Question 110196: This question is totally confusing, please help, I have no ideal on where to start.
Daniel has $575 in one-dollar, five dollar, and ten dollar bills. All together he has 95 bills. The number of one dollar bills plus the number of ten dollar bills is five more than twice the number of five dollar bills. How many of each type does he have?
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website! You need to create expressions that describe both the number of bills and the value of those bills.
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We want to start by defining the variables and since the question is asking how many of each type of bill he has, that gives us a good basis to do this. So:
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Let o = the number of ones
Let f = the number of fives, and
Let t = the number of tens.
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One piece of information we have is that he has a total of 95 bills and that he only has ones, fives, and tens. So we can write:
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Eq 1) 
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Now, the value of the one dollar bills that he has is also o, because each one dollar bill has a value of one dollar. The value of the five dollar bills is 5f, because each five has a value of five dollars, and similarly the value of his ten dollar bills is 10t. Since we know that the total amount of money he has is $575, we can write:
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Eq 2) 
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So far, so good. Now, twice the number of five dollar bills is 2f, and five more than that is 2f + 5, and we are told that is equal to the number of ones plus the number of tens, o + t. Now we can write:
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 , but we need to rearrange this so that this equation is in the same form as the other two by subtracting 2f from both sides, thus:
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Eq 3) 
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So now we have three linear equations in three variables. Let's see if there is a solution.
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Step 1, subtract Eq 3) from Eq 1), term by term:
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 , and simplify
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Now we know that Daniel has 30 fives and we can substitute this information into Eq 1) and Eq 2), thus:
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Eq 4)  , which simplifies to  ,
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Eq 5)  , which simplifies to 
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Solving Eq 4) for o, we get  , and then substituting into Eq 5) we get:
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 , then solve for t:
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And now we know that Daniel has 40 tens. Substitute that into Eq 5) to get:
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Giving us the last piece of the puzzle: Daniel has 25 ones.
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Now we have to check the answer.
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25 ones plus 30 fives plus 40 tens is, in fact, 95 bills.
$25 in ones, plus $5 times 30 = $150 in fives, plus $10 times 40 = $400 in tens is 25 + 150 + 400 which does, in fact, equal $575. So the answer checks, and perhaps you are a little less confused.
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