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

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by BTGmoderatorLU » Wed Oct 31, 2018 2:52 pm

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Source: Manhattan Prep

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 eggs of the eggs hidden were not red.

The OA is E.
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by fskilnik@GMATH » Wed Oct 31, 2018 6:47 pm
BTGmoderatorLU wrote:Source: Manhattan Prep

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 eggs 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.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
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by Jay@ManhattanReview » Wed Oct 31, 2018 10:51 pm
BTGmoderatorLU wrote:Source: Manhattan Prep

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 eggs of the eggs hidden were not red.

The OA is E.
Given: A number of eggs dyed various colors were hidden for an egg hunt.

Question: How many eggs in total were hidden?

Let's take each statement one by one.

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.

Say that integer is x, thus, the number of red eggs = x^2
Say the total number of eggs = N, thus, N = 24x.

Since we do not have the value x, we can't get N. Insufficient.

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

We do not know how many red eggs are there. Insufficient.

(1) and (2) together

From (1) and (2) together

N = 143 + x^2

=> 143 + x^2 = 24x

=> x = 11 or 13

Can't get the unique value of N. Insufficient.

The correct answer: E

Hope this helps!

-Jay
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