A bag contains 12 yellow balls of 400 grams each and 18 blue balls of 700 grams each. Some yellow and some blue balls are taken from this bag and thrown into a second empty bag. At the end of this process; The average of the weights of the balls in the first bag was 500 grams, and the average of the weights of the balls in the second bag was 600 grams. What is the number of blue balls thrown into the second bag?
A) 16
B) 14
C) 12
D) 10
E) 8
Let's call the number of yellow balls taken out and put in the second bag $y$, and the number of blue balls taken out and put in the second bag $b$. We want to find the value of $b$.
We can start by writing an equation for the average weight of the balls in the first bag:
$$\frac{12(400) + 18(700)}{30} = 500$$
This equation represents the total weight of all the balls in the first bag (12 yellow balls of 400 grams each and 18 blue balls of 700 grams each) divided by the total number of balls (12+18=30). Since we know this average weight is 500 grams, we can simplify the equation:
$$12(400) + 18(700) = 15000$$
Next, we can write a similar equation for the average weight of the balls in the second bag:
$$\frac{400y + 700b}{y+b} = 600$$
Here, we're only considering the balls that were taken from the original bag and put into the second bag. The fraction $\frac{y}{y+b}$ represents the proportion of those balls that were yellow, and the fraction $\frac{b}{y+b}$ represents the proportion that were blue. The total weight of the yellow balls is $400y$, and the total weight of the blue balls is $700b$, so we divide by the total number of balls in the second bag (which is $y+b$). Since we know this average weight is 600 grams, we can simplify the equation:
$$400y + 700b = 600(y+b)$$
Expanding the right side gives:
$$400y + 700b = 600y + 600b$$
Simplifying further:
$$100y = 100b$$
So, $y=b$. This means that the number of yellow balls put in the second bag is the same as the number of blue balls put in the second bag.
We know that a total of 30 balls were in the original bag, and some of them were put into the second bag. The total number of balls in the second bag is $y+b$, so:
$$y+b \leq 30$$
Since we know that $y=b$, this simplifies to:
$$2b \leq 30$$
Which means:
$$b \leq 15$$
We can't have a fraction of a ball, so $b$ must be a whole number. The answer choices are all whole numbers less than or equal to 15, so we can test each one by plugging it in to the equation we derived earlier:
$$400y + 700b = 600(y+b)$$
For example, if we plug in $b=14$, we get:
$$400y + 9800 = 600(y+14)$$
Which simplifies to:
$$200y = 1400$$
This tells us that $y=7$, which means that 7 yellow balls and 14 blue balls were put in the second bag. We can check that this satisfies both of the original conditions:
- The average weight of the balls in the first bag is $\frac{12(400) + 18(700)}{30} = 500$ grams.
- The average weight of the balls in the second bag is $\frac{400(7) + 700(14)}{7+14} = 600$ grams.
Therefore, the answer is $\boxed{\textbf{(B) }14}$.
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