2012 AMC 10A Problem 6

Question

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?

\dfrac{10}{3}

\dfrac{20}{3}

7

\dfrac{15}{2}

8

Let the two numbers be

x

and

y

Use

xy=9

and the reciprocal condition to relate

x

and

y

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.

Problem 6 in other years