2012 AMC 10A Problem 6

Problem 6 of 25EasierAlgebra

The product of two positive numbers is

9.

The reciprocal of one of these numbers is

4

times the reciprocal of the other number. What is the sum of the two numbers?

Show solution

Solution

Let the two numbers be

x

and

y

such that

xy = 9 \text{ and } \dfrac{1}{x} = \dfrac{4}{y}.

We get that

y = \dfrac{9}{x}

\dfrac{1}{x} = \dfrac{4x}{9}.

x = \dfrac{3}{2},

since

x

is positive. Then

y = 9 \div \dfrac{3}{2} = 6.

The desired sum is then

6 + \dfrac{3}{2} = \dfrac{15}{2}.

Thus, D is the correct answer.

Tagged: system of equations · rational equation