AAGT1
 

:smile:
Thank you Luigi, Richard, Alex, and Numbers (welcome to M/Forum)

Luigi:Pl. refer to my AAGT1. These are from 'Can you solve these' Series 2 by Davis Wells. In all there are 3 series. Editor Jean Slack. Publishers Tarquin Pub. I got them from FOYLES Lon.

Background of David Wells:
Born 1940, and has the rare distinction of being a Cambridge scholar in maths and failing his degree.
While at University he was British under 21 Chess champion.
He is the inventor of games “Guerilla” and ‘Checkpoint Danger’
He is puzzle editor of ‘Games and Puzzles magazine’ and ‘Problem Solver’ for secondary pupils. He has published several books most of which I have.

Regards the Quiz I find there has been no response to Ques.A1
These are the correct answers leaving out Ques A1.
2) 26/41
3) one of the many fractions 169/408 or 84.5/204
4)99,999
5)18
6)53,67,71,89

My Comments:
I enjoyed the dialog this quiz generated.
Yes Alex the idea conveyed is that decimal 63414 is periodic in 26/41

I agree with you Richard that the actual percentage is greater than the one given.
Perhaps if it was written 63.41463414* ( dot symbol of recurring decimal) it would be more correct. This is an excellent point!
Consider if one more person was asked, the fraction would then be 27/42 which is 0.642857142 and this is greater than 0.63414* (dot)
We cant consider a fraction of a human being!

Numbers: you found no difficulty in accepting the fraction and I’m glad.

Since this problem generated a controversy I put down the Hint given by David Wells

Hint:‘The decimal 0.xyz xyz is equal to the fraction xyz/999
Similarly the decimal wxyz wxyz =wxyz/9999
Ans:41. As a fraction the percentage given is as 7046/11111 which cancels down to 26/41. So the smallest number of people who could be asked the question is 41 of whom 26 replied Yes”

Pl. Note: A1 has not been answered.
Hint: Restrict your search to fractions where the numerator is 1. Because the first decimal starts 0.02 the first fraction must be between 1/50 and 1/33
Now could you crack it out?
Mally :coffee:
P.S, Yes Alex you have it

[QUOTE=mfgoode]Regards the Quiz I find there has been no response to Ques.A1[/QUOTE]
Personally, I considered them trivial and a "warm-up" for the remaining questions.
[QUOTE]
I agree with you Richard that the actual percentage is greater than the one given.
Perhaps if it was written 63.41463414* ( dot symbol of recurring decimal) it would be more correct. This is an excellent point![/QUOTE]
The typographical character ellipsis ('…') is the proper method to indicate that something continues. Although I don't think that it is customary to use multiple ones, you did use it in problem A1. In the above quote, I see an asterisk.

I tried to read something into your use of the term EXACTLY. It looked like a trick to me. Had you simply said "63.41463414…% of the people", or even "63.41463414% of the people", I would interpret it to mean that you were looking for a fraction with the smallest denominator whose decimal expansion matches the digits given.

EXACTLY why was the particular wording chosen?

:)

[QUOTE=Wacky]Personally, I considered them trivial and a "warm-up" for the remaining questions.

The typographical character ellipsis ('…') is the proper method to indicate that something continues. Although I don't think that it is customary to use multiple ones, you did use it in problem A1. In the above quote, I see an asterisk.

I tried to read something into your use of the term EXACTLY. It looked like a trick to me. Had you simply said "63.41463414…% of the people", or even "63.41463414% of the people", I would interpret it to mean that you were looking for a fraction with the smallest denominator whose decimal expansion matches the digits given.

EXACTLY why was the particular wording chosen?

:)[/QUOTE]

:smile: : Thank you wacky for your observations and comments.

[QUOTE=Wacky]Personally, I considered them trivial and a "warm-up" for the remaining questions.]

Yes they were meant to be. The general rule, but not necessarily universal, is the practice to, as you say, 'warm up'. All text books and exam papers start with the simple problems first and then move onto 'stiffer' ones towards the end.
This rule was not followed in the quiz I compiled in my selection. The word 'trivial' is a relative term. I suppose thats why it was left out in the first
response? Still it is not answered, except by Alex, which has not been posted.

