# Can someone help me figure this problem?

Find the sum of the reciprocals of two real numbers, given that these numbers have a sum of 75 and a product of 25.

I just need the process (i don't really need the answer), so i will actually learn. Thanks

sort by: active | newest | oldest
Jack A Lopez says: 2 years ago
I think this kind of problem is intended to make use of two separate skills. The first is translating words into equations. I.e. if you call those two real numbers a and b, then:

a+b = 75, a*b=25, and s = (1/a) + (1/b)

That's the first step.  The second step is seeing patterns, or maybe you could call it remembering things you've seen before.  In this case the trick is that (1/a) and (1/b) have a common denominator, which is a*b.

s = (1/a) + (1/b) = b/(a*b) + a/(a*b) = (a+b)/(a*b)

So if you get that far, then you can do a couple of substitutions, namely:
a+b = 75 and  a*b=25

s = (a+b)/(a*b) = 75/25 =3

Anyway, I know aelias36 already linked to Mr. Trinh's solution, which included this,and included the sort of brute force way of doing things, namely solving for a and b via the quadratic equation, and then substituting a and b into  s = (1/a) + (1/b).

Your keywords included the word "challenge", which says to me that this is a problem in some sort of competitive math event, and for events like that, the math problems often come with built in short cuts, that the writers of the problems are probably expecting you to use.  So there always might be a trick, and it's not unreasonable to sort of look at the problem asking yourself, "Well, is there a trick to this?  Is there a shortcut?"

Anyway, I hope that gives you some insight into the "process" .  A lot of it is about seeing patterns, and remembering stuff you've seen before.

iceng in reply to Jack A Lopez2 years ago
Not bad at all for a post apocalyptic nuclear desert avatar :-)
Jack A Lopez in reply to iceng2 years ago
Thanks. I always knew being on the Math team would give me skills that would come in handy later in life. Uh... Sort of. ;-)
aelias36 says: 2 years ago
http://mvtrinh.wordpress.com/2011/05/16/25-and-75/
iceng in reply to aelias362 years ago
+1