume the times it will revolve is one. Since they are the same size it will not take any longer or any shorter of a distance to go around something of the same distance as its circumference. Check = grab two quarters, try it out.

6) Three kinds of apples are all mixed up in a basket. How many apples must you draw (without looking) from the basket to be sure of getting at least two of one kind? You have a box full of 3 different types of apples, you want to know how many times must you draw in order to draw two of the same kind. Does it matter how many apples there are in total? How many times can you pick apples from the box without picking the same one twice? Because there are 3 different kinds of apples, you can draw 3 times with a chance of coming up with a different apple each time. Now you have exhausted that option, the next apple you draw will have no choice but to be the same kind as one of the previous ones. Therefore, you see drawing 3 apples will give not be enough to ensure you draw one kind twice but drawing 4 apples will guarantee you draw one type of apple twice.

7) Suppose you have 40 blue socks and 40 brown socks in a drawer. How many socks must you take from the drawer (without looking) to be sure of getting (i) a pair of the same color, and (ii) a pair with different colors? i) If you draw 3 sox from the drawer two of them will be the same color. ii)You have to draw 40 sox from the drawer because even though it is extremely unlikely you could draw 40 of the same color sox in the drawer.

8)Reuben says, “Two days ago I was 20 years old. Later next year I will be 23 years old.” Explain how this is possible. Reubens birthday is on Dec. 31st. On Dec. 30th 2010 Reuben is 20 years old. On the 31st he turns 21. On January 1st he say "two days ago i was 20 years old next year I am 23." Next December 31st 2011 he turns 22 and LATER in 2012 he turns 23

9) A rope ladder hanging over the side of a boat has rungs one foot apart. Ten rungs are showing. If the tide rises 5 feet, how many rungs will be showing? Understand: Draw a picture of a boat that is able to be affected by the tides (must be floating on the ocean). Ten rungs of the ladder are showing. Variable: The tide rises 5 feet. Solution: the boat is still floating because no variable introduced suggests that the boat has sunk. Therefore - 10 rungs are still showing.

10) Suppose 1/2 of all people are chocolate eaters and 1/2 of all people are women. (1) Does it follow that one fourth of all women are chocolate eaters? (2) Does it follow that one half of all men are chocolate eaters? Explain. Understand: All people suggest a population of people. This can be interpreted as any group. Suddenly, half of all people are women. This forces the population to be half women and half ~women (not women). Half of all people are chocolate eaters. Half of the population (composed of half women, half ~women) are chocolate eaters. I picture this by imagining two different coloured circles (red and blue), one superimposed over the other. Each circle is bisected into 2 equal parts. The red circle represents women (shaded) and ~women (not shaded). The blue circle represents chocolate eaters (shaded) and ~chocolate eaters (not shaded). Notice that the circles, when one is rotated, the colour mix moves with it - the circles are not locked together; they do not necessarily spin together. 1. Does it follow that 1/4 of all women are chocolate eaters? No. It doesn't. This problem suggests no linkage between (chocolate eaters, ~chocolate eaters) and (women, ~women). 2. Does it follow that 1/2 of all men are chocolate eaters? No. Same as above.

11) A woman, her older brother, her son, and her daughter, are chess players. The worst player's twin, who is one of the four players, is the best player of the opposite sex. The worst player and the best player have the same age, who is the worst player? Understand: Draw the family. Assign them ages by changing their heights tallest(oldest). Assume that there is a twin. Logically infer: what is a twin, who in this group is capable of being a twin? Twins are born at the same time. The mother has an older brother, therefore the mother and the older brother cannot be the twins. The woman's children must be the twins. At this point, we know everyone's sex and everyone's age relative to the other members of the group. The characteristics defining who the best and worst players are are these: [the worst player has a twin] [the worst and best players have the same age]. From these, we can infer that the worst and best players are the children (because only they can have the same ages in the group). Unfortunately, that is as far as we can go in determining who are the best and worst players because the determining characteristics to not suggest which sex either explicitly is. This exhausts the two things we know of each member of the group (relative age and sex).

12) A Manhattan fellow had a girlfriend in the Bronx and a girlfriend in Brooklyn. He decided which girlfriend to visit by arriving randomly at the train station and taking the first of the Bronx or Brooklyn trains that arrived. The trains to Brooklyn and the Bronx each arrived regularly every 10 minutes. Not long after he began his scheme the man's Bronx girlfriend left him because he rarely visited. Give a (logical) explanation. The manhattan fellow is taking trains that are arriving, not departing. The question says nothing of the other girlfriend, who is probably equally as annoyed with the manhattan fellow because she never sees him either. The premises of the problem, specifically [(A manhattan fellow) decided which girlfriend to visit by arriving randomly at the train station and taking the first of the bronx or brooklyn trains that arrived] do not explicitly state that the trains that arrive at the station then proceed to depart the station. The logical explanation is that the premises of the problem do not suggest either girlfriend being satisfied.

13) If a clock takes 5 seconds to strike 5:00 (with 5 equally spaced chimes), how long does it take to strike 10:00 (with 10 equally spaced chimes)? The problem does not suggest exactly what the time delay between the 10:00-strike chimes is as the 5:00-strike chimes is stated (at 5 sec/5 chimes = 1 sec/chime). The chime at 10:00 could be a faster set of chimes, perhaps 10 chimes over 3 seconds?

