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A number of eggs dyed various colors were hidden for an egg hunt. How

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A number of eggs dyed various colors were hidden for an egg hunt. How  [#permalink]

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New post 17 Nov 2014, 12:49
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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.

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Re: A number of eggs dyed various colors were hidden for an egg hunt. How  [#permalink]

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New post 17 Nov 2014, 15:41
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Lets call that integer i, the number of red eggs r, and the number of non-red eggs z.

1. We know that r=i^2 and r+z=24*i. Two equations, three variables. NOT SUFFICIENT.
2. We know that z=143. NOT SUFFICIENT.

Combining the two, we get:
i^2+z=24*i => i^2+143=24*i. Standard quadratic equation, which we can solve by using a discriminant formula that gives us i=13 or i=11. Both answers work here, so still NOT SUFFICIENT.

Answer is E.
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Re: A number of eggs dyed various colors were hidden for an egg hunt. How  [#permalink]

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New post 17 Nov 2014, 23:29
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2
from first statement its red=a^2 , where a is an integer
& total hidden eggs=24a

as we can realize that one needs the value for a to solve this problem

coming to second statement it says 143 eggs of the total hidden eggs are not red
so, rest of the eggs are red except 143 eggs

hence, we can formulate it with the help of statement one that
24a-143=a^2
a^2-24a+143=0
a=13,11 No definite result found.

Answer is E
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Re: A number of eggs dyed various colors were hidden for an egg hunt. How  [#permalink]

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New post 18 Nov 2014, 08:10
Bunuel wrote:

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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Re: A number of eggs dyed various colors were hidden for an egg hunt. How  [#permalink]

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New post 31 Oct 2018, 19:57
Bunuel wrote:
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.

\({\rm{Total}}\,\,\left\{ \matrix{
\,{\rm{red}}\,\,:\,\,\,R \ge 1\,\,{\mathop{\rm int}} \hfill \cr
\,{\rm{not - red}}\,\,:\,\,\,N \ge 1\,\,{\mathop{\rm int}} \hfill \cr} \right.\)

\(? = R + N\)

Let´s go straight to (1+2): a BIFURCATION will prove the correct answer is (E).

\(\left( {1 + 2} \right)\,\,\left\{ \matrix{
N = 143 \hfill \cr
R = {M^2}\,\,,\,\,R + N = 24M\,\,\,\,\left( {M \ge 1\,\,{\mathop{\rm int}} } \right)\,\,\,\left( * \right) \hfill \cr} \right.\)

\(\left( * \right)\,\,\, \Rightarrow \,\,\,N = 24M - {M^2} \Rightarrow \,\,\,\,11 \cdot 13 = 143 = M\left( {24 - M} \right)\,\,\,\, \Rightarrow \,\,\,M = 11\,\,{\rm{or}}\,\,M = 13\)

\(\left\{ \matrix{
\,M = 11\,\,\,\,\, \Rightarrow \,\,\,\,\,R = {11^2}\,\,\,\,\, \Rightarrow \,\,\,\,\,? = {11^2} + 143 \hfill \cr
\,M = 13\,\,\,\,\, \Rightarrow \,\,\,\,\,R = {13^2}\,\,\,\,\, \Rightarrow \,\,\,\,\,? = {13^2} + 143 \hfill \cr} \right.\)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: A number of eggs dyed various colors were hidden for an egg hunt. How   [#permalink] 31 Oct 2018, 19:57
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