# Another Day with Algebra

I couldn’t get through the day without algebra. Being conversant with algebra means I can figure out the best buy in paper towels, and how much dirt I need to buy in order to fill a hole in my back yard.

Let’s start with the daunting exercise of figuring out the best buy on a favorite brand of paper towels. Suppose I’m faced with the following three choices:

Choice A:

16 family rolls (= 40 regular rolls) “quick size”

2 rolls, 143 2-ply sheets, total 124 ft^2, 11” x 5.7”

$64.25 ($2.81 / 100 sheets)

Choice B:

8 family rolls (= 20 regular rolls) “quick size”

2 rolls, 143 2-ply sheets, total 124 ft^2, 11” x 5.7”

$42.80 ($0.04/count)

Choice C:

12 mega rolls (= 26 regular rolls) “select-a-size”

12 rolls, 120 2-ply sheets per roll, total 648 ft^2, 11” x 5.7”

$41.00 ($0.03/count)

Companies market products to make money. They purposely try to confuse you about which choice is the best buy, in hopes that you’ll end up paying more. This is really annoying. Luckily, I am math literate, so I know I can prevail.

Look at the first two choices. They are actually in the same units – “family rolls” of “quick size”- so we don’t have to convert between units here to compare costs. Choice A has twice as much product as choice B, so 2 units of B equals 1 unit of A. 2 units of B cost 2*$42.80 = $85.60 and 1 unit of A costs $64.25. Clearly A is a better buy than B.

In order to look at the cost per sheet, start with choice A’s measure of “2 rolls, 143 2-ply sheets”. This is confusing. Does each roll have 143 sheets, or do the two rolls together have 143 sheets? Since 143 is odd, guess that 143 is the number of sheets on each roll. Then 143 *16 = 2288 sheets per package. If we take the total cost, and divide by the number of sheets, we come up with $64.25/2288 = $0.02808129 per sheet, which is approximately equal to the company’s advertised “$2.81 / 100 sheets”. Choice B has half as many sheets, so its cost is $42.80/1144 = $0.3741259 or approximately $0.04 per sheet, which they describe as “$0.04/count”.

So far, so good. Now we come to choose C. A mega roll sounds bigger than a family roll. Is it? Let’s see. Each mega roll has 120 sheets, whereas each family roll has 143 sheets. So, a mega roll is actually smaller than a family roll.

Is choice C a better buy than choice A? Well, 12 mega rolls have 120 sheets each, so there is a total of 12 * 120 = 1440 sheets. So choice C costs $41.00/1440 = $0.02847222 or approximately $0.03 per sheet. At $0.02808129 per sheet, choice A is a very slightly better price than choice C’s $0.02847222 per sheet. No one could fault you for choosing A over C here.

Wow, that was a lot of work just to figure out the best buy on paper towels. And yes, I have been known to whip out the calculator on my phone, while standing in the grocery store, to figure out the best buy on a product on the spot.

Which brings us to another household use of algebra. In my yard, I need to fill in a hole dug by some animal (which is no longer living there). The hole has a diameter of 24 inches and is 6 feet deep. How much dirt should I buy to fill it?

First I need to calculate the volume of the hole. I need to convert to a single set of units, so we’ll use feet. The hole has a diameter of 2 feet, giving it a radius of 1 foot. From algebra, I know that volume V = pi*r^2*h, or:

V = pi * 1^2 * 6 = 6 * pi = 18.9 ft^3.

Now I need to buy the dirt, which can be sold in cubic feet and/or quarts. If the store has 2 cubic foot bags of dirt, you’ll need at least 18.9/2 = 9.45 bags of dirt to fill the hole. Just buy 10 to be safe.

Suppose the bags are labelled in quarts. A little web research reveals that 1 cubic foot of dirt equals 25.75 dry quarts. Therefore, I will need 18.9 ft^3 * 25.75 quarts/ft^3 = 486.675 quarts of dirt. If the store sells 50-quart bags of dirt, you will need 486.675 quarts/(50 quarts/bag) = 9.7335 bags of dirt. Again, just buy 10. And voilà, algebra has saved the day again.

Everyone needs to be proficient in basic algebra. It’s there to help you figure out the best buy on a product, and how much of a product you need to complete a task. This is absolutely critical when buying a lot of a product you don’t normally buy, like dirt to fill a hole. A little math can protect you from colossal miscalculation, and having to make yet another trip to the store.