2017 AMC 10A Problem 5

1. Give the numbers names, $x$ and $y$. "The sum is $4$ times the product" means
$$x + y = 4xy.$$
2. The question asks for $\frac{1}{x} + \frac{1}{y}$. To add fractions, use a common bottom, which is $xy$:
$$\frac{1}{x} + \frac{1}{y} = \frac{y}{xy} + \frac{x}{xy} = \frac{x + y}{xy}.$$
3. The top, $x + y$, is exactly what step 1 tells us about. Replace it with $4xy$:
$$\frac{x + y}{xy} = \frac{4xy}{xy} = 4.$$

The answer is $\boxed{\textbf{(C)}\ 4}$.

The sum of two reciprocals is always "sum of the numbers over product of the numbers." So whenever a problem links the sum and the product, you can get the sum of reciprocals straight away, without knowing the numbers. Many pairs of numbers fit this problem, and they all give $4$.

Try a concrete pair. If the two numbers are equal, say both are $x$, then the condition becomes $2x = 4x^2$. Dividing by $2x$ (it is not zero) gives $x = \tfrac12$. Check: $\tfrac12 + \tfrac12 = 1$ and $4 \cdot \tfrac12 \cdot \tfrac12 = 1$. The reciprocals are $2$ and $2$, which add to $4$.

Call the numbers $x$ and $y$. The condition says $x + y = 4xy$.

Combine the reciprocals over a common denominator:
$$\frac{1}{x} + \frac{1}{y} = \frac{y + x}{xy} = \frac{x + y}{xy}.$$
Substituting $x + y = 4xy$ gives
$$\frac{4xy}{xy} = 4.$$

The answer is $\boxed{\textbf{(C)}\ 4}$.