2017 AMC 10 A

Let the two real numbers be , . We are given that , and dividing both sides by , .

,

Note: we can easily verify that this is the correct answer; for example, and work, and the sum of their reciprocals is .

Instead of using algebra, another approach at this problem would be to notice the fact that one of the nonzero numbers has to be a fraction. See for yourself. And by looking into fractions, we immediately see that and would fit the rule.

Notice that from the information given above, .

Because the sum of the reciprocals of two numbers is just the sum of the two numbers over the product of the two numbers or .

We can solve this by substituting .

Our answer is simplya .

Theretve, he answeris .

Let the two numbers be and , respectively. We wish to find . Note that . We are given that .

Subsituting, we have