The price, $C, of printing n copies of a book is partly constant and varies partly as n. The selling price of each copy of the book is fixed as $b. If 630 copies were printed and sold, there would be a loss of 10%. If 980 copies were printed and sold, there would be a profit of 12%. How many copies would have to be printed and sold in order to break even (i.e. neither gain nor lower)
Let's break down the problem step-by-step:
Step 1: Define the variables
Let's define the cost of printing as C, the number of copies printed as n, and the selling price of each copy as b.
Step 2: Define the cost equation
The given information states that the cost of printing is partly constant and varies partly as n. So, we can express the cost equation as:
C = a + bn, where a is the constant cost and bn represents the variable cost.
Step 3: Calculate the cost for 630 copies
According to the problem, if 630 copies were printed and sold, there would be a loss of 10%. This means that the selling price of each copy is 90% of the cost. So, the cost equation becomes:
0.9C = a + b(630)
Step 4: Calculate the cost for 980 copies
Similarly, if 980 copies were printed and sold, there would be a profit of 12%. This means that the selling price of each copy is 112% of the cost. So, the cost equation becomes:
1.12C = a + b(980)
Step 5: Eliminate the constants
To find the value of n at break-even point, we need to eliminate the constants (a) from the equations. We can do this by subtracting the two equations:
1.12C - 0.9C = a + b(980) - a - b(630)
Simplifying the equation gives:
0.22C = b(980 - 630)
Step 6: Simplify the equation further
0.22C = b(350)
Step 7: Calculate the cost equation at break-even point
At break-even point, there is no profit or loss, so the selling price per copy is equal to the cost per copy. Therefore, we can set the cost equation equal to the selling price:
C = b(n)
Step 8: Find the value of n at break-even point
Substituting the value of b from Step 6 into the equation in Step 7 gives:
C = 0.22C(n)
Dividing both sides of the equation by C gives:
1 = 0.22n
Dividing both sides of the equation by 0.22 gives:
n = 4.545
Since the number of copies must be a whole number, the minimum number of copies required to break even is 5.
Therefore, a minimum of 5 copies would have to be printed and sold in order to break even.
To find the number of copies that need to be printed and sold to break even, we need to set up two equations based on the given information.
Let's start by defining the cost equation based on the printing cost, which is partly constant and varies partly as the number of copies:
C = a + bn
Where:
C is the total cost of printing n copies
a is the constant cost (partly constant)
b is the variable cost per copy (varies partly as n)
According to the given information, if 630 copies were printed and sold, there would be a loss of 10%. This means that the cost C would be 10% higher than the selling price of 630 copies:
C = 1.1 * 630b
Similarly, if 980 copies were printed and sold, there would be a profit of 12%:
C = 0.88 * 980b
Now, we need to solve these two equations simultaneously to find the values of a and b. Let's equate the two expressions for C:
1.1 * 630b = 0.88 * 980b
Simplifying the equation:
1.1 * 630 = 0.88 * 980
693 = 862.4
This equation is not true, which means there is no consistent value for b. Therefore, there is an error in the given information, and it is not possible to determine the break-even point using this data.
To find the correct answer, you need additional information such as the actual selling price of each copy or the correct percentages of loss and profit.