题目信息
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?
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.
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正确答案：
A:Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Algebra Simultaneous equations
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 5a + 10b + 25c + 50d = 1,000, or a + 2b + 5c + 10d = 200. Determine the value of a.
a + b + c + d = 50
a + 2b + 5c + 10d = 200
Tip Note that each of a, b, c, and d must be a nonnegative integer, and so care must be taken in deducing non-sufficiency. For example, there are many real number pairs (x,y) that satisfy the equation 2x + y = 1, but if each of x and y must be a nonnegative integer, then x = 0 and y = 1 is the only solution.
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. 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, b = 20, c = 10, and d = 10; NOT sufficient.
The correct answer is A;statement 1 alone is sufficient.
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 5a + 10b + 25c + 50d = 1,000, or a + 2b + 5c + 10d = 200. Determine the value of a.
a + b + c + d = 50
a + 2b + 5c + 10d = 200
Tip Note that each of a, b, c, and d must be a nonnegative integer, and so care must be taken in deducing non-sufficiency. For example, there are many real number pairs (x,y) that satisfy the equation 2x + y = 1, but if each of x and y must be a nonnegative integer, then x = 0 and y = 1 is the only solution.
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. 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, b = 20, c = 10, and d = 10; NOT sufficient.
The correct answer is A;statement 1 alone is sufficient.
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