|Math 110/003 - Group 10|
Hi guys! Do you want to work with us (Group 9) for the group project? :) Our emails are on our group page. -- Ellen | firstname.lastname@example.org
EllenTsang 07:21, 18 November 2010 (UTC)
Homework 3 - Third Part - Problem Solving Skills - Due date: October 13, 2010
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.
In this problem, two different methods of telling time are used, one is by hours and minutes and the other is just minutes. Since an hour is 60 min, and the problem states that it took an hour and 20 min, this would mean it took 80 min. Therefore, it took 80 min to travel from the terminal to the airport and 80 min to travel from the terminal to the airport each way traveling at a speed of 30mi/hr.
2. A lady did not have her driver's license with her when she failed to stop at a stop sign and then went three blocks down a one-way street the wrong way. A policeman saw her, but he did not stop her. Explain.
This problem states that a lady did not have her drivers license, failed to stop at a stop sign, and wen three blocks down a one-way street the wrong way without getting stopped - which all are assumed to be things that you should pay attention to while driving. This problem however never mentioned that the woman was driving, so she could have been walking or running. Therefore, the driving rules would not apply to her and she was not breaking the law, so that is why the policeman did not need to stop her.
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.
Since we already know that the labels are incorrect, we could select a fruit the box that says APPLES & ORANGES because that way if an apple is selected, we would know that that box is apples, and the one that says oranges would have to be APPLES & ORANGES because it is labeled incorrectly so it cannot be oranges.
Box1 : APPLES & ORANGES (must be APPLES because it cannot be APPLES & ORANGES, and apples can't go in the ORANGES box)
Box2 : ORANGES (must be named APPLES & ORANGES because: it cannot be apples since an apple was selected from original APPLES & ORANGES box (box1), and cannot be ORANGES because it is labelled incorrectly)
Box3 : APPLES (must be ORANGES because: it cannot be APPLES because it is labelled incorrectly, and the original ORANGES box (box2) is now APPLES & ORANGES)
4. I am the brother of the blind fiddler, but brothers I have none. How can this be?
Being a brother could mean that he is the brother of a boy or a girl. Therefore, since he does not have any brothers, the blind fiddler must be his sister.
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?
Since the quarter has to go all the way around the other quarter, if you try it, you will find that it revolves twice. One rotation will get it halfway around the quarter, and the other rotation will bring it to its original position.
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?
Considering if there were only 6 apples in the basket with two of each kind, then after six draws one can be sure of getting at least two of one kind.
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 the socks were drawn one by one and the pair had to be drawn consecutively, there is a possibility that that may never happen if the socks drawn come out alternatively (considering that once the sock is drawn it is not put back in.
ii) Again, using the same method in part i (drawing the socks one at a time) if all the socks drawn up until the 40th sock were the same color then by the 41st pick you would have drawn a pair of mismatched socks.
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.
Reuben was born on January 1st at 12am. Then two days later it would be a new year, he would have turned 21 from 20. Then later in the year he celebrates his birthday again at 12am, turning 22. Then it becomes the next year which means he is 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 five feet, how many rungs will be showing?
If the boat is at least 15 feet tall then all the rungs on the ladder would show, as well if the ladder is hung horizontally then all the rungs would show as well.
10. Suppose one-half of all people are chocolate eaters and one-half of all people are women. (i) Does it follow that one-fourth of all people are women chocolate eaters? (ii) Does it follow that one-half of all men are chocolate eaters? Explain. Patterns (i) and (ii) will not follow, as we are using the same group of "all people" where one half are all women and one half are all chocolate eaters does not make this exclusive. Meaning that the chocolate eaters can be included in the group of women and vice versa. If the question was phrased as "one half of all people are women while the OTHER half are chocolate eaters" then patterns (i) and (ii) may follow.
10. Suppose one-half of all people are chocolate eaters and one-half of all people are women. (i) Does it follow that one-fourth of all people are women chocolate eaters? (ii) Does it follow that one-half of all men are chocolate eaters? Explain.
(i) It does not follow that ¼ of all of the people are women chocolate eaters because the facts above state that ½ of the total people are women, it does not mean that they all eat chocolate. (ii) Similarly, it does not follow that ½ of men are chocolate eaters. Being a man or woman is independent of preference for chocolate.
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, and the best player are of opposite sex. The worst player and the best player have the same age. If this is possible, who is the worst player?
This is not possible because the facts are inconsistent such that the situation doesn’t make sense with the question.
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.
