S95-03

Official Solution:


We need to determine the number of incidents reported in 2005. The problem tells us that 1,000 incidents were reported in 2003. In 2004, \(x\) percent fewer incidents were reported, so \((100 - x) \times 1,000\) incidents were reported. Since a percentage is equivalent to a fraction with a denominator of 100, we can express the number of incidents reported in 2004 as \(\frac{100 - x}{100} \times 1,000\). In 2005, \(y\) percent more incidents were reported than in 2004. This can be expressed as \(\frac{100 + y}{100}(\frac{100 - x}{100} \times 1,000)\). It is this target expression that we must find a value for.

Multiply the numerators and denominators: \(\frac{100 + y}{100}(\frac{100 - x}{100} \times 1,000) = \frac{1,000(100 + y )(100 - x)}{10,000}\).

Cancel 1,000 from the numerator and denominator and FOIL what remains: \(\frac{(100 + y)(100 - x)}{10} = \frac{10,000 - 100x + 100y - xy}{10}\).

Evaluating this expression will require either solving for \(x\) and \(y\) or finding a way to substitute a number for all variable terms.

Statement 1 says that \(xy = 50\). This does not allow us to solve for either \(x\) or \(y\). We can substitute it into our target expression to get \(\frac{10,000 - 100x + 100y - 50}{10}\), but doing so does not eliminate the need to solve for \(x\) and \(y\). Statement 1 is NOT sufficient to answer the question. Eliminate answer choices A and D. The correct answer choice is B, C, or E.

Statement 2 says that \(y - x - \frac{xy}{100} = 4.5\). Note that this equation has an \(x\) term, a \(y\) term, and a term containing \(xy\). Thus, substitution will likely be possible.

Divide the original expression by \(10\) to eliminate the fraction: \(\frac{10,000 - 100x + 100y - xy}{10} = 1,000 - 10x + 10y - \frac{xy}{10}\).

Now factor out an additional \(10\) and reorder the terms in the expression: \(10(100 + y - x - \frac{xy}{100})\).

The left side of the equation in statement 2 is contained in this expression. Thus, we know that we can find a value for the expression. Statement 2 alone provides sufficient information to answer the question.


Answer: B