A bit of fun, see if you need a tutor? (2 Viewers)

[oasfree]

Honestly you guys got close but did not quite solve it. Some got it right but by the wrong way, therefore invalid. Some just got it wrong straight out. What I mean is it is still wrong if you get the right answer using the wrong method. I would like once again show the original question which was a multiple choice format
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Anna walks to school in 20 minutes. Anna can cycle to the school in 5 minutes. How much faster can Anna cycle than walk?

B 3
C 4

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I keep only 2 reasonable choices here. What it means is that you only need to guess. I suppose you either pick B or C. And if you hit what the author had in mind, you win. Otherwise you lose. I change "How much faster" to "How many times faster" after I see the ambiguity. Just by looking at the choices given, I know that the question is asking "How many times faster" rather than a pure "difference" in speed which could not be determined at all when the distance is not provided. It is clearly poor English wording. So I changed the question to "How many times faster can Anna cycle than walk?" just to remove the obvious ambiguity/fault. So now we have only one clear question to answer.

So the poor kids facing this selective school test had to pick B or C. Only one is right. When I asked people to solve it, even the ones that got the right result got it wrong because invalid reasoning was used. And this shows the limitation of multiple choices. You cannot show that you understand the matter. You cannot show that you think the author of the question got it wrong. In this case I think the author got the idea and answer right, but the person used bad wording. So I altered the wording to clear that up.

I just want to give more people a chance to comment. In my solution I will first eliminate the wrong reasonings that people have been trapped using proof by contradiction. Then I will solve the only left over interpretation of the question by algebraic method to show how algebra can hit it right on the head and clear the confusion once for all. I think when you set out to completely solve a problem, even a simple problem can become quite a challenge.

First, I want to remove the wrong reasonings

1/ You can wrongly use 20/5 = 4. Your answer will be 4x but it's wrong. The question is referring to speed, you try to solve it by a ratio of time. This is a disconnection here. You cannot validly jump between speed and time without showing a valid connection

2/ You can translate to speed and still get it wrong. Assuming the distance is D

D/5 divided by D/20 = D/5 x 20/D = 20/5 = 4

So the answer is 4x speed which is wrong. Why?

Let's imagine that cycling time is same as walking time, hence the speeds are also the same. Just pick any number say 5 minutes

D/5 divided by D/5 = D/5 x 5/D = 1

Now you have the answer "Anna can cycle 1 time faster than walk". This is clearly a contradiction when the time and speed are the same. People have made a mistake by leaving the word "faster" out.

Clearly, there is a big possibility is that the right answer is 3x

1/ I will not accept 3x as the valid answer if you arrive at 3x this way

20 - 5 = 15 (the difference in times)
15 / 5 = 3

While the answer (3x) is right and the reasoning seems on the right track, it fails to address the fact that the question is about speed, not time. As speed is inversely proportional to time, you really blow it!

2/ Therefore the correct answer must be calculated from looking at the difference in speeds and it is in term of speed. And here it is in algebra

Let D be the distance
Let Sw be the speed of walking
Let Sc be the speed of cycling
Let Tw be the time for walking
Let Tc be the time for cycling

1) Sw = D/Tw = D/20
2) Sc = D/Tc = D/5

The difference in speed between cycling and walking is Sc - Sw

3) Sc - Sw = D/5 - D/20 = 4D/20 - D/20 = 3D/20 = 3 x D/20

Now look at step 1 and 3, we know D/20 is Sw, we derive

4) Sc - Sw = 3 x Sw

Look at the question again "How many times faster can Anna cycle than walk?"

Equation 4 clearly shows the answer. The difference in speed between cycling and walking is exactly 3 times the speed of walking. In correct English comprehension, this "difference in speed" is the "faster" bit the question is asking.

BTW: a simpler problem that illustrates the exact same English comprehension is
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Anna has 4 dollars. John has 1 dollar. How many times more money does Anna has than John?"
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If you solve it by 4/1 = 4, you are screwed! because using same method, if both of them has 1 dollar each. You end it up doing 1/1 = 1. So Anna has 1 time more money than John when, in fact, they have same money! That's a contradiction. You have to solve it by (4-1)/1 = 3 because of the word "more".

I hope this whole exercise shows that we need the clarity of reasoning, the accuracy and the completeness when we are serious about problem solving.
So many people were stumped by this question regardless of their qualification and age! The habit to rush and avoid critical thinking screw us all.

