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Tough and Tricky questions: Word Problems.

A number of eggs dyed various colors were hidden for an egg hunt. How many eggs in total were hidden?

(1) The number of red eggs hidden is the square of an integer, while the total number of eggs hidden is 24 times that integer.

(2) Exactly 143 of the eggs hidden were not red.

Kudos for a correct solution. Official Solution:A number of eggs dyed various colors were hidden for an egg hunt. How many eggs in total were hidden? Initially, we can't really rephrase the question. We are asked for the total number of eggs hidden for a hunt.

Statement 1: INSUFFICIENT. This statement tells us 2 facts. Using \(x\) to represent the unknown integer, we can write the following:

Red Eggs \(= x^2\)

Total Eggs \(= 24x\)

However, we have no way of determining x, so this statement is not enough.

Statement 2: INSUFFICIENT. This statement tells us the following:

Non-Red Eggs \(= 143\)

By itself, we cannot hope to know how many eggs were hidden in all.

Statements 1 and 2 together: INSUFFICIENT. We know the following:

\(\text{Red} + \text{Non-Red} = \text{Total}\)

\(x^2 + 143 = 24x\)

We can rearrange this quadratic equation, setting one side equal to 0:

\(x^2 - 24x + 143 = 0\)

At this point, we can stop if we study the equation closely. The factored form of the equation must be as follows:

\((x - ...)(x - ...) = 0\)

The reason is that the middle term (\(-24x\)) is negative, while the constant term (143) is positive. This means that the factored form on the left must have two minus signs.

As a result, we expect two positive solutions for \(x\). In fact, we could have just one positive solution, if the equation factors into something like this: \((x - ...)^2 = 0\). However, that would require the constant term (in this case, 143) to be a perfect square, since \(x\) is an integer. (For instance, if the original equation were \(x^2 - 24x + 144 = 0\), it would factor to \((x - 12)^2 = 0\), and \(x\) would have just one possible value, 12.) Thus, there are two possible values of \(x\).

Alternatively, we could simply factor \(x^2 - 24x + 143 = 0\). Since \(143 = 11 \times 13\), we have the following:

\((x - 11)(x - 13) = 0\)

\(x = 11\) or \(x = 13\).

Thus, there are two possible values for \(x\), leading to two possible total numbers of eggs. Even together, the statements are not sufficient.

Answer: E.

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