Official Solution:Aunt Marge is distributing 20 pieces of candy among her nephews and nieces. If Robert receives 2 more candies than Kate, Bill gets 6 fewer than Mary, and Mary gets 2 more candies than Robert, how many candies does Kate receive? A. 2
B. 4
C. 6
D. 8
E. 10
We are given the following equations:
• \(R = K + 2\) (i)
• \(B = M - 6\) (ii)
• \(M = R + 2\) (iii)
We are also given that \(K + R + B + M = 20\). Since we are asked to find the number of candies Kate receives, let's try to express each variable in terms of \(K\), substitute these expressions into \(K + R + B + M = 20\), and solve for \(K\).
From (i), we already have an expression for \(R\) in terms of \(K\): \(R = K + 2\).
Next, substituting the expression for \(R\) from (i) into (iii), we find \(M\) in terms of \(K\): \(M = R + 2 = (K + 2) + 2 = K + 4\).
Then, substituting the expression for \(M\) into (ii), we find \(B\) in terms of \(K\): \(B = M - 6 = (K + 4) - 6 = K - 2\).
Now, we substitute the expressions for \(R, B,\) and \(M\) in terms of \(K\) into the equation \(K + R + B + M = 20\):
\(K + (K + 2) + (K - 2) + (K + 4) = 20\)
Simplifying, we get \(4K + 4 = 20\), so \(4K = 16\), and finally, \(K = 4\).
Therefore, Kate receives 4 pieces of candy.
Answer: B
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