Wednesday, October 14, 2009
Learn The Math - Part 2
Tip #198 - Learn The Math – Part 2. In the last post, I was discussing how important it is to learn how to do some basic math to help you save money. Last time we discussed calculating percentages so that it is easy to figure out how much you will save on sale items in a store. In the past, I’ve discussed calculating price per unit. Today, let’s discuss fractions. Think back to fourth grade or so. Fractions are made up with a numerator (the number on top) and a denominator (number on the bottom). So in the fraction ½, 1 is the numerator and 2 is the denominator. Fractions are useful when we are using recipes. Cooking from scratch is a great way to save money. And bulk cooking is a great way to save time and money. So suppose you decide rather than going to the bakery to buy a dozen muffins for your daughter’ soccer breakfast, you are going to make them instead. By making them from scratch, you will save several dollars.
The recipe you choose makes one dozen muffins. But you decide since you are already heating up the oven and have all of the ingredients out, you will make some muffins for your family as well to have for breakfast over the next few days saving more time and more money. So you need to double the muffin recipe. How do we double ¼ teaspoon of something? What is twice 2/3rds of a cup? Let’s attack those fractions.
When adding fractions, it can be very simple. If the denominator is the same, all you do is add the two numerators together and the denominator stays the same. Luckily in recipes, since we are doubling, tripling, or quadrupling measurements, the fractions we are using are the same and therefore have the same denominator. So if the recipe calls for 1/3 cup milk and we are doubling the recipe, then we do 1/3 + 1/3 which equals 2/3. Of course we could always fill a measuring cup to the 1/3 line two times but that takes longer than filling it to the 2/3 line. And when doing bulk cooking every time cutting step helps.
If the recipe calls for ¼ teaspoon of salt and we are doubling the recipe then we would add ¼ plus ¼ to get 2/4. Huh? There is no measurement for 2/4. Aaah, think back again to fourth grade. This is where we reduce the fraction. Since the numerator and the denominator are both divisible by 2, we can reduce the fraction by dividing it by 2/2. (any number over itself is the same as 1. So 3/3 is the same as 1; 4/4 is the same as 1, etc. And any number divided by 1 is the same as the original number so it’s always safe to divide a fraction by a number over itself.) Two divided by 2 is 1 for the numerator and 4 divided by 2 is 2 for the denominator so we get ½. Therefore, ¼ + ¼ becomes 2/4 which becomes ½. Again we can do a ¼ teaspoon two times but that takes longer than doing ½ teaspoon once. Also, if you are going to triple or quadruple a recipe, it would become too laborious to do ¼ teaspoon three or four times, not to mention the chance of losing count and messing up the recipe.
Let’s do one more example. Suppose this muffin recipe calls for 2/3 cup of oil and again we are doubling the recipe. We add the numerators 2 + 2 to get 4 and we keep the denominator the same. So the answer is 4/3. Now I know there is no 4/3 line in a measuring cup. What do we do? Whenever the numerator is bigger than the denominator we divide the numerator by the denominator. So 4/3 is 4 divided by 3. Three goes into 4 one time with 1/3 left over. So 4/3 cups is the same as 1 cup plus 1/3 cup.
We’re going to do one more, harder example, and then we’ll be ready to quadruple those brownie recipes! Suppose you have a recipe that calls for ½ cup of white sugar and ¼ cup of brown sugar. You pull out your ingredients and realize you don’t have any brown sugar on hand. But you know from past experience that you can substitute the white sugar for the brown sugar. How do you add fractions with two different denominators? Well, you need to make the denominators the same before you add them. Just like you reduced the fraction 2/4 in the earlier example (by dividing the numerator and denominator by 2) to get ½, you can do the opposite now. So you are actually multiplying the fraction by any number to get a new fraction of the same worth. When multiplying you multiply both the numerator and the denominator. So ½ times 2 is 2/4. (1 x 2 = 2 for the numerator and 2x2 = 4 for the denominator). ½ times 3 is 3/6. ½ times 4 is 4 /8.
Now back to the problem. You want to add ½ cup sugar plus ¼ cup of sugar. We need to look at the denominators. Since they are not the same we cannot just add the numerators. We need to make the denominators the same. We do this by finding a number that both denominators divide into evenly. Two and 4 both divide into 4 evenly. So we will make the denominators both 4. Since one of the fractions already has a 4 in it, we only need to change the other. We need to make ½ have a 4 in the denominator. To get 4 on the bottom we need to multiply the fraction by 2/2. Remember any number over itself equals 1. And any number multiplied by 1 is itself so we can multiply any fraction by any number over itself to keep the original number have the same worth or meaning. So to change ½ to have a 4 in the denominator we will multiply ½ by 2/2 and we get 2/4. Now we can add the ¼ cup of brown sugar (which we are using white sugar instead) plus 2/4 cups of white sugar to get ¾ cups of white sugar.
In Real Life (IRL) – I hope I didn’t make this fraction lesson too complicated. I wanted to spell everything out in case some readers completely forgot how to add, multiply, or divide fractions. But once you do it several times, it really becomes second nature. When I am adding ½ cup of sugar plus ¼ cup of sugar I instinctively know that it is ¾ of a cup. And I don’t need to sit down with paper and pencil to calculate it. And the nice thing is that recipes always use the same measurements over and over again. It’s always ¼, 1/3, ½, 2/3, and ¾. And once in while there’s a 1/8 in there. But it makes it really easy and second nature to add them together and do one big measurement when doubling, tripling, and quadrupling measurements.
My real life experience hint is that when you are doubling (or higher) a recipe, make sure you write down the new measurement you are using. I often get lazy and just say in my head, “oh, I’ll just double it without writing it down.” And it never fails I always use one of the original measurements in the recipe rather than the doubled amount. And then I have flat muffins with half the baking powder used. Not good! Of course, if you catch the mistake in time you can always add in the extra amount of the ingredient. It’s when you halve a recipe and you throw in too much of something that you are in trouble. So be careful and write down the new measurements for the doubled recipe. And let’s go bake!