14) One day in the maternity ward, the name tags for four girl babies became mixed up. (i) In how many different ways could two of the babies be tagged correctly and two of the babies be tagged incorrectly? (ii) In how many different ways could three of the babies be tagged correctly and one baby be tagged incorrectly? Identify the babies (w,x,y,z). Identify the names (w,x,y,z). (Baby,Name). Only one of each baby and one of each name tag exists. When correctly tagged, (w,w), (x,x), (y,y), (z,z). i. two babies tagged correct, two incorrect. if (w,w) and (x,x), then (y,z) and (z,y). Cases: 6 unique cases because w|x,y,z x|y,z y|z z|(none) 2. If three of the babies are correctly tagged then the last baby cannot be uncorrectly tagged (can only be correctly tagged), because each the babies and the name tags are mutually exclusive when paired (babies cannot share name tags, only one of each baby and name tag exist). Three correctly tagged babies would result in one tag left for one baby. As 3/4 tags were correct, it follows that the 1/4 tag remaining must be correct. Case: (x,x) correct! (w,w) correct! (y,y) correct! z is tagged with what? (z,z) because z is the only baby remaining. (z,z) is correct!

15) Alex says to you, “I'll bet you any amount of money that if I shuffle this deck of cards, there will always be as many red cards in the first half of the deck as there are black cards in the second half of the deck.” Should you accept his bet? Tell Alex to get a job because you should not take this bet. There are 52 cards in a deck and if you split the deck in 2 there are 26 cards. If you have 12 black cards and 14 red cards in the first half you will automatically have 14 black cards and 12 red cards. This happens because there are 26 black and red cards and that is the whole deck. And if you split the deck in half and count 12 black cards you know in the other half there will be 14 black cards and that works for red cards aswell. 16. Suppose that each daughter in your family has the same number of brothers as she has sisters, and each son in your family has twice as many sisters as he has brothers. How many sons and daughters are in the family? Well this is quite a large family. There are 4 daughters and 3 brothers. Look at the specifications in the question and they fit.

17. The zero point on a bathroom scale is set incorrectly, but otherwise the scale is accurate. It shows 60 kg when Dan stands on the scale, 50 kg when Sarah stands on the scale, but 105 kg when Dan and Sarah both stand on the scale. Does the scale read too high or too low? Explain. The only conclusion I can see is that even if the scale was set too low or too high, the combined weights (which may be false) should still add up if the scale is still set to its wrong setting.

18)Alice takes one-third of the pennies from a large jar. Then Bret takes one-third of the remaining pennies from the jar. Finally, Carla takes one-third of the remaining pennies from the jar, leaving 40 pennies in the jar. How many pennies were in the jar at the start? If you have x pennies, you remove a third of the x, then you're left with y pennies. You subtract a third of the y pennies, which leaves you with z pennies. subtracting a third of the z pennies will leave you with 40 pennies. You can make an equation, to find x,y,z. You keep plugging in values into each previous equation to find these values. I worked backwards once I got the answers and saw that I ended up with 40. 135 pennies were there at the start.

19)One morning each member of Angela's family drank an eight-ounce cup of coffee and milk, with the (nonzero) amounts of coffee and milk varying from cup to cup. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. What is the least number of people in the family? Since the number of people that drink coffee in the morning drink the same amount of liquid, you can make a formula where m is milk in oz,and c is coffee oz and n is the number of people. m divided by four (because angela drinks a quarter of the total milk) plus c divided by six (she drinks a sixth of the total coffee) multiplied by n which equals the total amount of milk plus coffee. the number of people is at least 5. the only way you can have a positive answer is by making equal to five since you cannot have a negative number of people!

20)Of two clocks next to each other, one runs 5 min per hour fast and the other runs 5 min per hour slow. At midnight the clocks show the same time. At what time are they are one hour apart? When it is midnight, after each hour the fast clock will gain five minutes more than it normally would. 1:05 becomes 2:10, which goes to 3:15, etc. This is the same for the slow clock except it goes backwards. After an hour it should have gone to 1 oclock but it is only at 12:55. At 6:30 for the fast clock it will only be 5:30 on the slow clock.

21)Sven placed exactly in the middle among all runners in a race. Dan was slower than Sven, in 10th place, and Lars was in 16th place. How many runners were in the race? If Sven is right in the middle, there has to be equal numbers of people before him or after him. If Dan is slower than Sven in 10th place, Sven could be ninth. With equal numbers on both sides that equals seventeen including Lars, who is 16th.

22)During a vacation, it rained on 13 days, but when it rained in the morning, the afternoon was sunny, and every rainy afternoon was preceded by a sunny morning. There were 11 sunny mornings and 12 sunny afternoons. How long was the vacation? You can make rainy mornings, rainy afternoons and sunny in both three variables: x,y,z respectively. Using the terms given, you can make a system of equations. This is the only way that seems possible. The wording was confusing and it was not clear whether or not there were entire days that were sunny.

23)Suppose you overhear the following conversation: Paul: How old are your three children? Paula: The product of their ages is 36 and the sum of their ages is the same as today's date. Paul: That is not enough information. Paula: The oldest child also has red hair. If you were Paul could you determine the ages of Paula's children? Explain. Paula is clearly a nut. 1)We do not know the date she is referring to. 2)Red hair, besides being unattractive, does not tell give us numerical information about their ages.

24. Two candles of equal length were lit at the same time. One candle took 6 hr to burn out and the other candle took 3 hr to burn out. After how much time was one candle exactly twice as long as the other candle? Make both candles = 12 cm to make it easy. And also lets call the 6hr candle A and the 3 hr candle B. Now we must determine the cm per hour so divide 12 by 6 to get 2cm/h and 12 by 3 to get 4cm/h. Now in the first hour candle A becomes 10cm and B becomes 8 cm. In the second hour candle A becomes 8cm and candle B becomes 4cm. So it takes two hours for it to equal half.

25. Two candles of length L and L + 1 were lit at 6:00 and 4:30, respectively. At 8:30 they had the same length. The longer candle died at 10:30 and the shorter candle died at 10:00. Find L. link title