## Ask Professor Puzzler

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"How do you add 8 cubic yards and 8 cubic feet?" ~Anon.

Since you're asking the question, you're probably deduced that you can't simply add 3 and 3 to get 6. If you did that, you'd then have the question of whether the units in the result are cubic yards or cubic feet. The simple rule is that you can't add two quantities which have different units.

If two units are the same *type* of unit (for example, they're both distances, or they're both times), then you can rewrite one of them so they have the *same* unit, and then you can add them.

In this case, both units are *volumes*, which is an amount of 3-dimensional space something takes up. We want to either convert 3 cubic yards into cubic feet, or we want to convert cubic feet into cubic yards. I'm going to convert cubic yards into cubic feet.

If you have 8 cubic yards, you can picture that as a cube. Conveniently, 8 = 2^{3}, so you can picture it as a cube that has a width of 2 yards, a height of 2 yards, and length of 2 yards. Length times width times height = 2 yards x 2 yards x 2 yards = 8 yards^{3}.

But we know how to convert 2 yards into feet; there are 3 feet in a yard. So it turns out that our cube's dimensions are 6 feet by 6 feet by 6 feet. So Length times width times height = 6 x 6 x 6 = 216 feet^{3}.

Now that we have written both units in cubic feet, we can add them: 216 feet^{3} + 8 feet^{3} = 224 cubic feet.

That answers your question, but we've really only touched the surface of converting units - you can find a more detailed study unit here: Conversions factors and unit conversions.