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#### Kerim

##### Member
The water tap A fills a pool in x hours.
The water tap B fills the same pool in y hours.
The values of x and y are integers greater than 1.
Both water taps fill the pool in z hours.

The question is:
If x is given, find the value of y (actually all possible values) that lets the value of z be an integer too.
In other words, find y=f(x) so that the result z is also an integer.

Have fun... but please let me know if you already solved it or perhaps saw it somewhere.

Kerim

#### tcmtech

##### Banned
In what reality does this ever happen when filling a pool or anything else for that matter?

#### Ratchit

##### Well-Known Member
The water tap A fills a pool in x hours.
The water tap B fills the same pool in y hours.
The values of x and y are integers greater than 1.
Both water taps fill the pool in z hours.

The question is:
If x is given, find the value of y (actually all possible values) that lets the value of z be an integer too.
In other words, find y=f(x) so that the result z is also an integer.

Have fun... but please let me know if you already solved it or perhaps saw it somewhere.

Kerim
I never saw this before.

Ratch

#### Ratchit

##### Well-Known Member
In what reality does this ever happen when filling a pool or anything else for that matter?
This is a puzzle. That means it is usually a specific problem that has no relevance to the real world, and exists only for amusement and entertainment.

Ratch

#### Kerim

##### Member
Well done, Ratch.

The complete y=f(x) could be expressed as:
y = x * (x - n ) / n , where n is a divisor of x.

As you know, your solution is for n = 1 which is always a divisor of x.

Kerim

#### Tesla23

##### Member
Well done, Ratch.

The complete y=f(x) could be expressed as:
y = x * (x - n ) / n , where n is a divisor of x.

As you know, your solution is for n = 1 which is always a divisor of x.

Kerim
I don't think this is the complete solution, for example, if x=6 your solution gives y=30, 12, 6 (for divisors 1,2,3), but y=3 is also a solution.

#### Ratchit

##### Well-Known Member
I don't think this is the complete solution, for example, if x=6 your solution gives y=30, 12, 6 (for divisors 1,2,3), but y=3 is also a solution.

Ratch

#### kubeek

##### Well-Known Member
The water tap A fills a pool in x hours.
The water tap B fills the same pool in y hours.
The values of x and y are integers greater than 1.
Both water taps fill the pool in z hours.
Ambiguous question, the way I see it X=Y=Z, as they both do it in Z hours.

#### tcmtech

##### Banned
Ambiguous question, the way I see it X=Y=Z, as they both do it in Z hours.
That's the way I see it as well. Too much ambiguity involved. There is no defined ratios or anything to factor any values from. Tap flow rates are undefined, pool volume is undefined, and time is undefined. Thus there is no fixed definable real answer or solution.

My answer is once you figure out how to do math without numbers then show me how to spell words and make sentences without using letters and when you're done with that I want a design for a solid brick house made entirely out of wood. Good luck.

#### The Electrician

##### Active Member
Well done, Ratch.

The complete y=f(x) could be expressed as:
y = x * (x - n ) / n , where n is a divisor of x.

As you know, your solution is for n = 1 which is always a divisor of x.

Kerim
If n must be a divisor of x then this misses the solutions where y < x. If you let n range from 1 to x-1, keeping only those values for y where y is an integer, all the solutions will be obtained.

Brute force gives a bunch of solutions, showing the ones where y < x:

#### Ratchit

##### Well-Known Member
Ambiguous question, the way I see it X=Y=Z, as they both do it in Z hours.
I did use in my solution, and also pointed out the rate of both spigots on "together". That showed the interpretation, and should have removed any ambiguity.

Ratch

#### Ratchit

##### Well-Known Member
That's the way I see it as well. Too much ambiguity involved. There is no defined ratios or anything to factor any values from. Tap flow rates are undefined, pool volume is undefined, and time is undefined. Thus there is no fixed definable real answer or solution.
Rates, not ratios are given and used. A different flow rate is given for each spigot. Pool volume is irrelevant. There are many pairs of answers.

My answer is once you figure out how to do math without numbers then show me how to spell words and make sentences without using letters and when you're done with that I want a design for a solid brick house made entirely out of wood. Good luck.
That is not what this puzzle is doing.