 

[oasfree]

oasfree, the selective schools test does not require algebraic thinking.

in fact, i didnt even learn algebra until about year 8.

if you actually have an answer, your not allowed to use algebra to prove it because people in the test shouldnt need to use algebra in order to solve the problem.

It was a multiple choice problem. You only need to pick a choice. But I present it this way to see if people really understand the problem. You actually learn algebra since you were in grade 2! Algebra is just a way of thinking. In grade 2 you surely had problems like "John has 5 fruits. John gave away some fruits. He has 2 left. How many did he give away". That's a classic algebra problem for grade 2 kids.

 

[Takuto]

lol that isnt algebra.. it can be worked out using simple arithmetics. 5-2. even then its sooo basic that its negligible. i dont think the year 6'ers can jump from that simple problem to ur working out above. LOL

and seriously, leaving out that its a multiple choice. LOL ? of course we'll get it wrong.. cause there are a few ways to answer it..

 

[oasfree]

lol that isnt algebra.. it can be worked out using simple arithmetics. 5-2. even then its sooo basic that its negligible. i dont think the year 6'ers can jump from that simple problem to ur working out above. LOL

and seriously, leaving out that its a multiple choice. LOL ? of course we'll get it wrong.. cause there are a few ways to answer it..

You should research more on what algebra is. Even kids at year 1-2 have done algebra in a very informal way. It's a way of thinking that treats some entity (written or unwritten) as the unknown to be discovered. Then the entity is entered into a relationship. Resolution of the problem is by way of understanding how these entities are balanced by the constraints. Modern formal algebra as you do in HSC is a formal mathematical logic that is implemented as a language. This mathematical language is now embedded in all programming languages. If you ever learn computer languages, you will realise this embedding.

In this problem I don't care what answer you guys give as long as you can back it up logically. In fact I reckon it was unfair for the problem to be used as a multiple choice question as in its original form, there was English wording problem. I did remove that ambiguity/fault in the English to make it clearer. In the second variant, I think the problem is a very common applied mathematics problem that people talk about every day. But to solve it correctly and completely requires discipline. When I found that so many people failed on this including adults with University degrees in computer science and mathematics, I was astounded. I want to show people that completeness is important to deal with problems that are more than what meet the eyes. At University level, you guys will start to see how completeness is the final verification proof for all rules and formulas.

This grade 1-2 problem is a classic example to show the flaw of indiscipline thinking
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Anna has 4 dollars. John has 1 dollar. How many times more money does Anna has than John?"
---

I was completely amazed to find educated adults failing to solve it. They jumped immediately to answer 4/1 = 4 and declared it a no brainer until I showed them the contradiction in their thoughts. It turns out that sloppy language comprehension and careless usage of English has damaged their ability to comprehend even kiddy stuff.

If you hire a professional electrician to wire up your house, you pay big money because you cannot trust your untrained mate to do it. If NASAS hire a mathematician to calculate the path for the launch of a spaceship, for the same math, they still want a very discipline professional to sending it to empty space.

What I learn from simple every day problems is that nothing is easy when you want to do it properly. For example, I find even simple job like lawn mowing isn't easy when I try to do it properly. To reach any level of perfection and completeness, it's hard. This lack of perfectionism is now affecting all students and teachers and workers.

 

[Takuto]

the kids are just taught a method to straight out 5-2 it when they are shwoing working.

not like they are going to 2 + x = 5 then move the 2 over, hhah.

i dont know abt u but i dont count that as doing algebra.. u dont even learn elementary algebra til year 8 or something

 

[Sayangliss]

I say the answer is C- so Anna can cycle 4times faster then she can walk.
My logic was that for every 1minute of distance she traveled on the bike, she needs 4minutes to travel the same distance by walking.
Like that dollar question, Anna has 4x more money then John has - John needs an addition of 3 more dollars, but its really 4 times of his own amount to match Annas'.

Not very algebraic though, and I think previous users had that logic too.... if that is logic.. = =

 

[Takuto]

actually since the op finally told us its a MC and removed the ambiguity, the answer now has to be 3x faster.

20 mins - 5 mins = 15 mins saved

15 mins / 5 mins = 3

3x faster.



4x AS FAST using your guys working would be a perfectly valid answer as well

 

[georgefren]

no. your second question is still incorrect- she has four times the money of john.

the thing is, your trying to use english incorrectly. if you had asked how "much" more money, it would be three dollars. you asked "how many times more" which has to be four.

