Bunuel wrote:
Tough and Tricky questions: Word Problems.
At a certain circus, every child was given either two or three balloons. How many children received three balloons?
(1) At the circus, 40 percent of the children received two balloons.
(2) A total of 360 balloons were given out to children at the circus.
Kudos for a correct solution. Official Solution:At a certain circus, every child was given either two or three balloons. How many children received three balloons? We must determine the number of children who received 3 balloons, given that every child received either 2 or 3 balloons.
Statement 1 says that \(40%\), or \(\frac{2}{5}\), of the children received 2 balloons. This means that \((100 - 40)% = 60% = \frac{3}{5}\) of the children received 3 balloons. However, since we lack any information about the total number of balloons (or about the total number of children), it is not possible to solve for the number of children who received 3 balloons. There could be 2 children with 2 balloons and 3 children with 3 balloons, or 2,000 with 2 balloons and 3,000 with 3 balloons. 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 a total of 360 balloons were given out to children at the circus. However, since we have no information about how many children got 3 balloons and how many got 2 balloons, we cannot determine a unique value for the number of children who received either quantity of balloons. There could be 180 children with 2 balloons and 0 children with 3 (giving \(2 \times 180 = 360\) balloons), or there could be 0 children with 2 balloons and 120 with 3 (giving \(3 \times 120 = 360\) balloons). Statement 2 is NOT sufficient to answer the question. Eliminate answer choice B. The correct answer choice is either C or E.
Taking the statements together, we have the following facts: \(\frac{2}{5}\) of the children got 2 balloons, \(\frac{3}{5}\) of the children got 3 balloons, and there were 360 total balloons given out to children. If we label the number of children \(x\), then the total number of balloons is \(2(\frac{2}{5}x) + 3(\frac{3}{5}x)\) -- that is, 2 balloons for \(\frac{2}{5}\) of the children and 3 balloons for the other \(\frac{3}{5}\).
Setting this expression equal to 360, we have: \(2(\frac{2}{5}x) + 3(\frac{3}{5}x) = 360\). This is a single-variable equation, and so we can solve for \(x\), the total number of children. Once we have \(x\), we will be able to solve for the number of children with 3 balloons: \(\frac{3}{5}x\). Therefore, we have enough information to answer the question.
Answer: C.
When S-1 and S-2 combined are combined, we have the equation \(2(\frac{2}{5}x) + 3(\frac{3}{5}x) = 360\). Here x turns out to be in a fraction. (1800/13). But the number of children cannot be a fraction. So does it mean the Answer is E?