1490.16 – Buying Sand

A large sack of sand weighs 50 pounds and costs $3.25, while a small sack of sand weighs 30 pounds and costs $2.29. You want to purchase at least 170 pounds of sand to make the greatest sand castle ever. What is the cheapest way to buy all that sand?

Solution

The cheapest way to buy 170 pounds of sand in this situation is to buy 180 pounds. Large sacks of sand get you more sand for your money than small sacks, so you should buy as many as you can without going over:

50 lbs$3.25≈15.4 lbs$1.0030 lbs$2.29≈13.1 lbs$1.00\begin{aligned}\dfrac{50 \text{ lbs}}{\$3.25} &\approx \dfrac{15.4 \text{ lbs}}{\$1.00} \\[1em]\dfrac{30\text{ lbs}}{\$2.29} &\approx \dfrac{13.1\text{ lbs}}{\$1.00} \end{aligned}

Three large sacks will get you 150 pounds of sand for $9.75. At this point, the next sack you purchase will put you above 170 pounds, whether it be large or small, so it’s cheapest to buy a small sack. You have now bought 180 pounds of sand for three payments of $3.25 and one of $2.29, which comes out to $12.04. You can always find something to do with the extra sand.