2. 外语
  3. A $10 bill (1,000 cents) was replaced with 50 coins having the same total value. The only coins used were 5-cent coins, 10-cent

A $10 bill (1,000 cents) was replaced with 50 coins having the same total value. The only coins used were 5-cent coins, 10-cent

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问题 A $10 bill (1,000 cents) was replaced with 50 coins having the same total value. The only coins used were 5-cent coins, 10-cent coins, 25-cent coins, and 50-cent coins. How many 5-cent coins were used?
(1) Exactly 10 of the coins were 25-cent coins and exactly 10 of the coins were 50-cent coins.
(2) The number of 10-cent coins was twice the number of 5-cent coins.
选项 A、Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B、Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C、BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D、EACH statement ALONE is sufficient.
E、Statements (1) and (2) TOGETHER are NOT sufficient.
答案A
解析 Let a b, c, and d be the number, respectively, of 5-cent, 10-cent, 25-cent, and 50-cent coins. We are given that a + b + c+d= 50 and Sa + 10b + 25c + 50d = 1,000, or a + 2b + 5 c + 10d= 200. Determine the value of a.
a + b + c+d=50
a + 2b + 5c+10d=200
(1)    We are given that c = 10 and d = 10. Substituting c = 10 and d=10 into the two equations displayed above and combining terms gives a + b = 30 and a + 2b = 50. Subtracting these last two equations gives b = 20, and hence it follows that a = 10; SUFFICIENT.
(2)    We are given that b = 2a. Substituting b -2a into the two equations displayed above and combining terms gives a +2a+ c+d= 50 and a + 4a + 5c + 10d= 200, which are equivalent to the following two equations.
3a + c + d=50
a + c + 2d=40
Subtracting these two equations gives 2a — d= 10, or 2a = d + 10. Since 2a is an even integer, d must be an even integer. At this point it is probably simplest to choose various nonnegative even integers for d to determine whether solutions for a, b, c, and d exist that have different values for a. Note that it is not enough to find different nonnegative integer solutions to 2a = d+ 10, since we must also ensure that c and d are nonnegative integers. If d= 8, then 2a = 8 + 10 = 18, and we have a = 9, b= 18, c= 15, and d=8. However, if d= 10, then 2a = 10 + 10 = 20, and we have a =10,6 = 20, c = 10, and d= 10; NOT sufficient.
The correct answer is A;
statement 1 alone is sufficient.
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