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Question 526482: Hello,
I'm stuck on this word problem. At the Halloween event there were 335 people in costume. There were 31 fewer vampires than witches and four times as many ghosts as all of the vampires and witches together. How many ghosts were there?
Thanks for your help.
Found 2 solutions by oberobic, bucky:
Answer by oberobic(2304)   (Show Source):
You can put this solution on YOUR website! We must assume that there only ghosts, vampires, and witches at the costume party.
Or we must assume the given total is only ghosts, vampires, and witches.
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g = number of ghosts
v = number of vampires
w = number of witches
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g + v + w = 335
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v + 31 = w
or
v = w - 31
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g = 4*(v+w)
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substitute for v
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g = 4*(w-31 +w)
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g = 4w - 124 +4w
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g = 8w -124
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we now have all three variables defined in terms of 'w' so we can substitute these into the total equation
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(8w-124) + (w-31) + w = 335
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collect terms
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10w -155 = 335
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10w = 490
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w = 49
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v = w -31
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v = 49-31
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v = 18
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g = 4*(49+18)
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g = 268
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Check the total
18 + 49 + 268 = 335
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Answer: The party was attended by 268 ghosts, 18 vampires, and 49 witches.
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Done.
Answer by bucky(2189) (Show Source):
You can put this solution on YOUR website! Have no fear of this scary problem. You can do it.
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Begin by letting V represent the unknown number of vampires, W represent the unknown number of witches, and G the unknown number of ghosts. You know that there were a total of 335 people attending the event. Presuming that the costumes were only vampires, witches, and ghosts, you can add the unknown numbers together and get set it equal to the number of attendees. This gives you the following first equation:
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Next you are told that there were 31 fewer vampires than witches. So you know that if you took away 31 from the number of witches, it would equal the number of vampires. You can write this in equation form as:
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Substitute the right side (W - 31) for V in the first equation and you have:
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You can simplify this equation by adding the two W's together to get 2W and by adding +31 to both sides to cancel or get rid of the -31 on the left side. When you do those two things the equation now becomes:
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Finally, you are told that there are 4 times as many ghosts as there are vampires and witches combined. So if you added the number of vampires and witches together and multiplied by 4 the answer would equal the number of ghosts. In equation form this is:
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You already said earlier that . So, substitute W - 31 for V the equation immediately above and you have:
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Combine the W terms inside the parentheses and this equation becomes:
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Now do the distributed multiplication on the left side by multiplying 4 times each of the terms inside the parentheses and you get:
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Return to the equation and substitute for G its equivalent 8W - 124 and you have:
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Now add the two W terms and also add + 124 to both sides to cancel or eliminate the -124 on the left side and you have:
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Solve for W by dividing both sides by 10 to get:
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At last ... you know that there were 49 witches at the event. Now recall that way earlier you were given the fact that there are 31 fewer vampires than witches. Therefore, if you subtract 31 from the 49 witches you find that there were 18 vampires.
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And at long last, if you add the number of vampires to the number of witches and multiply that total by 4, you should arrive at the number of ghosts. So you can write:
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Substitute the number of vampires and witches:
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Add the terms in the parentheses:
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and do the multiplication and you find:
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You now know that at the event there were 18 vampires, 49 witches, and 268 ghosts. As a double check, you can add these three numbers and you find that they do add up to 335, the number of people attending the event.
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Hope this helps you to get over being afraid of problems like this one.
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