Answer:
<em>The most he can pay for each of the remaining ladders if he is to obtain a 48% markup goal is </em><em>$36.357</em><em>.</em>
Explanation:
- <em>A hardware buyer plans to purchase 75 ladders which will retail for $35 each. </em>This means that he is going to sell each ladder for $35, regardless of how much he paid for them before.
- <em>He has already placed an order for 48 ladders at $16.50 each</em>. So out of the 75 ladders he is planning to purchase and then sell, he bought 48 at a reduced price of only $16.50 each. Now if we subtract those numbers, as shown below, we determine that he still needs to buy 27 more ladders at an unknown price:
<em>(75 - 48) ladders = 27 ladders</em>
- <em>What is the most he can pay for each of the remaining ladders if he is to obtain a 48% markup goal? </em>So we need to determine the maximum price for each of the remaining 27 ladders if he is to obtain a 48% markup goal.
<em>Markup </em><em>in this case is just a measure of the ratio (in %) between the profit made </em>(by selling 75 ladders for $35 each)<em> to the cost paid </em>(by buying 48 of them at $16.5 each and 27 of them at an unknown price). Its formula is as follows:

Now, we already have our markup goal of 48%, so we can substitute that number into the formula, and divide both sides of the equation by 100:

We know that Cost is the sum of what he paid for the 48 ladders, <em>plus </em>what he is to pay for the remaining 27 ladders, so it should look something like this:


Where '<em>x</em>' is the maximum price he can pay for each of the remaining 27 ladders if he is to obtain a 48% markup goal.
We should also know that Profit is what he gets by selling the 75 ladders, <em>minus </em>the cost paid for them. It should look something like this:
![Profit=(75ladders*35\frac{dollars}{ladder} )-[(792dollars)+(27ladders*x)]](https://tex.z-dn.net/?f=Profit%3D%2875ladders%2A35%5Cfrac%7Bdollars%7D%7Bladder%7D%20%29-%5B%28792dollars%29%2B%2827ladders%2Ax%29%5D)
![Profit=(2625dollars)-[(792dollars)+(27ladders*x)]\\Profit=(2625dollars)-(792dollars)-(27ladders*x)\\Profit=(1833dollars)-(27ladders*x)](https://tex.z-dn.net/?f=Profit%3D%282625dollars%29-%5B%28792dollars%29%2B%2827ladders%2Ax%29%5D%5C%5CProfit%3D%282625dollars%29-%28792dollars%29-%2827ladders%2Ax%29%5C%5CProfit%3D%281833dollars%29-%2827ladders%2Ax%29)
Next, we want to substitute what we have so far for Cost and Profit into the worked Markup formula we had written before, and solve the equation by isolating our '<em>x</em>'. To do that, let's follow these steps:

![\frac{(1833dollars)-(27ladders*x)}{(792 dollars)+(27ladders*x)}=0.48\\1833dollars-27ladders*x=0.48*[792 dollars+27ladders*x]\\1833dollars-27ladders*x=380.16dollars+12.96ladders*x](https://tex.z-dn.net/?f=%5Cfrac%7B%281833dollars%29-%2827ladders%2Ax%29%7D%7B%28792%20dollars%29%2B%2827ladders%2Ax%29%7D%3D0.48%5C%5C1833dollars-27ladders%2Ax%3D0.48%2A%5B792%20dollars%2B27ladders%2Ax%5D%5C%5C1833dollars-27ladders%2Ax%3D380.16dollars%2B12.96ladders%2Ax)
At this point, <em>we want to transfer the 'x' terms to one side of the equation, and the other terms to the other side</em>, so we get to the answer:

Finally, we divide both sides of the equation by 39.96 ladders:

So the most he can pay for each of the remaining ladders if he is to obtain a 48% markup goal is $36.357. That means that even if he buys the remaining ladders for a higher price than what he is willing to sell them for, he still obtains a 48% markup goal.