Task
A fuel oil dealer buys 20,000 gallons of heating oil at \$2.65 per gallon and another 14,000 gallons at \$3.00 per gallon. (The oil is the same grade and quality, but the price varies due to the market.) He has a contract to sell up to 35,000 gallons of oil next month at \$3.25 per gallon, but wants to use as much cash as possible immediately for future investments. To raise cash, he can sell some of his oil to another distributor, who will pay \$2.75 per gallon now. How much investment money can the dealer raise now by selling oil and still be able to break even after selling the remainder next month?
IM Commentary
The task is a modeling problem which ties in to financial decisions faced routinely by businesses, namely the balance between maintaining inventory and raising short-term capital for investment or re-investment in developing the business.
Solution
The dealer has spent:
$$
20,000 \text{ gal.} \times \frac{$2.65}{\text{gal.}} + 14,000 \text{ gal.} \times \frac{$3.00}{\text{gal.}} = $95,000.
$$
He has 34,000 gallons, and if he sells $x$ gallons now and $(34,000 − x)$ gallons next month, he will get
$$
\begin{align}
x \text{ gal.} \times \frac{$2.75}{\text{gal.}} + (34,000 - x) \text{ gal.} \times \frac{$3.25}{\text{gal.}} &= 34,000 \times $3.25 - x \times $0.50 \\
&= $\left(110,500 - \frac{x}{2} \right).
\end{align}
$$
Thus, the dealer breaks even if
$$
\left( 110,500 - \frac{x}{2} \right) = 95,000,
$$
or
$$
x = 31,000.
$$
He can sell 31,000 gallons now, raising a total of \$85,250 for investment, and break even by selling the remaining 3,000 gallons next month.