If for instance the Bronx train arrives at 10:00, 10:10, 10:20, and the Brooklyn train arrives at 9:59, 10:09, 10:19 then he would take preference over the Brooklyn train and never visit because it arrives earlier.
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)?
If you consider the time between chimes then you know it takes 45/4 seconds to strike 10:00.
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?
(i) There are six ways that two of the four babies can be correctly tagged. (ii) There are no ways that ¾ of the four babies can be correctly tagged and 1 incorrectly.
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?
Alex is correct, you should not accept his bet because naturally the amount of red cards is 50/50 to the amount of black cards in the deck.
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?
D = Daughter S = Son
Daughter has same number of brothers as sisters. Therefore, D = S + 1
Son has twice as many brothers as sisters. Therefore, S - 1 = D/2
Combine the equations together to solve for S:
2(S+1 = D/2) = 2S - 2 = D
Since we already know that D = S+1, S + 1 = 2S - 2
To solve for S, isolate S: 2S-S = 1+2 S = 3
We can find D by substituting S into one of the above equations:
Using the equation D = S + 1, D = 3 + 1 D = 4
It can therefore be seen that there are four daughters and three sons.
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.
This problem can easily be solved by considering what happens if Dan stands on the scale while sarah is already on it. The scale will read 50 kg with Sarah on the scale and then 110 kg when Dan gets on as well. Since this reading is 5 kg higher than the total reading in the example (105 kg), the scale reads 5kg too high.
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?
In this situation, Alice leaves two thirds of a whole, Bret leaves two thirds of two thirds (4/9), and Carla leaves two thirds of four ninths (16/81).
Adding the fractions of the total amount left in the jar, we find that it equals 8/27. Now we need to determine what value of x satisfies the equation that (8/27)x = 40
This can be done by dividing 40 by 8/27. The value for this is 135. Therefore, the jar started with 135 pennies originally.
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?
M = Total Milk C = Total Coffee A = Angela's Drink F = Family
A = (1/4)M + (1/6)C F = (3/4)M + (5/6)C
The least number of people in the family is the number of people required to drink the remaining 5/6 of coffee (5 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?
F = Fast Clock S = Slow Clock
Both F and S = 5 minutes.
One Hour = 60 Minutes
60 = x (F + S)
aka. the clocks become 10 minutes apart every hour.
60/x = 10
60/10 = x
x = 6
The clocks are therefore an hour apart after six hours.
Basic Skills Project:
As a group we plan to contribute to the inequalities section. By creating subsections of: solving linear inequalities quadratic rational radical trigonometric exponential logarithmic inequalities
and making examples of each of them we can make it clearer for other students as well as adding other useful online links.
An example can be shown with: Subsection 1: Linear inequalities
online reference: http://www.purplemath.com/modules/ineqgrph.htm
video explanation: http://www.youtube.com/watch?v=0X-bMeIN53I
- not sure if this is how you make the homework page ..
Basic Skills Project:[ http://wiki.ubc.ca/Course:MATH110/003/Groups/Group_10/Basic_skills_project]
Hey guys, I suggest to organise your pages in the wiki in a way that makes a bit more sense and keep it tidy. What about you write the solutions of these problems on your page and we keep the discussion page for discussions?
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?
since Sven placed exactly in the middle among all runners and Dan came in slower than Sven in 10th place this would mean that Sven would have to be in 9th place or higher. Let's say that Sven came in 9th place this would mean that there are a total of 17 runners since 9 is midway of 1 and 17. This shows that there were a total of 17 runners in the race since Sven coming in at 8th place would mean a total of 15 runners. However, since Lars came in at 16th place this would mean that there are more than 15 runners.
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?
There would be a total of 23 days in the vacation since 11+12=23
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.
No, it would not be possible for Paul to determine the ages of each specific child since there is a lack of information given. In addition, qualitative information such as the oldest child having red hair would have no bearing on the situation.
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?
One candle was exactly twice as long as the other exactly 2 hours after being lit. this is because one candle took twice as long to burn out than the other which would mean that 2/3 of the 6 hour candle=4 and 1/3 of the 3 hour candle=2
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.
In order to solve this problem we used the equation length/time to determine how long it would take for a candle to burn out. The candle with length L took 4 hours to burn out (10:00-6:00) while the candle L+1 took 6 hours to burn out (10:30-4:30). If we apply this to the formula we get L/4 and L+1/6. In order to find L we must equate these to each other L/4=L+1/6. This would equal 6L=4L+4. 2L=4 which means the length of L=2.