 Posers and Puzzles

1. 25 Sep '07 15:27
Our lucky man Herman just inherited 90 million dollars and his rich uncle said to take some out and invest the rest for income. His uncle said to take out some percentage number, a natural number from 1 to 50 percent. But only choose the ones that a bank can pay you exactly without leftovers, such as 0,1456 cents.
So Herman does some quick calcs and sees that if he takes off 10 percent of the 90 mil, he gets 9 mil and that is 81 mil V 9 mil, a 9 to 1 ratio. Is there a formula that can show which cases from 1 to 50 percent removed from the 90 mil will lead to exact solutions? For instance, if you want to have a ten to one ratio, that requires you to remove 8,181,181,81 but that leads to a bit of excess the bank would require you to pay. So how do you calculate these ratios? That is to say, the ratio between what you have left to invest V what you have taken out. It is interesting that if you take out ten percent and have an exact amount, \$9,000,000.00 that the ratio is not 10:1 but 9:1, comparing the two numbers after the withdrawal.
2. 25 Sep '07 15:37
I don't understand what relavence the final ratio has to do with the question
3. 25 Sep '07 16:25
Originally posted by preachingforjesus
I don't understand what relavence the final ratio has to do with the question
The rich uncle has math quirk where he wants Herman to select only ratios that give exact money back from the bank. Like I said, the second scenerio doesn't work, a ratio of 10:1, because it would require the bank giving Herman \$1,818,181,81.1818 dollars, which the bank cannot do exactly. The first scenerio where he takes out 9 mil exactly, of course the bank can pay exactly, which is a 9:1 ratio. So how many out of the ratio set from 1 to 1 through 50:1 will have exact payouts and is there a formula that can predict that?
4. 25 Sep '07 20:02
Presumably this boils down to what fractions of 9,000,000,000 give an integer answer?

In which case we need to find all the factors of 9,000,000,000.

Then when we have found each factor F we have all the fractions 1/F, 2/F, 3/F etc to consider up to 1/2.

Then finally eliminate the fractions we have counted more than once (eg we cannot count 5/50 and 1/20)

Seems like number-crunching ... is there an elegant way to do this?
5. 25 Sep '07 20:301 edit
Originally posted by wolfgang59
Presumably this boils down to what fractions of 9,000,000,000 give an integer answer?

In which case we need to find all the factors of 9,000,000,000.

Then when we have found each factor F we have all the fractions 1/F, 2/F, 3/F etc to consider up to 1/2.

Then finally eliminate the fractions we have counted more than once (eg we cannot count 5/50 and 1/20)

Seems like number-crunching ... is there an elegant way to do this?
It's easy to find those factors.

3x3x(2x5)^9

That gives 1/3, 1/2 and 1/5
6. 25 Sep '07 20:38
90 million is 9E7 not 9. But how do you go from a given ratio of numbers and work back to the 90 mil figure and find out what the withdrawal would have to have been? Like 13:1, since you don't know what the numbers should be, how would you work it out and is there a formula that would let you calculate it directly given the 90E6 number as the starting point?
7. 25 Sep '07 20:43
Originally posted by AThousandYoung
It's easy to find those factors.

3x3x(2x5)^9

That gives 1/3, 1/2 and 1/5
All factors?
2
3
5
6
9
10
12
15
......

which gives 1/6, 1/9, 1/10, 1/12 ........ et cetera
8. 25 Sep '07 20:46
Originally posted by sonhouse
90 million is 9E7 not 9. But how do you go from a given ratio of numbers and work back to the 90 mil figure and find out what the withdrawal would have to have been? Like 13:1, since you don't know what the numbers should be, how would you work it out and is there a formula that would let you calculate it directly given the 90E6 number as the starting point?
getting from the fraction to the ratio is trivial so lets keep to fractions!!

and the number we are looking at is 9,000 million (cents)

Apart from number crunching can any Number Theorist give an answer?
9. 25 Sep '07 20:50
Originally posted by wolfgang59
All factors?
2
3
5
6
9
10
12
15
......

which gives 1/6, 1/9, 1/10, 1/12 ........ et cetera
It's more restrictive than a factor problem, the 10:1 ratio for instance, I already figured out is \$81,818,818,81.8181818181...etc., V \$8,181,181.8181818181... See the problem? it is not an exact solution, given dollars to pennies, the bank cannot give out an exact amount to cover the withdrawal. The problem is finding out which of the set of ratios can give exact money. For instance, 3:1 is \$67,500,000.00000000 with \$22,500,000.00000 withdrawn. The bank can pay that without any leftover or rounding needed.
10. 25 Sep '07 20:57
Originally posted by sonhouse
It's more restrictive than a factor problem, the 10:1 ratio for instance, I already figured out is \$81,818,818,81.8181818181...etc., V \$8,181,181.8181818181... See the problem? it is not an exact solution, given dollars to pennies, the bank cannot give out an exact amount to cover the withdrawal. The problem is finding out which of the set of ratios can giv ...[text shortened]... ith \$22,500,000.00000 withdrawn. The bank can pay that without any leftover or rounding needed.
Same problem.

"For instance, 3:1 is \$67,500,000.00000000 with \$22,500,000.00000 withdrawn"

is the same as 1/4

we need to find fractions of 9,000,000,000 cents that give an integer number of cents
11. 25 Sep '07 21:00
Originally posted by sonhouse
It's more restrictive than a factor problem, the 10:1 ratio for instance, I already figured out is \$81,818,818,81.8181818181...etc., V \$8,181,181.8181818181... See the problem? it is not an exact solution, given dollars to pennies, the bank cannot give out an exact amount to cover the withdrawal. The problem is finding out which of the set of ratios can giv ...[text shortened]... ith \$22,500,000.00000 withdrawn. The bank can pay that without any leftover or rounding needed.
To be more general

any ratio a:b is the same as the fraction a/(a+b)

any fraction x/y is the same as ratio x🙁y-x)

much easier to consider the fractions which leads us to look for factors as in my initial reply
12. 25 Sep '07 21:001 edit
Originally posted by sonhouse
It's more restrictive than a factor problem, the 10:1 ratio for instance, I already figured out is \$81,818,818,81.8181818181...etc., V \$8,181,181.8181818181... See the problem? it is not an exact solution, given dollars to pennies, the bank cannot give out an exact amount to cover the withdrawal. The problem is finding out which of the set of ratios can giv ...[text shortened]... ith \$22,500,000.00000 withdrawn. The bank can pay that without any leftover or rounding needed.
I think they are saying that with a factor of 10, for example, you get a 9:1 ratio. The ratios and factors match up one-to-one.

EDIT: Ahh, beaten to the punch.
13. 25 Sep '07 21:01
damn smileys!!! 🙄 😕 😛 😉 😳
14. 25 Sep '07 21:02
Originally posted by sven1000
I think they are saying that with a factor of 10, for example, you get a 9:1 ratio. The ratios and factors match up one-to-one.

EDIT: Ahh, beaten to the punch.
For an astrophysicist you type slow!

😀
15. 25 Sep '07 21:05
Originally posted by wolfgang59
For an astrophysicist you type slow!

😀
Yeah, to add to the humiliation my research is computer simulation. 🙁

I need one of those direct brain-to-computer links.