Math for Grownups (14 page)

Suppose Graham knows that he needs 5∕4 cup of olive oil, but he doesn’t want to measure out ¼ cup five times. He can convert that improper fraction to a mixed number. He should just divide the numerator by the denominator (the top number by the bottom one): 5 / 4 = 1. That 1 is the whole number. But there’s a 1 left over also, right? That’s the remainder. Pop that remainder on top of the original denominator (which was 4), and you end up with ¼. Add them together and you get 1¼ cups.

Why would you need to use improper fractions? Let’s say you need 2½ cups of flour, but you have only a ½-cup measure. How many times do you need to dip your ½-cup measure into the flour canister to get 2½ cups? Think of 2½ cups as an improper fraction:

2 (the whole number) • 2 (the denominator) = 4 + 1 (the numerator) = 5∕2 (put the new number over the old denominator)

That’s 5 half-cups of flour.

Have you ever wondered how the chefs in television contests come up with such delicious dishes—without a recipe? Well, they’re masters of their craft, of course, but they also depend on some basic formulas, many of which are shown as ratios.

A ratio is a way of showing the relationship between two numbers. In
Chapter 4
, we talked about scale drawings, and we said that often in such drawings, ¼" = 1'. That’s a ratio. Every ¼" on the drawing corresponds to 1' in the real world. Because ratios are usually written with a colon, we could have represented this ratio as 1': ¼".

We often use the word
part
when we’re talking about ratios. That’s because we use ratios to show the relationship between different things. For example, Joe likes a stiff drink after work each day. If he mixes three ounces of Scotch with one ounce of water, the ratio is 3:1.

Here’s another example. Felice finally has her own place in the Big Apple. She plunked down a crazy amount of cash for her studio apartment, which means she’ll be on a tight budget from now to eternity.

But that’s okay with her. If she can make it here, she can make it anywhere.

Felice has a futon, which does double duty as both bed and couch. Her grandmother gave her a card table and four folding chairs. And she picked up a pot, a skillet, and a set of mismatched dishes at a flea market in The Village. The can opener and a handful of flatware were left behind by a former roommate.

In other words, she’s all set, as long as she doesn’t need to make any fancy meals. For now, she’ll live on leftover pizza, salads, and canned soup.

But these menus get old quickly. Before she knows it, Felice is ready for something different. She can’t afford to buy any specialty cooking tools—not even a measuring cup—and she still needs to eat on the cheap.

Long-grain white rice is a great option. It’s versatile, and she can make it with everything she has on hand. Her grandmother not only gave her a place to eat but also taught her how to make rice—using a 2:1 ratio, or 2 parts water to 1 part rice.

That means Felice can use a regular coffee mug to measure her rice and water. It doesn’t matter how big the coffee mug is, as long as she keeps the ratio the same. She fills her mug with rice and pours it into the pot. Then, using the same mug, she adds 2 mugs of water. She brings it all to a boil, turns down the heat, and covers it with the lid. In 15 to 20 minutes, she has fluffy white rice to serve with sautéed veggies.

Yum!

But what if she wants to entertain? Changing a ratio is simple. To double the recipe, she merely doubles each part: 4 parts water to 2 parts rice. And what if one day she isn’t as hungry as usual? She can simply reduce the ratio: 1 part water to ½ part rice.

Here’s hoping that rice doesn’t get too old too soon.

A ratio is just a way to show the relationship between numbers. Mathematically speaking, ratios compare only two numbers.

Ratios can be written in the following three different ways.

• 2:1

•

• 2 to 1

Ratios can also be written as fractions, which means they can also be written as percents.

All you need to do is divide the denominator into the numerator and change the answer to a percent, like this:

3:4
=
¾

3 / 4
=
0.75

0.75
=
75%

In other words, ratios are more flexible than the most adept chef or baker. Despite the fact that, mathematically speaking, a ratio shows the relationship between two numbers, in the kitchen, cooks may use more complex “ratios” with more than two numbers.

Technically, these aren’t real ratios because they can’t be written as fractions or percents, but they do show the relationships among ingredients. For example, the typical ratio for baking biscuits is 3:2:1—that is, 3 parts flour, 2 parts liquid, 1 part fat. How does this work in the kitchen?

If you had 3 cups of flour, you would need 2 cups of liquid and 1 cup of fat. But what if you had 6 cups of flour? Well, 6 = 2 • 3, so you would need to multiply each of your other numbers in the “ratio” by 2.

2 • 2
=
4

1 • 2
=
2

You would need 4 cups of liquid and 2 cups of fat.

