greg1313 said:Which fractions can I make?! From what?? :)
I'm stuck, too but I'll give it a shot.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?
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 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)
cshao123 said:Thank you! How would you go about proving you can't make any fractions below 1/2?
I like Serena said:We start with 1/1 which satisfies the condition.
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.
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.
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.
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.
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.
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.