 

[Takuto]

isn't he right?

1x more of $1 would be $2

2x $3

3x $4

4x $5

?

 

[addikaye03]

lol that isnt algebra.. it can be worked out using simple arithmetics. 5-2. even then its sooo basic that its negligible. i dont think the year 6'ers can jump from that simple problem to ur working out above. LOL

and seriously, leaving out that its a multiple choice. LOL ? of course we'll get it wrong.. cause there are a few ways to answer it..

it kinda is, remember when we used to get Q like 2+ box=7, ultimately the same as having a pronumeral e.g 2+x=7 therefore x=5.

 

[Takuto]

yer but we aren't taught the algebraic relationship where u like move the 2 to the other side.

i would have counted up to do that back then . lolhhah

 

[Sayangliss]

no. your second question is still incorrect- she has four times the money of john.

what? lol, that is my answer. or are you saying that that is that the incorrect answer?

1x more of $1 would be $2

2x $3

3x $4

4x $5

This is very logical, but it just makes the question all the more confusing. If you think of $2 as double the amount of $1... double signifies 2x.
>.>!! but wait, I think I get what you mean. Hmm

 

[Takuto]

double DOES signify 2x ... 'as much'

but it signifies 1x 'more'

theres a diff between 'as much' and 'more' hhah.

 

[addikaye03]

yer but we aren't taught the algebraic relationship where u like move the 2 to the other side.

i would have counted up to do that back then . lolhhah

yer i spose ur right. and so would i lol

 

[oasfree]

Do you guys notice that there is a series of math books for primary schools called "Targeting mental math" for primary school? That one is tough series. It asks kids to try to perform mental calculations. It's tough at first but kids get used to imagining and improving the ability to toss imaginary variables inside their mind. My kid was using this series from grade 1. By grade3, the kid started to beat me in memory and calculation speed. I had to resort to my experience to reach answers quicker because I started to lose on speed as the kid had a younger and trouble-free mind. By end of grade 3, the kid started to beat me consistently on tough coding problems mapping between numbers and words or letters. My kid now has mental ability to compute so fast and accurately which was support by music training as well (A++ in recent grade exam). I am no longer a match for the kid. The poor school teachers are no longer a match for accuracy and speed. My kid will be doing OC class this year. From what I have learned from teaching my own kid, young minds are so much superior in computational ability. As I get older, I am losing that speed and memory speed. You guys out there in HS are supposed to be much better in memory and speed if you bother at all to train the mind to approach math mentally. If you try that you will find the need for written math is only for clarity and completeness. Smart kids manage to work out the answer and tick the multiple choice questions faster than any normal kid. That's I think the theory behind the high pressure and rapid pace OC and selective school tests. They test your memory, your computational ability, your combinatorial ability, you algebraic ability, ... assuming you can do all that without the need to write it down.

2 + x = 5
x = 5 - 2

This example is just formalism. Mental math will deliver the answer 1000 times faster than that. You have not even got time to write the first equation, the answer already appeared in the mind when you toss the variables around inside the brain. That's why in some IQ test, they won't allow any pen and paper at all. You just stare at the computer screen and type or click in your answers.

I suppose tutors and teachers are slower than the kids but no one would have the gut to admit that. Teachers mainly offer experience, knowledge and strategy. I am sure surely getting slower every year (even though I am still quite fast) as my mental ability gets over the hill. My tip for you guys is to try to train the mind to do mental math. In exam pressure, a balance ability is mental math and written math will deliver the best and most accurate performance.

 

[oasfree]

no. your second question is still incorrect- she has four times the money of john.

the thing is, your trying to use english incorrectly. if you had asked how "much" more money, it would be three dollars. you asked "how many times more" which has to be four.

Wrong! "How much more" means something else. How about you try this?
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Anna has $4 and John has $3. How much more money does Anna have than John?
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That asks for the difference only, 4 - 3 = 1

Now, look at this
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Anna has $4 and John has $3. How many times of money does Anna have compared to John?
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The answer is 4:3 or 4/3. This question ask about ratio.

Now finally look at this
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Anna has $4 and John has $3. How many times MORE money does Anna have than John?

The answer is (4 - 3) / 3 = 1/3

This is because, if John also has $4, then the answer would be

(4-4 )/3 = 0/3 = 0

That is Anna has ZERO times more money than John. That's logically correct.