Ratch

#### Kerim

##### Member
I don't think this is the complete solution, for example, if x=6 your solution gives y=30, 12, 6 (for divisors 1,2,3), but y=3 is also a solution.
You are totally right, Tesla23. And I wonder how we can define the complete solution of y=f(x)

In y = f(x) = x * (x - n ) / n, even if n is not a divisor of x, there is a possibility that n is a divisor of x * (x - n).
And your extra solution shows exactly this case. If x=6 and n=4, y = 6 * (6-4) / 4 = 6/2 * (6-4)/2 = 3 * 1 = 3.

Anyway, the idea behind this puzzle is showing how playing with integers only could be much harder than working with decimal numbers.

Edited:
If we re-write
y = x * (x - n ) / n
as
n = x * (x - n ) / y
we can deduce that a solution exists if y is also a divisor of x.

So perhaps the complete answer of f(x) could be:
y = x * (x - n ) / n
where
n OR y is a divisor of x.
What do you think?

Kerim

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#### tcmtech

##### Banned
Rates, not ratios are given and used. A different flow rate is given for each spigot. Pool volume is irrelevant. There are many pairs of answers.

That is not what this puzzle is doing.

Ratch

I don't consider an undefinable question to be a puzzle.

As shown in post 10 when the basic formula is put together there is no defined answer but an infinite number of workable combinations that fit this no solution is the correct or incorrect one.

If X,Yand Z are all positive integers then the answer is that X,Y and Z are any positive values greater than zero going to infinity that fit the formula.

#### Ratchit

##### Well-Known Member
I don't consider an undefinable question to be a puzzle.
I don't either. I said that a puzzle is a problem that has no practical use except for amusement and entertainment. Just because the solution has no single definable pair of numbers does not disqualify it from being a puzzle. I don't consider the question to be undefinable. Perhaps it could be explained better, but it is not undefinable.

As shown in post 10 when the basic formula is put together there is no defined answer but an infinite number of workable combinations that fit this no solution is the correct or incorrect one.
The request was to find y=f(x) so that the result z is also an integer. The fact that many pairs of numbers satisfy that criteria is irrelevant.

If X,Yand Z are all positive integers then the answer is that X,Y and Z are any positive values greater than zero going to infinity that fit the formula.
The hard part is to find the relationship between two numbers that satisfy the conditions.

Ratch

#### Kerim

##### Member
For instance, the idea of this problem came to me first when I was trying, last week, to teach a kid at the 5th grade how to solve a numerical exercise using fractional numbers.
To make things, at the beginning, rather easy for him, I tried to find out suitable integers for x and y, so that the answer of the pool exercise (above) could be simple (hence also an integer).

That is all

Kerim

#### The Electrician

##### Active Member
This is just a typical rate problem of the type that terrorized beginning algebra problems in high school: https://www.algebra.com/algebra/hom...ems/HOW-TO-Solve-Rate-of-Work-Problems.lesson

Kerim modified the problem to require integer solutions, but it's still a classic rate problem.

I wouldn't say that none of the infinite solutions is correct; I would say they are all correct--they satisfy the problem parameters.

#### Tesla23

##### Member
So perhaps the complete answer of f(x) could be:
y = x * (x - n ) / n
where
n OR y is a divisor of x.
What do you think?
Kerim
Here is how you can find the complete solution:

you can rewrite 1/x + 1/y = 1/z as
(y+x)(x-z) = x^2

so the answers are related to the divisors of x^2. Also, as y>0 and z>0,
y+x > x and x-z < x, so this means we can start with divisors larger than √x^2 for (y+x).

As an example, for x = 6, if we look at all the divisors of 36, and associating (x+y) with the divisor:

you can see that only the divisors (x+y) > 6 have y > 0.

Restricting to divisors >6 for (x+y) gives;

which are all the solutions.

The number of solutions is the number of distinct divisors of x^2 which are greater than x.
So if x is prime you will only have one solution.

I'm not quite sure how 5th grade kids will cope!

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#### Kerim

##### Member
I'm not quite sure how 5th grade kids will cope!
You are right.

It became an exercise for me only when I tried to simplify his problem by using integers for its given data (x and y) and for its result (z) as well.

Kerim

#### atferrari

##### Well-Known Member
Both water taps fill the pool in z hours.
I know this thread has gone a long way already. The sentence in bold, does it actually mean "both taps deliverying at the same time"? Ratchit did interpret it that way, right?

If the above is true, how could it be that once rate x is given, I could get multiple options for y? Sorry, a bit at lost.

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