# Which fractions can you make (with proof)?

• MHB
• cshao123
In summary: Ah ok that’s nice! Any ideas how to prove no fractions are left out?We can prove by induction that all fractions between 1/2 and 1/1 can be reached using the given rules.Base case: 1/2 and 1/1 are both reachable.Inductive step: Assume that we have reached all fractions between 1/2 and 1/1 using the given rules. Then, according to the rules, we can make (x+1)/(2x+1) by adding 1/1 to the set of reachable fractions. This fraction satisfies 1/2 < (x+1)/(2x+1) < 1 since x >= 1. Thus, all fractions between
cshao123
Suppose you have the fraction 1/1. If you can make a fraction x/y, you can also make y/(2x). Also, if you can make x/y and a/b where GCD(x,y)=GCD(a,b)=1, you can make (x+a)/(y+b). Which fractions can you make?

Which fractions can I make?! From what?? :)

greg1313 said:
Which fractions can I make?! From what?? :)

Sorry I didn't word this very well! So you can make 1/2 by using the y/2x rule, and continue using the rules in the question to make more fractions.

cshao123 said:
Suppose you have the fraction 1/1. If you can make a fraction x/y, you can also make y/2x. Also, if you can make x/y and a/b where GCD(x,y)=GCD(a,b)=1, you can make (x+a)/(y+b). Which fractions can you make?
I'm stuck, too but I'll give it a shot.

Okay, so we have 1/1 = x/y right? So are we taking x = 1 and y = 1? But where did the x/(2y) thing come in? Under what rules do we admit 1/(2 * 1) = 1/2.

Guessing time. You have GCD(x, y) = 1. Now, x = 1, y = 1 works... 1/1. So does x = 1, y = 2 mean the fraction 1/2 follows the rules? That would imply that any combination of x, y where GCD(x, y) = 1 holds true. So it's true for any x and y that are relatively prime. That would mean that we can choose x = 6, and y = 11 and create the fraction 6/11?

Am I on the right track here?

-Dan

cshao123 said:
Suppose you have the fraction 1/1. If you can make a fraction x/y, you can also make y/(2x). Also, if you can make x/y and a/b where GCD(x,y)=GCD(a,b)=1, you can make (x+a)/(y+b). Which fractions can you make?

Hi cshao123! Welcome to MHB! (Wave)

Starting with 1/1, we can find 1/(2.1)=1/2.
Then we can add them (1+1)/(2+1)=2/3.
Invert again for 3/(2.2)=3/4.
Keep adding 1/1 for 4/5, 5/6, 6/7, 7/8, ...

Anyway, it looks like we can make all fractions $x/y$ with $\frac y2 \le x \le y$, doesn't it?
We can verify that those production rules will only generate new fractions within those bounds.
It's a bit more difficult to check that we won't 'miss' any fractions.
At least I can find all of them up y=13. (Thinking)

I like Serena said:
Hi cshao123! Welcome to MHB! (Wave)

Starting with 1/1, we can find 1/(2.1)=1/2.
Then we can add them (1+1)/(2+1)=2/3.
Invert again for 3/(2.2)=3/4.
Keep adding 1/1 for 4/5, 5/6, 6/7, 7/8, ...

Anyway, it looks like we can make all fractions $x/y$ with $\frac y2 \le x \le y$, doesn't it?
We can verify that those production rules will only generate new fractions within those bounds.
It's a bit more difficult to check that we won't 'miss' any fractions.
At least I can find all of them up y=13. (Thinking)

Thank you! How would you go about proving you can't make any fractions below 1/2?

cshao123 said:
Thank you! How would you go about proving you can't make any fractions below 1/2?

Suppose we have created a set of possible fractions $x/y$ that all satisfy $1/2\le x/y \le 1$ (the initial set does).
Then $1 \le y/x \le 2$, so that $1/2 \le y/(2x) \le 1$, which again satisfies the condition.
And with $b/2 \le a \le b$ and $y/2 \le x \le y$, we have $b/2 + y/2 \le a + x \le b + y$, so that $1/2 \le (a+x)/(b+y) \le 1$, which again satisfies the condition.
So all fractions that can be generated are between 1/2 and 1/1.

I like Serena said:

Suppose we have created a set of possible fractions $x/y$ that all satisfy $1/2\le x/y \le 1$ (the initial set does).
Then $1 \le y/x \le 2$, so that $1/2 \le y/(2x) \le 1$, which again satisfies the condition.
And with $b/2 \le a \le b$ and $y/2 \le x \le y$, we have $b/2 + y/2 \le a + x \le b + y$, so that $1/2 \le (a+x)/(b+y) \le 1$, which again satisfies the condition.
So all fractions that can be generated are between 1/2 and 1/1.

Ah ok that’s nice! Any ideas how to prove no fractions are left out?

## 1. What are fractions and how are they formed?

Fractions are a mathematical representation of a part of a whole. They are formed by dividing a whole into equal parts and expressing the number of parts taken as a numerator over the total number of parts as the denominator. For example, 1/2 represents one out of two equal parts of a whole.

## 2. Can any number be written as a fraction?

Yes, any number can be written as a fraction. This is because fractions represent a part of a whole, so any number can be expressed as a fraction by dividing it by a suitable whole number. For example, the number 3 can be written as 3/1, which is a fraction.

## 3. How do you add or subtract fractions?

To add or subtract fractions, you need to have a common denominator. This means that the denominators of the fractions must be the same. If they are not, you need to find the lowest common denominator by finding the lowest number that both denominators can divide into evenly. Then, you can add or subtract the numerators and keep the common denominator to get the final answer.

## 4. Can fractions be simplified?

Yes, fractions can be simplified by dividing the numerator and denominator by their greatest common factor. This results in an equivalent fraction that has a smaller numerator and denominator. For example, the fraction 4/8 can be simplified to 1/2 by dividing both numbers by 4.

## 5. How can I prove which fractions can be made from a given set of numbers?

To prove which fractions can be made from a given set of numbers, you can use the rules of arithmetic operations and the concept of equivalent fractions. By using these rules, you can manipulate the given numbers to form different fractions and show that they are equivalent to the given set of numbers. This can be done by multiplying or dividing both the numerator and denominator by the same number to get an equivalent fraction.

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