the numbers M and N are different numbers selected from the first 25 counting numbers: If M is larger than N, what is the smallest value that M * N over M-N could have?
I thought in this problem M and N had to be the same numbers so how could we do 1*2 over 25-1. Also, M has to be larger than N........still confused.
In the first line of your problem you said
"M and N are different numbers".
when I formed 25 - 1 for M-N , wasn't my M > N ???
ok, thanks. I was just thinking I had to use the same number for M & M on the top and bottom of the fraction. I understand now. Thanks so much!
To find the smallest value that M * N over M-N could have, we can consider all possible pairs of numbers (M, N).
Since M and N are different numbers selected from the first 25 counting numbers, we know that the smallest value for M is 2 and the smallest value for N is 1 (since they must be different).
Let's try to substitute M = 2 and N = 1 into the expression M * N over M-N:
(2 * 1) / (2 - 1) = 2 / 1 = 2
So, when M = 2 and N = 1, the expression evaluates to 2.
Now, let's check if there is any other pair of numbers that can give a smaller value.
If M = 3, the smallest possible value for N would be 1, but the expression would evaluate to:
(3 * 1) / (3 - 1) = 3 / 2 = 1.5
Since 1.5 is greater than 2, we conclude that the smallest value that M * N over M-N could have is 2, and it occurs when M = 2 and N = 1.
the smallest value of MN/(M-N) is obtained when
MN is as small as possible and M-N is as large as possible
since M and N are from the first 25 natural numbers
the smallest product is 1x2 or 2
and the largest difference is 25-1 or 24
the smallest value of MN/(M-N) is 2/24 or 1/12