Alphonso and Karen started out with the same number of apples. Karen gave twelve of her apples to Alphonso. Next, Karen gave half of her remaining apples to Alphonso. If Alphonso now has four times as many apples as Karen, how many apples does Karen now have?
(A) 12
(B) 24
(C) 36
(D) 48
(E) 72
Suppose that Alphonso and Karen each start with $n$ apples.

After Karen gives 12 apples to Alphonso, Karen has $n-12$ apples and Alphonso has $n+12$
apples.

After Karen gives half of her remaining apples (that is, $\frac{1}{2}(n-12)$ apples) to Alphonso, she has $\frac{1}{2}(n-12)=\frac{1}{2} n-6$ apples and Alphonso has $n+12+\frac{1}{2}(n-12)=\frac{3}{2} n+6$ apples.

Since Alphonso now has four times as many as Karen, $4\left(\frac{1}{2} n-6\right)=\frac{3}{2} n+6$ or $2 n-24=\frac{3}{2} n+6$ so $\frac{1}{2} n=30$ or $n=60$.

This means that Karen now has $\frac{1}{2}(60-12)=24$ apples.

ANSWER: (B)