A3. Twelve people, including at least one child, bought twelve loaves of bread at a bakery. Each man bought two loaves, each woman bought half a loaf, and each child bought a quarter of a loaf. How many men, women, and children bought bread?
(A) 4 men, 6 women, 4 children
(B) 4 men, 5 women, 2 children
(C) 5 men, 6 women, 1 child
(D) 5 men, 1 woman, 6 children
(E) It is impossible to determine.
A3. Let $a$ be the number of men, $b$ be the number of women, and $c$ be the number of children. It is given that $a+b+c=12$ and $a \cdot 2+b \cdot \frac{1}{2}+c \cdot \frac{1}{4}=12$. From both equations, it follows that $6 b+7 c=48$. Since $7 c=6(8-b)$, the number $c$ must be divisible by 6, so $c$ can only be $6$. Then $b=1$ and $a=5$.