AMC12 2006 B

AMC12 2006 B · Q14

AMC12 2006 B · Q14. It mainly tests Money / coins, Divisibility & factors.

Elmo makes $N$ sandwiches for a fundraiser. For each sandwich he uses $B$ globs of peanut butter at $4$ cents per glob and $J$ blobs of jam at $5$ cents per glob. The cost of the peanut butter and jam to make all the sandwiches is $2.53$. Assume that $B$, $J$ and $N$ are all positive integers with $N>1$. What is the cost of the jam Elmo uses to make the sandwiches?

Elmo 为筹款活动做了 $N$ 个三明治。每个三明治使用 $B$ 团花生酱（每团 4 美分）和 $J$ 团果酱（每团 5 美分）。制作所有三明治的花生酱和果酱总成本为 $2.53$ 美元。假设 $B$、$J$ 和 $N$ 都是正整数，且 $N>1$。Elmo 用于制作三明治的果酱成本是多少？

(A) $1.05$ $1.05$

(B) $1.25$ $1.25$

(D) $1.65$ $1.65$

(E) $1.85$ $1.85$

Answer

Correct choice: (D)

正确答案：（D）

Solution

From the given, we know that $253=N(4B+5J)$ (The numbers are in cents) since $253=11\cdot23$, and since $N$ is an integer, then $4B+5J=11$ or $23$. It is easily deduced that $11$ is impossible to make with $B$ and $J$ integers, so $N=11$ and $4B+5J=23$. Then, it can be guessed and checked quite simply that if $B=2$ and $J=3$, then $4B+5J=4(2)+5(3)=23$. The problem asks for the total cost of jam, or $N(5J)=11(15)=165$ cents, or $1.65\implies\mathrm{(D)}$

由题意可得 $253=N(4B+5J)$ （单位为美分）因为 $253=11\cdot23$，且 $N$ 为整数，所以 $4B+5J=11$ 或 $23$。容易推出 $11$ 不可能由整数 $B$ 和 $J$ 组成，因此 $N=11$ 且 $4B+5J=23$。然后可简单试算得 $B=2$、$J=3$ 时，$4B+5J=4(2)+5(3)=23$。题目问果酱的总成本，即 $N(5J)=11(15)=165$ 美分，即 $1.65\implies\mathrm{(D)}$

Topics