Additional Math Pages & Resources

Thursday, December 3, 2009

Reducing and Reworking Ratios

Ratio is a word used in the math universe to mean several things:
  • comparative value of two or more amounts, such as 4 cups of water to 1 can of lemonade concentrate.
  • relationship of a part to a whole, such as she took 1/4 of the quiche for lunch.
  • you can also use a colon between the numbers to express a ratio, like this 1:4
Ratios can be reduced (like fractions). The 4 cups to 1 can of lemonade be converted to other units and reduced. But if the concentrate only comes in cans, there's not much value in simplifying to other, smallest terms.

The terms used in a ratio are not commutative or associative - they have to be in the right order. She wouldn't take 4/1 quiches for lunch.  Notice that the ratio does not imply anything about the value, ingredients, calories or even the size or weight of the quiche.

A common use of ratios is when making coffee - a frequently-quoted ratio of ground coffee to water is two tablespoons coffee to six ounces (by volume) of water.

Let's say a Coffee Thermos holds 100 ounces of brewed coffee. That's  about 16 times 6 ounces. If you need two tablespoons of ground coffee for 6 ounces, you need 32 tablespoons for 100 ounces.

You are making a bigger amount of coffee than a cupful, but the ratio stays the same.

(To be precise, it would be 16.66 times 6 but we don't need that level of precision, as the water amounts can vary slightly too. Some of the water you put in remains in the grounds and doesn't make it into the thermos anyway.)

If you want to, you can measure the ground coffee (and many other cooking ingredients) by weight. Ratios still apply. You have to be sure that you correctly convert the amounts being changed so the ratios remain equivalent.

Why would you want to do this? Well, it might be easier to pour a pound of coffee into a big pot than to do 32 tablespoon scoops.

How much does a tablespoon weigh?  It depends on the coffee. But let's say about 4-5 grams. So replacing our 32 tablespoons volume measure with 5 grams weight measure looks like this:

5 gm per tablespoon x 32 = 160 grams by weight

100 ounces of water weighs about 2800 grams.

Now our ratio is "cleaner" because it's in the same units.

160:2800 can be simplified to 16:280 or 8:140 or 4:70 or 2:35 or 1:17

Remember, this is by weight, so simply measure the weight of the water in your coffee cup, and add 1/17th as much coffee.

Conveniently enough, 6 ounces of water is equal to 170 grams. So you add 10 grams of coffee, and you are set. How many tablespoons is that? Two.

I think the days of the coffee cup holding 6 ounces are over. Adjust the quantities upwards since your mug is bound to be larger! Twelve ounce cups hold 340 grams of water so you can add 4 tablespoons of coffee. That will assure you don't fall asleep reading my blog!

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