Course:MATH110/Archive/2010-2011/003/Groups/Group 15/Homework 3

Question 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.

1hour and 20min is identical to 80min, there is nothing else to explain.

Question 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.

Since the policeman never had the opportunity to check the driver’s license, he wouldn’t have known that she did not have it on her. Therefore, that mistake was irrelevant to the question why the policeman didn’t stop her at her antics. There can be a few possibilities why the policeman didn’t stop the driver despite seeing her apparently breaking road etiquettes (failing to stop at a stop sign and going down a one-way street the wrong way). The policeman could have been preoccupied with another road incident and therefore didn’t manage to stop the female driver in time. Or it could be that execution of road policies was not usually strictly carried out in that vicinity (or city, or country). Or it could be that the woman was not driving at all- there isn’t any indication in the question that the woman was in fact conducting a vehicle when she did all those things.

Reading the question “superficially”, as demonstrated by the solutions we have seen above, it is very easy to automatically assume that the woman broke the road codes while she was driving. We expect to be given the context in the beginning of any paragraph of texts. In this example, the first sentence seems to set the scene for the readers, “The lady driver did not have her driver’s license with her…”, to be immediately followed by a series of driving behaviours that we all know are breaking the road code. We link the lady and her driver’s license to the assumption that “she must be behind the wheel”. However, as one should be able to see, that piece of information was there to mislead instead of inform. I found that to understand the problem (Polya’s Step 1) really useful in thinking up solutions for this problem. Specifically it helped me to re-read the question carefully and especially not taking anything in it for granted. As we all know, a pedestrian who fails to stop at a stop sign (presumably those signs designed for vehicles), and “going” (walking) down a one way street the wrong way (wrong only for vehicles) does not get stopped by the police!

Question 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.

First, you simply pick the box that marked "apple and oranges". And it contend apple. Then you know that the box is only contain apples only. Now you know that the box that marked oranges cant not be the box that contain apply. So the "oranges box is contain apples and oranges. ==

Question 4

I am the brother of the blind fiddler, but brothers I have none. How can this be?

Since I have no brother, then the blind fiddler must be my's sister.

Question 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?

I must at least draw 4 times, that's because the 4th apple are the same as one of the first three i draw.

Question 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)pair of the same color When we draw 3 socks without looking, we can be sure that there are a pair of the same color since there are only 2 kinds of color.

ii)pair of different color When we draw a total of 41 socks, we can be sure that we have two socks that are different coloured. On the 41st draw, there must be a different color sock since one of the color is all drawn out.

Question 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 in december 31. and spoke on january 1.

Question 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?

Assuming that the boat remains in the water as the tide rises, 10 rungs will be showing. This is because as the tide rises the boat will be levitated with the rise.

It is easy to be distracted by the presence of actual figures in the problem. It would have been the easiest route to solve the problem – one has the figures, one simply needs to do some mathematical calculations. However, not only does that approach miss the point of the question totally, it also lacks creativity on the part of the problem-solver. Step 2 of the Polya method emphasises that creativity is one of the most important element (even before organisation and experience) one should adopt while planning for a strategy to tackle a problem.

Question 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?

There is no worst player, because it is not possible at all. Since we notices that the worst player are the twin and the best player is same age and opposite sex of the twin. The woman, her older brother or her son, and her daughter could be twins... Without further information, we could not figure whos who.

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.

The Manhattan fellow likely has a daily schedule along with a set time allotted to visiting his girlfriends. The trains also have a schedule so statistically speaking there will be a point of where both schedules meet up in favor of the Brooklyn train.

Question 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)?

First of all one has to consider the possibilities that, for whatever reason, the said clock is programmed in such a way that it only chimes at 5:00 and thus will never chime at 10:00, in which case this question will not be valid in the first place. That would have been easy. A little research into chiming clocks, however, reveals that if they chime at a particular hour, chances are they chime on every hour as well. Although “reading the question carefully” and “eliminating alternative possibilities” of Poyla’s Step 1 doesn’t quite apply here, at least one can definitely eliminate those possibilities before moving on to the others.

So, perhaps one is to move on to analysing the question in a more literal manner to see if it is going to bring back any valid solutions. If the clock takes 5 seconds to strike 5 chimes, we can easily visualise that each chime happens on the second – first second, first chime, second second, second chime, etc. To strike 10 chimes, then, one assumes that the first chime would be on the first second it starts chiming. Therefore, 10 chimes will take 10 seconds. However, one has no evidence from the data given (Step 1-Understanding the Problem) and needs to ask if there is enough information for a correct solution to be reached. We have been given the information in the question that the 10 chimes for 10:00 are equally spaced, but equally spaced at what value? We do not know that. Therefore, we cannot confidently come to a solution without doubts on the back of our heads until we have all the information we need to solve this problem.

