AMC 8 2023 Problem 14
AMC8 2023 年第 14 题
特殊数与不定方程★★★☆☆
Nicolas is planning to send a package to his friend Anton, who is a stamp collector. To pay for the postage, Nicolas would like to cover the package with a large number of stamps. Suppose he has a collection of 5-cent, 10-cent, and 25-cent stamps, with exactly 20 of each type. What is the greatest number of stamps Nicolas can use to make exactly $7.10 in postage?
(Note: The amount $7.10 corresponds to 7 dollars and 10 cents. One dollar is worth 100 cents.)
Nicolas计划将一个包裹寄给爱好集邮的朋友Anton．为了支付邮资，Nicolas准备在包裹上贴大量的邮票．假设他有5美分，10美分，25美分的邮票，每种邮票恰好各有20张. Nicolas最多可以使用多少张邮票来恰好支付$7.10的邮资？ (注意：金额$7.10是指7美元10美分, 1美元是100美分.)
- A.45
- B.46
- C.51
- D.54
- E.55
正确答案 · Correct Answer
E
解析 · Solution
Solution 1
Let's use the most stamps to make $7.10. We have 20 of each stamp, 5 -cent (nickels), 10 -cent (dimes), and 25 -cent (quarters). If we want the highest number of stamps, we must have the highest number of the smaller value stamps (like the coins above). We can use 20 nickels and 20 dimes to bring our total cost to 7.10 - 3.00 = 4.10. However, when we try to use quarters, the 25 cents don't fit evenly, so we have to give back 15 cents to make the quarter amount $4.25$. The most efficient way to do this is to give back a 10 -cent (dime) stamp and a 5 -cent (nickel) stamp to have 38 stamps used so far. Now, we just use quarters to get a grand total of 38 + 17 = 55., SohumUttamchandani, wuwang2002, TaeKim, Cxrupptedpat, InterstellerApex, mahika99, RyanZ4552
The value of his entire stamp collection is 8 dollars. To make $7.10 with stamps, he should remove 90 cents worth of stamps with as few stamps as possible. To do this, he should start by removing as many 25 cent stamps as possible as they have the greatest denomination. He can remove at most 3 of these stamps. He still has to remove cents worth of stamps. This can be done with one 5 and 10 cent stamp. In total, he has stamps in his entire collection. As a result, the maximum number of stamps he can use is .
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