The word MORE makes a whole world of difference. And that's why selective school tests try to make a fine distinction between the sharpest minds and the normal minds. This is the kind of math in the old schools. Modern math is generally simplified to avoid hurting the lazier mind. If you ever try a Singaporean or Taiwanese math test, you will find that they screw kids badly. Every question has a twist (trap on contradiction or conditions) to make sure kids must be on the alert at all time. American math tests from some advanced states like Florida and New York also have same philosophy. They put in a whole bunch of questions that only critical minds can avoid the traps.

Do you guys know that if kids only need to score about 65% across GA, math and English to earn a place in selective schools? They fail mainly on the 45% of tricky questions that require a fine comprehension that weeds out all the traps and satisfy all constraints. That's why only the smartest kid in the whole class of 30 get a chance in OC. Then the smartest 2-3 among 30 kids in the later year get a chance to go to selective schools.

 

[Lazarus]

I have to say, I still agree with georgefren.

You are phrasing the question in the following way:

"How many times more money does Anna have than John?

But this question can only be interpreted in two ways; that is, as being equivalent to one of the following two questions:

(1) "How much more money does Anna have than John?"

(2) "Fill in the blanks - Anna has ____ times as much money as John."

Strictly speaking, the original question does not make sense, unless it is understood as meaning the same as either the first or second questions I have posed.

You ascribe much significance to the use of the words "times" and "more" in the original question. But that question is not phrased using proper English, and that is the reason why your analysis of those words appears to be contorted. When the question is rephrased in a sensible way, either the word "times" or the word "more" appears, but not both.

Clearly, if Anna has $4 and John has $1, Anna has three dollars more than John, and Anna has four times as much money as John.

Going back to the cycling question, there are two possible interpretations:

(1) "How much faster can Anna cycle than she can walk?"

(2) "Fill in the blanks - Anna can cycle ____ times as fast as she can walk."

The answers are, respectively, 300% faster and four times as fast.

If the author of the question in the selective schools test intended the correct answer to be a different one, in my respectful opinion he or she phrased the question incorrectly, because it does not ask what you are suggesting it asks.

 

[oasfree]

I have to say, I still agree with georgefren.

You are phrasing the question in the following way:

"How many times more money does Anna have than John?

But this question can only be interpreted in two ways; that is, as being equivalent to one of the following two questions:

(1) "How much more money does Anna have than John?"

(2) "Fill in the blanks - Anna has ____ times as much money as John."

Strictly speaking, the original question does not make sense, unless it is understood as meaning the same as either the first or second questions I have posed.

You ascribe much significance to the use of the words "times" and "more" in the original question. But that question is not phrased using proper English, and that is the reason why your analysis of those words appears to be contorted. When the question is rephrased in a sensible way, either the word "times" or the word "more" appears, but not both.

Clearly, if Anna has $4 and John has $1, Anna has three dollars more than John, and Anna has four times as much money as John.

Going back to the cycling question, there are two possible interpretations:

(1) "How much faster can Anna cycle than she can walk?"

(2) "Fill in the blanks - Anna can cycle ____ times as fast as she can walk."

The answers are, respectively, 300% faster and four times as fast.

If the author of the question in the selective schools test intended the correct answer to be a different one, in my respectful opinion he or she phrased the question incorrectly, because it does not ask what you are suggesting it asks.

As the original question asked "How much faster can Anna cycle than she can walk?", your view indicates that the answer is 300% faster. That means 3 times faster. Which is the right answer in my opinion. So the author's question was kind of OK. I only suggested that changing the question to "How many times faster can Anna cycle than walk?" to improve the question. If this does not improve it, then it's Ok to go back to the original question. The reason why I suggest to change the question because I think this question "How much faster can Anna cycle than she can walk?" refers to the raw amount of difference in speeds. But when I look at the choices of 3 and 4, they clearly indicate that it is about a multiple of the slower speed.

The question was not clear. Perhaps my rephrasing wasn't clear either. But I don't know how to rephrase it to make it any clearer. One thing is clear is that the question is not about "Fill in the blanks - Anna can cycle ____ times as fast as she can walk". The use of the word "faster" is completely different to "as fast as".

 

[Takuto]

But this question can only be interpreted in two ways; that is, as being equivalent to one of the following two questions:

(1) "How much more money does Anna have than John?"

(2) "Fill in the blanks - Anna has ____ times as much money as John."

"Fill in the blanks - Anna has ____ times more money than John" <- why can't it be interpreted like that? LOL