reciprocals haha good show. and done

[dolor]

The sum of two positive integers is 9. What is the least possible sum of their reciprocals?

[Acidic]

the two integers would have to be 4 and 5. hence the sum being 9/20, which is the smallest.

[Dillinger4]

Yes i would agree with Acidic. 4 + 5 = 9 --> 4/1 + 5/1 --> 1/4 + 1/5 --> 9/20.

[davebat]

i dont know anything about maths and dont know what a reciprocals is but why cant the 2 numbers be 2+7 = 9?
why do they have to be 4 + 5

[jemidiah]

1 + 8 = 1/1 + 1/8 = 9/8
2 + 7 = 1/2 + 1/7 = 9/14
3 + 6 = 1/3 + 1/6 = 9/18
4 + 5 = 1/4 + 1/5 = 9/20

Smallest is one with largest denominator, hence 4 + 5.

[Acidic]

Originally posted by davebat
...dont know what a reciprocals is

a reciprocal of a number is that number to the power -1
so the reciprocal of 2 is 1/2, of 10 is 1/10 of n is 1/n or n^-1

[fundu]

In general we should say that:

1/x + 1/y = (x+y)/xy

Now minimising 1/x + 1/y is equivalent to minimising (x+y)/xy

As (x+y) is constant, the problem reduces to maximizing xy.

We know (x+y)^2 = (x-y)^2 + 4xy

Thus if (x+y) is given, we can get xy to me maximum if x=y because in that case only (x-y)^2 will be zero, which is least for a square.

In short, if sum of two numbers is constant, their product is maximum when they are equal.

So making x and y towards equal, we will get 4 and 5.

This concept can be generalised however big the number is.


Similarly, vice versa, if product of two numbers is constant, their sum is minimum when they are equal.