AMC 8 1996 Problem 7
AMC8 1996 年第 7 题
数列与规律★★☆☆☆
Brent has goldfish that quadruple (become four times as many) every month, and Gretel has goldfish that double every month. If Brent has 4 goldfish at the same time that Gretel has 128 goldfish, then in how many months from that time will they have the same number of goldfish?
Brent 的金鱼数量每个月翻四倍，Gretel 的金鱼数量每个月翻一倍。如果在 Brent 有 4 条金鱼的同时 Gretel 有 128 条金鱼，那么从那时起多少个月后他们的金鱼数量将相等？
- A.4
- B.5
- C.6
- D.7
- E.8
正确答案 · Correct Answer
B
解析 · Solution
Solution 1
Call this month "Month 0". Make a table of the fish that Brent and Gretel have each month.
Month / Brent / Gretel: 0 / 4 / 128; 1 / 16 / 256; 2 / 64 / 512; 3 / 256 / 1024; 4 / 1024 / 2048; 5 / 4096 / 4096.
You could create a similar table without doing all of the calculations by converting all the goldfish into powers of 2. In this table, you could increase Brent's goldfish by two powers of 2, while increasing Gretel's fish by one power of 2.
Either way, in 5 months they will have the same number of fish, giving an answer of .
Create two equations.
Brent starts with 4 goldfish and they quadruple each month, giving us or .
Gretel starts with 128 goldfish and they double each month, giving us or .
Using substitution, . Thus 2x+2=x+7 giving us x=5, which is .
继续练习