2008 AMC 10B Problems/Problem 8
Contents
Problem
A class collects 50 dollars to buy flowers for a classmate who is in the hospital. Roses cost 3 dollars each, and carnations cost 2 dollars each. No other flowers are to be used. How many different bouquets could be purchased for exactly 50 dollars?
Solution 1
The cost of a rose is odd, hence we need an even number of roses. Let there be roses for some . Then we have dollars left. We can always reach the sum exactly by buying carnations. Of course, the number of roses must be such that the number of carnations is non-negative. We get the inequality which solves to . must be an integer, so there are possible values of , and each gives us one solution.
Solution 2
Let and be the number of roses and carnations bought. The equation should be . Since is an odd number, the product of must be even and smaller than . You can try nonnegative even integers for and you will end up with the numbers , , , , , , , , and . There are numbers in total, so the answer is .
Solution 3
Let represent the number of roses, and let represent the number of carnations. Then, we get the linear Diophantine equation, . Using the Euclidean algorithm, we get the initial solutions to be and , meaning the complete solution will be, ,
The solution range for which both and are positive is . There are possible values for .
2008 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
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