AAGT 1
 

:rolleyes:
As a sequel to’ Elementary Number quiz’ which was well received I am spurred on to send off some more from my formidable personal collection of puzzles and math recreational problems.
I have selected these puzzles from Series two ‘Can you solve these’ by David Wells. Out of the 112 problems given in this series I have chosen just 6 Arithmetical ones which are suitable for us. These are based on sound math ideas and thinking. I have also prepared from the series, problems in Algebra ,Geometry and Trig. for a later date. Hence I call it AAGT Quiz. So lets get started with AAGT 1 which deals with Arithmetic. If any one needs hints they are yours for the asking. Complete solutions will be given later as I have worked out most of them and have the correct answers.
AAGT 1.
A 1: What fractions when you turn it into decimal starts like this,
(a) 0.0204081632…….
(b) 0.0103092781……

A 2: EXACTLY 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?

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

A 4:What is the difference between ‘999999 thousand’ and 999899001 ?

A 5: (suitable for programming)
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

A 6 ; What are the prime factors of the number , 22,438,769 ?
Have a good day, :smile:
Mally. :coffee:

2 - 3,170,731,707
3 - 83/200
4 - 998,999,109,999
5 - n = 19
6 - 22438769 = 53 × 67 × 71 × 89

BTW, may you post author, title and editor of some of your books??

Luigi

[QUOTE=ET_]2 - 3,170,731,707[/QUOTE]

Luigi,
I don't think so.

"EXACTLY 63.41463414 %" is "EXACTLY 0.6341463414 of the participants"
and that is "6341463414/10000000000" which reduces to
"3170731707/5000000000 of the participants"

Your number is the number that replied affirmatively. But the question asks for the minimum number polled.

Since there is no way for a fractional participant to reply, the number of participants must be a multiple of 5000000000.

That number is less than the current world population, but if you exclude the infants who are too young to reply, I doubt that there are enough left.

Wouldn't you like to have that kind of market penetration.

The continued fraction expansion of .6341463414 is 0, 1, 1, 1, 2, 1, 2, 1, 9380862 so I strongly suspect the desired fraction is 26/41 = 0.6341463414634146341463414634...
Hence a positive multiple of 41 people must have been polled.

Maybe Mally wanted to imply that 63414 is periodic.

Mally: I overlooked that you did not include the 1/1 term in the harmonic numbers, so that answer is wrong.

Alex

[QUOTE=ET_]4 - 998,999,109,999[/QUOTE]

How do you interpret "999999 thousand" as almost 10^12?
I interpret it to be (10^6-1) * 1000. (<10^9)

999,999,000 - 999,899,001 = 999,000 - 899,001 = 100,000 -1 = 99,999

[QUOTE=akruppa]The continued fraction expansion of .6341463414 is 0, 1, 1, 1, 2, 1, 2, 1, 9380862 so I strongly suspect the desired fraction is 26/41 = 0.6341463414634146341463414634...
Hence a positive multiple of 41 people must have been polled.
[/QUOTE]
But that is NOT what he ask. I agree that it is a more reasonable answer, but it is not "EXACTLY 63.41463414 %"

Particularly in light of the question immediately above it, I don't think that we can consider it a mistake that he phrased it as he did.

[QUOTE=akruppa]
Mally: I overlooked that you did not include the 1/1 term in the harmonic numbers, so that answer is wrong.

Alex[/QUOTE]

My answer is written as: Sum (1/n) (for n=2 to 19) > 2,5

Luigi

[QUOTE=Wacky]How do you interpret "999999 thousand" as almost 10^12?
I interpret it to be (10^6-1) * 1000. (<10^9)

999,999,000 - 999,899,001 = 999,000 - 899,001 = 100,000 -1 = 99,999[/QUOTE]

I'd like to see if you were responsive! :razz: :whistle:

My fault :blush: I'd take my time reading my own answers BEFORE publishing them.

Luigi

[QUOTE=Wacky]But that is NOT what he ask. I agree that it is a more reasonable answer, but it is not "EXACTLY 63.41463414 %"

Particularly in light of the question immediately above it, I don't think that we can consider it a mistake that he phrased it as he did.[/QUOTE]

Yes, the answer to the question strictly as stated (with fully capitalised EXACTLY and all) isn't 26/41. But, as you noted, this answer fails a simple plausibility check. In such cases I think it is not unreasonable to assume that the author of the puzzle goofed.

Alex

Plausibility Check
 

Question 2.

The answer is 41, of whom 26 said yes.

How much more plausible can it get? :showoff:

[QUOTE=Numbers]Question 2.

The answer is 41, of whom 26 said yes.

How much more plausible can it get?[/QUOTE]

You have missed the point. 26 of 41 is = 63.41463414634146341463414634... %. That is a bit more than "EXACTLY 63.41463414 %" which was requested.
To get the EXACT answer, you would have to have sampled almost EVERY person on Earth. Since there are a number of individuals who would be unable to answer, it is quite likely that you could not have obtained the required number of responses.

Go back and read the dialog between Alex and myself.