Quote:[The typographical character ellipsis ('…') is the proper method to indicate that something continues. Although I don't think that it is customary to use multiple ones, you did use it in problem A1. In the above quote, I see an asterisk.]Unquote.

:sad: Yes. You are correct in ('...'). IMHO multiple ones are used and I have used it in A1. Thats exactly how David Wells printed it and its not my make up.
I used the non conventional * for recurring but qualified it with explaining that it was a dot symbol for recurring decimals.
If you know how to write the dot on the rt. hand top of a number which is popularly used for recurring decimal please let me know as I am not aware of how to print it from My MS office keyboard or if there is another recognised symbol that is used instead?

Quote[I tried to read something into your use of the term EXACTLY. It looked like a trick to me. Had you simply said "63.41463414…% of the people", or even "63.41463414% of the people", I would interpret it to mean that you were looking for a fraction with the smallest denominator whose decimal expansion matches the digits given.]unquote.

No wacky I am not an oriental yogi doing the Indian rope trick!
Please go back to my original Thread and you will find it is exactly as you say- 'Exactly 63.41463414%' This is how David Wells wrote it and thats what I copied. May I add that ('...') is not the same as the symbol written for the recurring decimal. The first term you use denotes a continuation of numbers one by one in a series like 12345..... Whereas the dot that is used is for a recurring block of digits like .63414.
Still wacky I might be wrong but by the preceding discussion I have learnt some finer points of math symbolism.
I would like to reiterate that the wording of the entire set of 6 questions is not by me but by David Wells. He does not normally answer questions on correction of his puzzles. I know because tho' he gives his address he does not acknowledge them
Thank you once again.
Mally :coffee:

[QUOTE=mfgoode]I suppose thats why it was left out in the first
response? Still it is not answered, except by Alex, which has not been posted.[/QUOTE]
If you wish it posted:
A1:[SPOILER] 1/49 [/SPOILER]
A2:[SPOILER] 1/97 [/SPOILER]

[QUOTE]If you know how to write the dot on the rt. hand top of a number which is popularly used for recurring decimal please let me know[/QUOTE]
Short of having a font which includes the particular glyph, I don't know how to reproduce it in any standard manner. Some programs are able to do sub- and super- scripts, but that is not standard and usually, at best, only approximates the desired effect.

As for the notation itself, I am not familiar with it. My initial reaction is that it is ambiguous. Does 0.010203* mean 0.01020303… or 0.0102033333… or something else?

[QUOTE] Quote[I tried to read something into your use of the term EXACTLY…

I would like to reiterate that the wording of the entire set of 6 questions is not by me but by David Wells.
[/QUOTE]

Sorry, please substitute "the author's" for "your" above. In any case, I do not feel that the problems are stated in a sufficiently precise manner.

But I do think that they are good problems. And I thank you for bringing them to us.

AAGT 1
 

[QUOTE=Wacky]If you wish it posted:
A1:[SPOILER] 1/49 [/SPOILER]
A2:[SPOILER] 1/97 [/SPOILER]

As for the notation itself, I am not familiar with it. My initial reaction is that it is ambiguous. Does 0.010203* mean 0.01020303… or 0.0102033333… or something else?

Sorry, please substitute "the author's" for "your" above. In any case, I do not feel that the problems are stated in a sufficiently precise manner.

But I do think that they are good problems. And I thank you for bringing them to us.[/QUOTE]
:smile:
Thank you for the interest you take in such elementary problems compared to all the advanced ones our members are doing in finding primes.
No Im not annoyed and dont need an apology.

The meaning is not ambiguous. if you get a 'run' like wxyz wxyz where wxyz repeats itself then the dot on the rt. hand corner of the number indicates that wxyz repeats and need only be written as wxyz' where Im putting the comma for the actual symbol which is a dot actually
Eg: 1/7 =142857 14287 142857 ....(to use your symbol . )
Then it need only be written 142857' using ' for the symbol.
Wacky I would like to request you to please post a thread on the common math symbols and their substitites which can be adapted to the standard keyboard. I see the symbols used by akkruppa and others which I cannot understand or interpret in the std. math notation.
Thank you
Mally :coffee:

I understand that you are using a symbol indicates that the preceeding group repeats (indefinitely). What I don't understand is where the group starts.

