Question 1121103: Susie buys 2 pieces of salmon, each weighing x pounds, and 1 piece of trout,y pounds, where x and y are integers. The salmon costs $3.50 per pound and the trout costs $5 per pound. If the total cost of the fish was $77, what could be the value of y?
Found 2 solutions by solver91311, greenestamps:
Answer by solver91311(24713)   (Show Source):
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2 times 3.50 times  is the amount spent on salmon. 5 times  is the amount spent on trout and the total amount spent is 77.
Which will be more convenient to represent as:
Since the weights in this problem must be positive integers, the smallest possible value for either or would be 1.
Clearly, if  , then  , and we have one possible answer.
We know that  because  is not evenly divisible by 5.
Note that for  to have an integer solution,  where  . But our previous work bounds the possible values for between 1 and 14, so the only two possible values are 7 and 14.
Hence, 12 lbs salmon and 7 lbs trout, or 2 lbs salmon and 14 lbs trout.
John
My calculator said it, I believe it, that settles it
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
The amount spent on salmon is 7x; the amount spent on trout is 5y. The total amount spent is 77:
Very informally....
Since x and y are positive integers, 7x is a multiple of 7. Then, since 77 is a multiple of 7, 5y must also be a multiple of 7.
Since 5 is not a multiple of 7, y must be. So the possible values of y are 7, 14, 21, ....
But y has to be less than 77/5 = 15.4; so the only two possible values for y are 7 and 14.
Answer: The two possible values for y are 7 and 14.
The same solution, using formal mathematics....
Since 5 is not a multiple of 7, y must be.
Then, as above, the constraints on the value of y lead to exactly two possible values of y -- 7 and 14.
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