Data Sufficiency: Simultaneous Equations
Tina and Ethan collect only red and black marbles. Together, they have a total of 100 (red and black) marbles. How many marbles does Tina currently have?
>(1) If Tina gives half of her red marbles to Ethan and Ethan gives half of his black marbles to Tina, then Tina would have a total of 20 more marbles than Ethan.
>(2) If Tina gives half of her black marbles as well as half of her red marbles to Ethan, then Ethan would have a total of 60 more marbles than Tina.
Incorrect.
[[snippet]]
According to Stat. (1), if Tina gives half of her red marbles to Ethan and Ethan gives half of his black marbles to Tina, then Tina would have a total of 20 more marbles than Ethan.
This does not help you. All you need to realize is that there is more than one possible scenario for Stat. (1).
Begin with an easy scenario where Tina has 60 marbles and Ethan has 40 (20 more than Ethan originally). Since Tina gives red marbles, and Ethan gives black marbles, the two quantities are unrelated, and you now can "play around" with the relative weights of red and black and make it so Ethan and Tina both give each other the same number of marbles, so the exchange does not change anything.
But it should also be clear that this is not the ONLY scenario. It will be possible to create a scenario where Tina has FEWER marbles than Ethan (for example) but gets so many black marbles in the exchange that she ends up 20 ahead.
There's no real need to plug in the actual numbers. You just need to realize that it's possible to play around with the relative weights of red and black to see that there's no way to fix the number of marbles at a single number, so Stat. (1) is insufficient. It could be, for example, that Tina already had 20 marbles more than Ethan before the exchange, or that they had the same number of marbles, and the exchange made the difference.
Stat.(1) → IS → BCE.
According to Stat. (2), if Tina gives half of her black marbles as well as half of her red marbles to Ethan, then Ethan would have a total of 60 more marbles than Tina.
You know that Half of Tina's red marbles + Half of Tina's black marbles = Half of Tina's marbles, so Tina gives Ethan half of her marbles and is left with half. That means
>$$E + \frac{T}{2} = \frac{T}{2} + 60$$
>$$E = 60$$
Since they have a total of 100 marbles,
>$$T = 100 - 60 = 40$$.
Stat.(2) → S → B.
Incorrect.
[[snippet]]
According to Stat. (2), if Tina gives half of her black marbles as well as half of her red marbles to Ethan, then Ethan would have a total of 60 more marbles than Tina.
You know that Half of Tina's red marbles + Half of Tina's black marbles = Half of Tina's marbles, so Tina gives Ethan half of her marbles and is left with half. That means
>$$E + \frac{T}{2} = \frac{T}{2} + 60$$
>$$E = 60$$
Since they have a total of 100 marbles,
>$$T = 100 - 60 = 40$$.
Stat.(2) → S.
Correct.
[[snippet]]
According to Stat. (1), if Tina gives half of her red marbles to Ethan and Ethan gives half of his black marbles to Tina, then Tina would have a total of 20 more marbles than Ethan.
This does not help you. All you need to realize is that there is more than one possible scenario for Stat. (1).
Begin with an easy scenario where Tina has 60 marbles and Ethan has 40 (20 more than Ethan originally). Since Tina gives red marbles, and Ethan gives black marbles, the two quantities are unrelated, and you now can "play around" with the relative weights of red and black and make it so Ethan and Tina both give each other the same number of marbles, so the exchange does not change anything.
But it should also be clear that this is not the ONLY scenario. It will be possible to create a scenario where Tina has FEWER marbles than Ethan (for example) but gets so many black marbles in the exchange that she ends up 20 ahead.
There's no real need to plug in the actual numbers. You just need to realize that it's possible to play around with the relative weights of red and black to see that there's no way to fix the number of marbles at a single number, so Stat. (1) is insufficient. It could be, for example, that Tina already had 20 marbles more than Ethan before the exchange, or that they had the same number of marbles, and the exchange made the difference.
Stat.(1) → IS → BCE.
Statement (1) ALONE is sufficient, but Statement (2) alone is not sufficient to answer the question asked.
Statement (2) ALONE is sufficient, but Statement (1) alone is not sufficient to answer the question asked.
BOTH Statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
EACH statement ALONE is sufficient to answer the question asked.
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
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