1. ## fraction problem

One day tom was playing with numbers.he wrote 11 fractions using all natural numbers from 1 to 22 exactly once, either as numerator or as denominator.how many of these fractions,at most,are integers?

2. ## Re: fraction problem

Originally Posted by mastermin346
One day tom was playing with numbers.he wrote 11 fractions using all natural numbers from 1 to 22 exactly once, either as numerator or as denominator.how many of these fractions,at most,are integers?
10.

3. ## Re: fraction problem

How could it be 10? If you look at the prime numbers 13, 17, 19, at most one of these can be paired with 1, so right off the bat, we know the answer is at most 9.

4. ## Re: fraction problem

Originally Posted by richard1234
How could it be 10? If you look at the prime numbers 13, 17, 19, at most one of these can be paired with 1, so right off the bat, we know the answer is at most 9.

How will I feel if I'm wrong now?

5. ## Re: fraction problem

Hmm, you might be correct. 17 could be paired with 19, didn't realize that.

6. ## Re: fraction problem

Sorry, I'm only joking around. I've had them all written down since before my first post on this thread.

and

there

are

only

10.

7. ## Re: fraction problem

The only way to get 10 is if one of 13,17, or 19 is paired with 1. WLOG suppose 13 is paired with 1.

13/1
17/19

We have 18 numbers left and want to find the max. number of integers we can obtain. Here is a possible solution.

22/11
21/7
20/10
18/9
16/8
15/5
14/2
12/4
6/3

It is indeed 10.

8. ## Re: fraction problem

Lol...but if you're "a tutor," you should at least explain why it's 10, or find a method to get 10 and show why 11 is impossible, right?

9. ## Re: fraction problem

According to me its 10 also.