Math for Grownups (16 page)

What if she used 4∕3 cups coconut and 1∕3 cup macadamia nuts? How many cups of white chocolate chips would she need?

She subtracts the numerators she has from the total she needs: 6 - 4 - 1 = 1. That gives her the answer, the number that goes in place of “?” in the previous equation.

Sweet success! She needs 1∕3 cup white chocolate chips.

“That settles it,” Betty says to her cat, Cookie, as she retrieves her 1∕3-cup measure from the cabinet. Soon the exotic smells of the islands transport her back to the beaches of Maui.

It’s almost like she’s still on vacation. Except for the sink full of dirty dishes.

It’s not always easy to tell which fractions are larger and which are smaller. That’s because it’s difficult to compare fractions with different denominators. You know that ¼ cup is smaller than ¾ cup, because you’re comparing like denominators. Just look at the numerator and see which is larger and which is smaller. But is 2∕3 a cup larger or smaller than ¾ cup?

Here, the denominators are not the same. Can you convert them so that they are the same?

Yes! The simple thing to do is to multiple one denominator by the other denominator to get a common denominator. So if you multiply
3 • 4, you’ll get 12. But because you changed the denominator, you also have to change the numerator by a corresponding amount. Here’s how. For the 2∕3 cup, you would have to multiply the denominator by 4 to get 12. So you multiply the numerator by the same number (4), and that keeps the ratio (remember ratios?) the same. Thus 2∕3 cup = 8∕12 cup.

You’re not quite done. For the ¾ cup, you have to do a similar operation. To get the common denominator of 12, you have to multiply the denominator of 4 by 3. So far, so good. But now you have to multiply the numerator, too: 3 • 3 = 9. So ¾ cup = 9∕12. If you compare 8∕12 with 9∕12,
you’ll see that ¾ cup is slightly larger than 2∕3 cup.

But what if you have more than two fractions with different denominators?

Suppose you’re making a cookie recipe that calls for ½ cup macadamia nuts, 2∕3 cup milk chocolate chips, and ¾ cup raisins. But you love dark chocolate, so you are going to replace all of those ingredients with dark chocolate chunks. How do you figure out the amount of dark chocolate chunks you’ll need? You’ll have to add ½ cup, 2∕3 cup, and ¾ cup. Now, you could multiply all those denominators together (2 • 3 • 4) to get a common denominator that will work for all of them. But that would give you a huge number you’d have to simplify. Another way to find your answer is this:

1. Find the smallest number that all of the denominators will divide into evenly. This is called the least common denominator, or LCD. (In the example above, 2, 3, and 4 all divide evenly into 12.)

2. Working with each fraction individually, divide this number by the denominator of the fraction. (In the first fraction of the example above, 12 / 2 = 6.)

3. Then multiply the numerator by that answer. (Because 1 • 6 = 6, ½ becomes 6∕12.)

4. Continue doing the same with the remaining fractions on the list.

5. Then add the numerators together and simplify (if needed) to get your answer.

In our example, the fractions become 6∕12, 8∕12, and 9∕12. Adding their numerators together gives us an answer of 23∕12. If you divide 23 by 12, you’ll end up with 21∕12 cups.

You look through the measuring cups in your cupboard and discover that you don’t have a 1∕12-cup measure. You look at the handy conversion chart (see the sidebar titled “Crafty Conversions”), and you discover that 16 tablespoons equals 1 cup. So how many tablespoons does 1∕12 cup equal?

Because 1 cup equals 16 tablespoons, you can multiply 1∕12 cup by 16 to find your answer.

You’ll need 11∕3 tablespoons.

Now, suppose you’re rummaging in your cupboards, and you notice you have leftover pistachios! You love love love pistachios, so you want to include them in the recipe. That means you have to decrease the dark chocolate chunks by whatever amount of pistachios you have.

You chop and measure, getting 1∕3 cup pistachios. You know you need 21∕12 cups of dark chocolate chunks to substitute for the other ingredients you like less well. So you have to subtract the 1∕3 cup pistachios from the 21∕12 cup total to find the new amount of dark chocolate chunks to add to the batter. Just as in adding fractions, to subtract fractions you have to find a common denominator. Then you subtract the numerators and you’re set. (Here we go!)

First, change 21∕12 to an improper fraction. That will make it easier to subtract.

Then set up the subtraction problem.

The common denominator is 12, because it’s the smallest number that both 12 and 3 divide into evenly. That means the first fraction will stay the same. To find the second fraction, multiply both the numerator and the denominator by 4.

Now subtract! (Just subtract the numerators. Just as in adding, the denominator stays the same.)

Want to know what this is as a mixed number? Change it! 12 goes into 21 one time with 9 left over.

And now you can simplify your fraction by dividing the numerator and denominator by 3.