Question 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?

That depends if we are using the Jokers in the deck (54 cards), if we are then I would accept his bet, if we are not (52 cards) then I would not accept his bet. This is because if we were using the jokers, they are both black cards so the total ratio of black to red is 28Black to 26Red as a result it is only possible to have a different amount of black cards to red cards in each half of the deck. However if there were no jokers in the deck, then there would be a perfect ratio of 26Black to 26Red cards. If you were to shuffle them and split them in half there will be different amounts of red and black in each stack, however the amount of one color in one stack will always be the same as the opposite color in the opposite stack. The scenario without jokers is explained below.

There are 52 cards in total and ${\tfrac {52}{2}}=26$ cards in each half. Half of all cards are red and half of all cards are black.

In the first half there are x red cards and y black cards.

x+y=26

where

x=26-y

and

y=26-x

In the second half there are the red cards which are not in the first half 26-x which is equal to y. And there are the black cards which are not in the first half 26-y which is equal to x. Therefore there will always be as many red cards in the first half of the deck as there are black cards in the second deck.

Question 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?

Assuem that the family has "x" number of girls and "y" number of boys

Then, according to the question we get two equations: X - 1 = y

and

2(y-1)= X

Let's solve these two equations by combining them together, then we get

X=4

Y=3

Therefore, this family has 4 girls and 3 boys.

Question 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.

First of all, we deduce two possible scenarios, one being a scale that reads too high and another, too low. Let’s have a closer look at both scenarios to see if they bring back a valid solution.

Suppose the scale reads too high. Dan’s real weight will then back less than 60kg, Sarah’s less than 50kg. The total of their weight is 60+50 kg = 110kg. However, when both of them stand on the scale, we know that it reads 105kg, which is lower than 110kg. 105 <110 points to the fact that the scale couldn’t have been reading too high. Instead, it seems to be under-reading their combined weight.

Now suppose the scale reads too low. Using the same line of analysis above, Dan’s real weight would have to be greater than 60kg, whereas Sarah’s would have been greater than 50kg. In this case, their real combined weight has to be greater than 110kg. The scale reads 105kg, which is less than 110kg and is, in other words, under-reading their weight, which validates the assumption that the scale reads too low.

Question 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? ]]

Let x represent the total number of pennies in the jar.

Let y represent the number of pennies Alice took from the jar.

Let z represent the number of pennies Bret took from the jar.

Let w represent the number of pennies Carla took from the jar.

From the information given, now we get 4 equations:

1) y = x/3 2) z = (1/3)(x - x/3) 3) w = (1/3)[x – (1/3)(x - x/3) – x/3] 4) x = y + z + w + 40

Plug the values into equation number 4 then we get--- (8/27)X =40 ---> X = 135

Therefore, there are 135 pennies at the start.

Question 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?

One clock runs 5 minutes faster per hour while the other runs 5 minutes slower.

After one hour, the two clocks will have a 10 minutes difference between each other.

There are 60 minutes per hour, therefore 60/10 = 6.

After 6 hours the two clocks will be one hour apart @ 6a.m.

Question 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?

There were 17 runners in the race. This is because Sven must be in less then 10th place (this is given). So assuming that he is in 9th place there are 8 people ahead of him, and because he is exactly in the middle there would have to be 8 people behind him all the way to 17th place (9+8) which would satisfy the given statement that there was somebody in 16th place. It would be impossible for Sven to place any lower then 9th place because then there would not be a 16th place for Lars to take.

Question 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?

The vacation was as long as the vacationers wanted it to be. Common sense tells us that vacationers are often on holidays well before they start, and most of them don't bother to collect themselves to normal work days after their vacations. I think they should just forget about the rains. Rain or shine, if one is at a place that pleases one, it shouldn't matter.

Question 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?

i)candle A = 6 hours

ii)candle B = 3 hours

So, Candle A takes twice more time to burn out than Candle B

Candle A: burns 1/6 per hour

Candle B: burns 1/3 per hour

Let the time it takes to burn out in hours be X

Then, Candle A = (6-x)/6

and Candle B = (3-x)/6

According to the info we get from the question, we get ---

2((x − 3) / 3) = (6 − x) / 6

6(6 − 2x) = 3(6 − x)

36 − 12x = 18 − 3x

18 = 9x

x = 2hrs

Therefore, the time it takes for one of the candle to be half of the other candle is 2 hours.

Question 25

Media:Example.ogg