## Thursday, November 25, 2010

### Cost Comparison Calculations with Conversions It's been an awfully long time since I last attended a math class. Six and a half years to be exact. Suffice it to say, my math skills have been a bit rusty as of late. When I was trying to figure out a cost comparison calculation last week to discover if powdered milk was cheaper than fresh milk, I just sat in front of the computer for a nice long while, hemming and hawing, not even knowing where to start, until eventually I managed to figure it out. And I used to be a math whiz!

I decided to share my calculations with you so that when you're faced with such a problem, you'll know the basic steps how to solve the problem and you'll be able to figure out which foods are cheaper. This cost comparison can work for any calculation, even weight to volume or dry to liquids, like if you want to figure out the price of instant hot cocoa vs homemade hot cocoa, or instant coffee vs Starbucks. ### Cost Comparison Calculations with Conversions

Before you start your calculations, you need to write down all the pertinent information that you know. You'll see below that I included how many tablespoons of powdered milk make 1 cup of reconstituted milk. For this, google was my friend and I discovered that 3 tablespoons of dry milk powder make 1 cup of milk.

However, I still needed to figure out how many tablespoons of dry milk would be in a pound. For this type of information, I love the site OnlineConversion.com. There's a special weight to volume cooking conversion, where you choose the food from a certain list, and compare quantities. There, I learned that 62.6 tablespoons of powdered milk made up one pound.
Then you might also need to know how many cups are in a gallon, or things like that. To get that information, I use OnlineConversion.com's common cooking conversions tool.
In this example, we'll be trying to figure out how much dry milk would need to cost per pound in order for it to be cheaper than fresh milk. To do this, we need to figure out how much the dry milk would need to cost per pound for it to be the same price as the fresh milk. If the price you find of the dry milk costs more than that, you know it's not worth it. If it's cheaper than that amount, you know it's a bargain.

To do this, we'll have to do some basic algebra with fractions and conversions.
First, we'll make an equation. On the right, we put the price that we do know- how much we pay for fresh milk per gallon. Locally, we pay \$4.86 per gallon of milk. (Yes, I know that is really high!)
On the other side of the equal sign, we'll put the price we want to know, represented by the variable x. We need to know how many dollars per pound we need for the price to equal \$4.86 per gallon.

At this point, I want to remind you that when it comes to fractions, 1 over 1 equals 1. When we have two things on opposite sides of an equals sign, and you put one on top of the other, you also get 1. Hence, 3 tablespoons of powdered milk over 1 cup equals one, and so on and so forth.
Another reminder is that we can multiply anything by 1 and it will remain the same thing. So we can also multiply by one of the fractions written below, because that doesn't change the quantity at all, as we're just multiplying by 1.

Ok, now we're back to the original equation. We have x dollars over pound. We want to get rid of that pound and replace it with gallon, somehow.
To do that, we first multiply that by 1 pound over 62.6 tablespoons of powdered milk. We can do that, because the fraction we're multiplying it by equals one. Note that we multiplied it by 1 pound over 62.6 tablespoons powdered milk instead of 62.6 tablespoons of powdered milk over one pound. We do it this direction because we're trying to get rid of the pound and replace it with gallon. When we have one thing in the numerator and one thing in the denominator, they will cancel each other out, as you will see.
We then multiply it by the other information we have from step 1, making sure that the you don't put two of the same signs on the same side of the fraction line. If in one place you have tablespoons powdered milk in the numerator, you should multiply it by the tablespoons powdered milk in the denominator. Same goes for cups.

So here we have the final equation. Remember that parenthesis is just another way of saying that we're multiplying. The second line is just a continuation of the first line- the paper just wasn't wide enough to fit it all on one line.

The next step that we do is separate the numerals from the rest, so we get fractions that are just numbers, and other fractions that are just measurements.

Now move all the fractions with numerals to the left and leave all the measurement fractions on the right. This makes it easier to ensure we don't miss any important numbers and mess up our calculations.

Now multiply the numbers together so you get one fraction with numerals. Then multiply all the measurements together so you get one fraction with measurements.

Now multiply the top numbers together, and then multiply the bottom numbers together. For the fraction with the measurements, you'll now be able to get rid of a lot of that mess. If you see pound in the numerator and pound in the denominator, you're essentially multiplying that by 1, so you can cross them both out. Same with tablespoons of powdered milk, and cups.

Now that you've done the multiplication and crossed out the quantities that were repeated in the numerator and the denominator, you're left with x times .7668 dollars per gallon equals 4.86 dollars per gallon.

With multiplication equations, if you have the same thing on both sides of the equals sign, you can cross them both off.

You're then left with x times .7668 equals 4.86

Now to find out the quantity of x, you divide both sides by .7668.

This gets you x = 6.34.

Ok, now in real terms, if your milk costs \$4.86 per gallon, as long as you're paying less than \$6.34 per pound for powdered milk, you're getting a good deal. Where I live, powdered milk costs \$3.15 a pound, which is definitely a good price- half the price of fresh milk! I'm buying 25 pounds of it!

If your milk costs a different amount, multiply that number by 1.3. If your milk is 1 dollar a gallon, so long as the milk powder costs less than \$1.30 per pound, then it's a good deal. If your milk is 2 dollars a gallon, as long as the powdered milk is less than \$2.60 a pound, you're getting a good deal.

You can use this same type of equation to figure out any cost comparisons that involve conversions. All you need to remember is to cancel out the quantity you don't want (in this case, pound) and replace it with the quantity that you do want (in this case, gallon) by multiplying by fractions that equal one and contain the quantity you want to eliminate on the other side of the divisor (fraction line).

Ok, peoples, in all honesty, was this math impossible to understand and follow, or did I do an ok job of explaining it? Would you be able to do the problem on your own now that I explained the steps to you?
Do you have an easy time doing cost comparisons with conversions or do you not bother? If you do, do you usually figure it out quickly, or does it take you a bunch of tries until you got it like it took me?
Have you ever bought powdered milk? How much does fresh milk cost where you live? How much does powdered milk cost? Do YOU know which is cheaper?

1. Thanks! I found this really handy!

2. Your conversion factor is .766 lbs of dry milk makes one gallon of milk so .766 lbs/gallon can be multiplied by the price that you pay for a container of dry milk and divided by the lbs of your dry milk container. That way as prices fluctuate, you only need this one calculation (since you found the conversion factor previously)
shortcut: (.766 lbs of dry milk/1gallon of milk made) x (_\$___ price paid for container of dry milk/ _lbs__ amount of dry milk) = \$ amount of the cost to make a gallon of milk from dry milk

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