SOLUTION: I got the question incorrect, but I do not know what I did wrong. The cost of manufacturing notebooks varies inversely as the number of notebooks. If it costs 50 cents a noteboo

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Question 158245: I got the question incorrect, but I do not know what I did wrong.
The cost of manufacturing notebooks varies inversely as the number of notebooks. If it costs 50 cents a notebook when 10,000 are manufactured, how much will it cost if 18,000 are made?
MY WORK:
50/10,000=x/18,000
1/200=x/18,000
200x=18,000
x=90 cents
Found 3 solutions by ilana, gonzo, Edwin McCravy:
Answer by ilana(307)   (Show Source): You can put this solution on YOUR website!
It says the cost varies inversely with the number of notebooks. That means that as the number of books increases, the cost per book decreases. That's your first clue that your answer is wrong. You set up a direct proportion, y=kx, rather than an inverse proportion, y=k/x. Since the cost is $.50 per book for 10,000 books, you can write the equation .50 = k/10000, so k = 5000. That means the inverse function is y = 5000/x. Now substitute 18000 for x to get y = 5000/18000 = 0.27777. So it costs 28 cents per notebook when 18,000 are manufactured.
Answer by gonzo(654)   (Show Source): You can put this solution on YOUR website!
if the cost varies inversely as the number of notebooks, then when the number of notebooks goes up, the cost must go down.
cost of manufacturing 10,000 notebooks is $.5 * 10,000 = $5,000
how much will it cost when 18,000 notebooks are manufactured?
good question.
it will be less, but how much less will be determined by what the cost will be when 18,000 notebooks are manufactured.
if the cost is $.25, then the cost to manufacture 18,000 notebooks is 18,000*$.25 = $4500 which, in this example, winds up being less than the cost to manufacture 10,000 notebooks.
that's because i halved the cost but didn't double the number of notebooks.
the actual reduction in cost would be less.
for argument sake, say $.40 per notebook.
then the cost to manufacture 18,000 notebooks would be 18,000*$.4 = $7,200.
your equation shows 50/10,000 = x/18,000
it really should be 10,000*50 = x*18,000
that means that 10,000*50 cents each = 18,000 * x cents each
that equation gets you x = 10,000*50/18,000 = 27.7777777777777
that just means that if you sell 10,000 units at 50 cents, then in order to sell 18000 units at the same total price, you have to sell the 18,000 units at 27.7777777777 cents apiece.
10000*50 = 27.777777*18000
50,000 = 50,000
the cost per unit * the number of units equation has to be preserved whether you are selling 10,000 units or 18,000 units.
saying that the cost is inversely proportional means that as the number of units goes up, the cost per unit goes down.
what is missing is how much down.
you don't have enough information to determine that.
usually they will say for an increase of x units, the cost per unit will drop y cents.
example:
for an increase of 8000 units the cost per unit will drop 15 cents.
check the question again and see if there is something missing in the information you have for it.
they might also tell you there is a fixed cost and an incremental cost per unit.
a fixed cost is the same regardless of the number of units produced.
the incremental cost per unit is applied to each unit manufactured.
example:
fixed cost is $5,000 and the incremental cost per unit is 50 cents.
manufacturing 10,000 units costs $5,000 fixed cost + ($.5*10000) = $5,000 incremental cost. total cost is $10,000 / 10,000 units = $1.00 a unit
take the same fixed cost and same incremental cost but manufacture 20,000 units.
cost is now $5,000 fixed cost plus ($.5*20000) = $10,000 incremental cost.
total cost is $15,000 / 20,000 units = $.75 a unit.
fixed cost remained the same and incremental cost per unit remained the same.
total cost went up, but the total cost per unit went down because the same fixed cost is being divided among 20,000 units rather than 10,000.
Answer by Edwin McCravy(20059)   (Show Source): You can put this solution on YOUR website!
I got the question incorrect, but I do not know what I did wrong.
The cost of manufacturing notebooks varies inversely as the number of notebooks. If it costs 50 cents a notebook when 10,000 are manufactured, how much will it cost if 18,000 are made?
MY WORK:



cents 


That method is OK for direct variation, but not
for inverse variation.  

Indirect variation is the "cheaper by the dozen" idea.

In direct variation, when one quantity goes up, the other goes up 
also, and when one goes down, the other goes down also.

But in indirect variation, when one quantity goes up, the other
goes down, and vice-versa.

So in this problem, when the production goes up from 10,000 to
18,000, you should expect the cost per notebook to go down.  So
you should expect an answer less than 50 cents, not more!

Here is how to handle this inverse proportion problem:

Start this:



Substitute  and 



Multiply both sides by 



Substitute this into 



Now substitute 



cents

or about 27.78 cents cost per notebook.

Edwin
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