Let's break this problem down into parts. We are given the following information:
Fixed costs (oven, marketing, and other costs): UGX60,000
Variable costs per cake: 500 + p, where p is the number of cakes made
We want to find:
The minimum number of cakes Daniel needs to make to avoid a loss (break-even point)
The selling price per cake to ensure no loss
To break even, the total revenue must be equal to the total costs. Let's denote the number of cakes Daniel needs to make as "x" and the selling price per cake as "s".
Total revenue: s * x
Total costs: UGX60,000 (fixed costs) + (500 + p) * x (variable costs)
To break even:
s * x = UGX60,000 + (500 + p) * x
Now, let's solve for x and s.
We can rearrange the equation to make it easier to solve:
s * x - (500 + p) * x = UGX60,000
x * (s - (500 + p)) = UGX60,000
Since we want to minimize the number of cakes (x), we need to maximize the term (s - (500 + p)). However, we also want to keep the selling price (s) reasonable. Let's assume that s = 500 + p + m, where m is the profit margin per cake.
x * (500 + p + m - (500 + p)) = UGX60,000
x * m = UGX60,000
Now, we can solve for x:
x = UGX60,000 / m
Since x must be a whole number (as we cannot make a fraction of a cake), we want to find the smallest possible value for m that still ensures no loss. As m is the profit margin per cake, a value of m = 1 would ensure that Daniel makes a profit of UGX1 per cake.
x = UGX60,000 / 1
x = 60,000
So, Daniel needs to make a minimum of 60,000 cakes to avoid a loss.
The selling price per cake (s) can be calculated using the profit margin (m):
s = 500 + p + m
s = 500 + 60,000 + 1
s = UGX60,501
Therefore, Daniel should make a minimum of 60,000 cakes and sell each cake at UGX60,501 to ensure no loss.
