By now, you probably have a good storehouse of area and volume formulas in your brain. You could find the area of a circle or the volume of a box in your sleep. Well, it's time to put those skills to good use! You can apply area and volume formulas to figure out which pizza size, cereal brand, ice cream scoop size, etc. is the best deal. And a number of Mathcounts problems ask you to do just that.
The problem will generally give you a number of choices, providing a length measurement and cost of each choice. It will then ask you to identify the choice that is the best deal (or, occasionally, the worst deal!). A typical problem of this category looks like the following:
Magic Box Inc. sells three kinds of boxes: red, green, and blue. All boxes are filled with magic dust and are cube-shaped. The red box has a side length of 1 in. and costs $1. The green box has a side length of 2 in. and costs $10. The blue box has a side length of 3 in. and costs $25. Which box will give you the best deal for magic dust?
First of all, understand that the best deal is the choice that gives you THE MOST FOR YOUR MONEY. The definition of "the most" depends on the specific problem. If the problem deals mainly with 2-dimensional shapes, such as circular pizzas or rectangular sheets of paper, you want the greatest area per dollar. On the other hand, if the problem deals with 3-dimensional shapes, such as semi-spherical ice cream scoops or rectangular-prism cereal boxes, you want the greatest volume per dollar.
To solve the problem, start by deciding whether you need to calculate area or volume. Then, calculate the area or volume of each choice. For instance, our red, green, and blue magic boxes have volumes of 1 in.3, 8 in.3, and 27 in.3, respectively.
Now divide each area or volume by the corresponding cost. This gives you the area per dollar or the volume per dollar.
Red box: 1 in3/$1 = 1 in3 per $
Green box: 8 in3/$10 = 4/5 in3 per $
Blue box: 27 in3/$25 = 27/25 in3 per $
Now select the choice that gives you the greatest area per dollar or volume per dollar. In this case, the blue box gives the greatest number of cubic inches per dollar, so it is the best deal. The green box gives the least number of cubic inches per dollar, so it is the worst deal.
IN SUMMARY: