Question 1105858: The cost of publication of each copy of a magazine is $1.30. It is sold to dealers for $1.50 per copy. the amount received for advertising is 20% of the amount received for all magazines sold beyond 100,000 . find the least number of magazines that can be published profitably , if 80% of the issues published are sold.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! C = 1.30 * x
S = 1.50 * y
x is the number published.
y is the number sold.
P = profit
number sold is 80% of number published.
formula for that is y = .8 * x
profit is 1.5 * number sold minus 1.3 * number published + 20% of the amount received for all magazines sold beyond 100,000.
amount received for all magazine sold over 100,000 would be 1.5 * (y - 100,000).
formula for profit becomes P = S - C + .2 * 1.5 * (y - 100,000).
you break even when the profit is 0.
formula becomes 0 = S - C + .2 * 1.5 * (y - 100,000).
since S = 1.5 * y and C = 1.3 * x, the formula becomes:
0 = 1.5 * y - 1.3 * x + .2 * 1.5 * (y - 100,000).
since y = .8 * x, this formula becomes:
0 = 1.5 * .8 * x - 1.3 * x + .2 * 1.5 * (.8 * x - 100,000).
simplify to get:
0 = 1.2 * x - 1.3 * x + .3 * (.8 * x - 100,000)
simplify further to get:
0 = 1.2 * x - 1.3 * x + .24 * x - 30,000.
combine like terms to get:
0 = .14 * x - 30,000.
add 30,000 to both sides of this equation to get:
30,000 = .14 * x
solve for x to get x = 214,285.7143.
that's the break even point.
to see if this is correct, solve the problem using x = 214,285.7143.
y = .8 * x = 171,428.5714.
profit = 1.5 * y - 1.3 * x + .2 * 1.5 * (y - 100,000).
replace x with 214,285.7143 and replace y with 171,428.5714 and you wind up with profit = 1.1 * 10^-8 which is equal to .000000011.
that's so small, it's effectively equal to 0.
the only reason it isn't 0 is most likely due to internal rounding in the calculator used.
if more than 214285.7143 are published, there will be a profit.
if less than 214285.7143 are published, there will be a loss.
for example:
assume 215,00 are published.
that means that .8 * 215,000 = 172,000 are sold.
profit is 1.5 * 172,000 - 1.3 * 215,000 + .2 * 1.5 * (172,000 - 100,000).
you get profit = 100.
now assume 210,000 are published.
that means that .8 * 210,000 = 168,000 are sold.
profit is equal to 1.5 * 168,000 - 1.3 * 210,000 + .2 * 1.5 * (168,000 - 100,000).
you get profit = -600.
-600 profit means a loss of 600.
looks like the formula is good.
since the number of magazine can't be a fractional amount, then round up and your solution is that the minimum number of magazines that have to be published in order to get a profit is equal to 214,286.
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