Thread: Fractions

Hi

I spent a lot of time trying to sovle this so I would really appreciate it if someone could just give me some hints what I am doing wrong.

Vicky.


There are positive intergers k, n, m such that

19/20 < 1/k + 1/n + 1/m < 1

What is the smallest possible value of k+n+m?


19 < 20 ( 1/k + 1/n + 1/m ) < 20

19 < 20 ( nm + km + kn/ knm ) < 20

nm + km + kn = try to make the value about 19.01


knm = 20

I did a lot of guessing and checking but now I am starting to see my solution doesn't make any sense..

Vicky.

Originally Posted by Vicky1997

Hi

I spent a lot of time trying to sovle this so I would really appreciate it if someone could just give me some hints what I am doing wrong.

Vicky.


There are positive intergers k, n, m such that

19/20 < 1/k + 1/n + 1/m < 1

What is the smallest possible value of k+n+m?

just a try





take the numbers 2,3,6



2+3+6 = 11 I think it is the smallest number

Originally Posted by Amer

2+3+6 = 11 I think it is the smallest number

Not quite. 10 + 6.6666... + 3.3333... = 20 (we have repeating decimals), and the sum of the fractions has to be less than 20.


01

Originally Posted by yeongil

Not quite. 10 + 6.6666... + 3.3333... = 20 (we have repeating decimals), and the sum of the fractions has to be less than 20.


01

the sum is less than 20

10+3.3333333333... + 6.66666666... = 19.999999999999... < 20 right

Originally Posted by Amer

the sum is less than 20

10+3.3333333333... + 6.66666666... = 19.999999999999... < 20 right

No. Go back to the fraction form:





Also, I believe it's been proven that 0.999.... (repeating forever) = 1.


01

Originally Posted by yeongil

No. Go back to the fraction form:





Also, I believe it's been proven that 0.999.... (repeating forever) = 1.


01

I think we should take 2,3

ok take 2,3,7

Originally Posted by Amer

I think we should take 2,3

ok take 2,3,7

Again no, you get a fraction less than 19/20 (about 0.80952380952381).

I'm starting to think that there is no solution.


01

Originally Posted by yeongil

Again no, you get a fraction less than 19/20 (about 0.80952380952381).

I'm starting to think that there is no solution.


01

the answer is 19.5238095238095

I take that back. You're right -- 2, 3, and 7 does work. I was using a spreadsheet when figuring out 1/k + 1/n + 1/m and I must have inputted the formula wrong. My bad.


01


EDIT: I found out what I did. I plugged in 2, 6 & 7 by mistake. Oops!

Originally Posted by yeongil

I take that back. You're right -- 2, 3, and 7 does work. I was using a spreadsheet when figuring out 1/k + 1/n + 1/m and I must have inputted the formula wrong. My bad.


01


EDIT: I found out what I did. I plugged in 2, 6 & 7 by mistake. Oops!

no problem we all make mistakes thanks

Thanks everyone for helping me out.

Vicky

Originally Posted by Amer

just a try





take the numbers 2,3,6



2+3+6 = 11 I think it is the smallest number

Is this the equation I should have set up?
And start plugging and checking the values?

Vicky.

Originally Posted by Vicky1997

Is this the equation I should have set up?
And start plugging and checking the values?

Vicky.

you can leave it but I prefer my equation



since the both right side and the left side are integers then I just start to guess numbers first take 2,2,2 that dose not work cuz the sum is 30

it is clear that if we choose two numbers (2) the sum of them will be more than 20 so try

2,3,3 dose not work too >20

try
2,3,4 dose not work too >20

2,4,4 =20 the same

3,3,3 < 19 so hold 2

2,3,5 >20

2,3,6 = 20 as you see

r
so
2,3,7 work

you can see that if we take bigger number the sum become smalle

Thank you.

Vicky.