The problem is as follows:

In an electronics factory, the owner calculates that the cost to produce his new model of portable TV is $26$ dollars. After meeting with the distributors, he agrees the sale price for his new product to be $25$ dollars each and additionally $8\%$ more for each TV set sold after $8000$ units. What is the least number of TV's he has to sell in order to make a profit?.

The answers are:

  • 16000
  • 15001
  • 16001
  • 15999
  • 17121

This problem has made me to go in circles on how to express it in a mathematical expression. I'm not sure if it does need to use of inequations.

What I tried to far is to think this way:

The first scenario is what if what he sells is $8000$ units, then this would become into:

$$\textrm{production cost:}\,26\frac{\$}{\textrm{unit}} \times 8000\,\textrm{units} = 208000\,\$$$

$$\textrm{sales:}\,25\frac{\$}{\textrm{unit}} \times 8000\,\textrm{units}=\,200000\,\$$$

Therefore there will be an offset of $8000\,\$$ as


So I thought what If I consider the second part of the problem which it says that he will receive an additional of $8\%$ after $8000$ units.

Therefore his new sale price will be $27\,\$$ because:

$$25+\frac{8}{100}\left(25\right )=27\,\$$$

So from this I thought that this can be used in the previous two relations. But how?.

I tried to establish this inequation:


But that's where I'm stuck at since it is not possible to obtain a reasonable result from this as one side will be negative and the other positive.

The logic I used was to add up $8000\,\$$ plus something which is the production cost must be less than what has been obtained from selling the first $8000$ units plus a quantity to be added to those $8000$.

However there seems to be an error in this approach. Can somebody help me to find the right way to solve this problem?


We know that the owner loses money for any production below $8000$ units, since it costs him more to produce his TV's than what he is receiving, so we consider production above $8000$ units.

Let's define $f:[8000,\infty)\rightarrow \mathbb{R}$ as follows: \begin{align} f(x)&= 25\cdot8000 + 27\cdot (x-8000)-26\cdot x\\ &=200000+27x-216000-26x\\ &=x-16000 \end{align} where $x$ represents the units sold. Notice that $25\cdot8000$ is the money he gets for the first $8000$ units sold, $27\cdot (x-8000)$ is the total money he gets for units sold after number $8000$, and $26x$ is the total amount of money it costs him to produce the units. So then $$f(x)=x-16000 > 0 \iff x > 16000.$$ To which we conclude that the owner has to sell at least $16,001$ units in order to make a profit.

  • $\begingroup$ Sorry about the delay into replying to this question, but I just returned from a break from studies. But can you explain me why are we subtracting $26x$ from the function you established?. I can get that we are summing up units but from then on I'm confused. Why is it $x-8000$ and not $8000-x$? If there are $x$ units wouldn't it mean that the remaining are $8000-x$. Can you help me clear these doubts? $\endgroup$ Jul 1 '18 at 3:37
  • 1
    $\begingroup$ Sure. The problem states that it costs $26$ dollars to produce each unit. Since $f(x)$ is a function that measures the profit, we have to subtract the cost of production, which will be given by $26 \cdot x$. Now, every unit sold after number $8000$ will be sold for $27$ dollars. Since $x$ is the number of units sold, $x-8000$ represents the number of units sold after the $8000$ mark. (We already established that $x$ should be greater than $8000$, for else there would be no profit.) I hope this clears up your doubts. $\endgroup$
    – Bergson
    Jul 2 '18 at 4:17
  • $\begingroup$ Thanks for the clarification. My remaining question (maybe obvious) but why the function you defined goes from $[8000,\infty)$ is it because the production function has to be above that mark? so that the owner makes a profit?. $\endgroup$ Jul 3 '18 at 3:03

He gets $25$ for the first $8000$ and $25\cdot 1.08=27$ for every one after that. Let him sell $n$. He receives $25\cdot 8000+27\cdot (n-8000)$ which has to be greater than $26 \cdot n$. These are equal at $n=16000$, so he needs to sell $16001$ to make a profit.


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