As for commonly used substitutions, I have no list. And I fear that any attempt to compile one would be rather incomplete because I tend to not even realize as unusual things that are personally common. So if you are confused about some terminology, please inquire. I'm sure that someone will be happy to give you an explanation.

Blocks of recurring digits
 

My understanding is that if what Mally refers to as a "run" but which I prefer to think of as a block of digits in the decimal expansion of a fraction repeats, there is in standard notation a dot over the top of the first and last digit in the block. Therefore, (using the symbol ' to indicate that the following digit has a dot over it) 0.0123412341234 could be written as 0.0'123'4.

However, as you have both found, it is not easy to represent a digit with a dot over it and use of the ellipsis ... (which is "usually" only three dots) has become more common. It is already used extensively in the notation for a series where, for example (2 x 1) + (2 x 2)... ...(2 x(y -1) + y means that the series continues indefinitely in the same manner until the term or terms found after the second ellipsis.

In the representation of the decimal expansion of a fraction, however, the elipsis would merely indicate that the expansion has not terminated and would not be interpreted by anyone I know of as meaning that any part of the preceding expansion should be repeated.

Moving on to what I think is a slightly more important point, I think the poster of a question (and that includes me, as I recently found to my cost) has a responsibility to ensure that their question is posed in such a way that it's meaning is unambiguous and clear. Falling back on the defence that "this is how it appears in my book", or as I did "this is how it was told to me" is disingenuous at best. Asking good questions is as much an art form as answering them, which is why quizzes are so popular and so much appreciated. Thank you, Mally, may we have the second round now please.

A 1: What fractions when you turn it into decimal starts like this,
(a) 0.0204081632…….
(b) 0.0103092781……
[spoiler] a) 1/49 (=2/98), b) 1/97[/spoiler]

A 2: [b]EXACTLY[/b] 63.41463414 % of the people asked if they used ‘scrubbo’ soap powder replied yes.
What is the SMALLEST number of people who could have been asked the question?
[spoiler] 5,000,000,000 - this is not the same as saying your calculator shows...[/spoiler]

A 3: Find a fraction which is greater than 7/17 but less than 5/12
[spoiler] so many to choose from - 169/408 is the midpoint[/spoiler]

A 4:What is the difference between ‘999999 thousand’ and 999899001 ?
[spoiler] 999999000-999899001 = 99999 [/spoiler]

A 5: How many terms of this series,
1/2 +1/3 +1/4 + 1/5 +1/6 +…….. are needed to make the sum of the series greater than 2 and ½
[spoiler] well I agree with the previously posted value of 19 but am not pleased with my method of arriving at that answer[/spoiler]
[spoiler]anyone get a nice solution not involving programs or lots of artihmatic?[/spoiler]

A 6: What are the prime factors of the number , 22,438,769?
[spoiler] 53x67x71x89 - again by an ugly method[/spoiler]

A 3: Find a fraction which is greater than 7/17 but less than 5/12

I am surprised nobody took the easy way out. While 169/408 certainly works, (5+7)/(17+12)=12/29 is much easier to find. Of course, proving this works generally probably is not easier than finding 169/408...

The Easy Way Out
 

Any two fractions can be thought of as being a/x, and b/y. What you did was convert them to fractions with the denominator x+y. This gave you (if you will forgive the rather ugly notation)

a((x+y)/x) /x+y and b((x+y)/y) / x+y

So any whole number between these two numerators will give a fraction that is > a/x, but < b/y.

Converting that to your case we get:

a((x+y)/x) = 11.9411… lets call this k

b((x+y)/y) = 12.0833… let’s call this t

And since t > a+b, a + b is an integer < t.

Of course, this is not a proof, this only helps us understand what happens. But it is now obvious that if y is a factor of x+y then t will be an integer so that t will not be < a+ b (it might = a+b, but it won't be < a+b).

So, your method works as long as y is not a factor of x+y, and as you point out, it is rather easy to do.

Oops
 

The second last paragraph of my previous post should obviously say:

Of course, this is not a proof, this only helps us understand what happens. But it is now obvious that if y is a factor of x+y then t will be an integer so that t will not be > a + b (it might = a+b, but it won't be > a+b).

Apologies for any confusion.