And of course you can reduce a recipe this way, too. If you had 1 cup of flour, you would need to reduce the other ingredients by 1∕3. (That’s because 3 = 6 / 3, or 6 • 1∕3.) To find out how much of the other ingredients you need, just multiply by 1∕3.

2 • ⅓
=
⅔

1 • ⅓
=
⅓

So you would need 2∕3 cup of liquid and 1∕3 cup of fat.

As any experienced cook will tell you, timing is often the most difficult skill to master in the kitchen. Nobody wants to sit down to a meal of overdone fish, cold broccoli, and room-temperature biscuits. (The butter should melt into the flaky layers, you know?)

Figuring out how long a dish should bake, roast, or boil is the first step to presenting a carefully choreographed dinner. And for many novice or not-so-frequent home chefs, a giant turkey is the most daunting of all entrées.

Sure, you can count on the pop-up timer. These come with some turkeys, or you can buy one separately. But you’ll still need to know when to put the bird in the oven—and when to start boiling the potatoes.

And there’s also the thawing time. Buying a frozen turkey means allowing time for it to defrost, which is probably a lot longer than you think!

But you don’t need Julia Child or a semester at Le Cordon Bleu to figure any of this out. Thawing times and cooking times depend on the turkey’s weight.

It’s your first Thanksgiving with your new husband, Tom. And your mother-in-law will arrive just in time for the 6:00
P.M.
dinner. She’s bringing pecan pie, stuffing, and homemade rolls. You’re in charge of all the rest—including the turkey. You’ve ordered a 12-pound bird, which you’ll need to thaw in the fridge before roasting. When should you pull it out of the deep freeze?

You know from your sister’s horror stories that you can’t cut corners by thawing the bird on the counter. Unless you want to host the Thanksgiving-dinner-when-everyone-got-
Salmonella
, your best bet is to defrost the turkey in the refrigerator. The United States Department of Agriculture (USDA) says to allow 5 hours of thawing time per pound. They oughta know, right?

You’ve bought a 12-pound turkey. How long should you allow for thawing?

It’s a simple problem, really. Just multiply the number of pounds by 5—the number of hours needed to thaw each pound.

12 • 5
=
60

So, you need to put the turkey in the fridge for 60 hours in order to thaw it. But let’s think a moment. Does this mean 60 hours before dinner is served? Nope. The USDA also says that serving raw poultry is a big no-no, so you’ll also need to roast the bird.

If your oven is set to 325°F, the USDA recommends roasting an unstuffed turkey for 2¾ to 3 hours. They’re the experts on avoiding food-borne illnesses, so you decide to follow their recommendations.

With a little time for resting—the turkey, not you—and carving, you estimate that it will take 3 to 3¼ hours to get the bird from the fridge to the table. You’ll need to add that to the thawing time in order to figure out when to pull the turkey out of the freezer.

60
+
3¼
=
63¼ hours

Clearly you’ll need more than a day, but how much more? There are 24 hours in a day. How many 24s are there in 63¼? You can use a calculator, but that could be confusing. Instead, try some mental math.

To make things easier, forget about the extra ¼ hour (or 15 minutes). You can add that on to the end. Working with whole numbers is much easier.

It looks like you’ll need at least 2 days. That’s because 24 times 2 is 48, which is less than the total time you have figured out. Will you need a third day? You can subtract to find out.

63
-
48
=
15

So, 2 days and 15 hours (plus the extra 15 minutes) ought to do it. But that doesn’t tell you
what
time
to start defrosting the turkey, does it?

Remember, your dinner starts at 6:00
P.M.
Fifteen hours before that is 3:00
A.M.
, and another 15 minutes before
that
is 2:45
A.M.
So you will have to take the turkey out of the freezer at 2:45
A.M.
on the Tuesday before Thanksgiving.

Because you’re doing all the cooking, you decide to let Tom get up to move the turkey from the freezer to the fridge. You set his alarm on Monday night and settle in for the last good night’s sleep of the week.

Why is it so difficult to subtract and add time? The answer is simple. While the decimal system of numbers is based on 10s (it is also known to math geeks as a base-10 system), time is based on 60s. In other words, there are 60 seconds in a minute and 60 minutes in an hour.

But it gets even worse. The number of hours in a day is 24, which is neither base-10 nor base-60.

There is an easy fix, though. Instead of thinking of minutes, hours, and days as numbers, imagine them on a clock face. If you think of the clock face as a circle divided into 12 wedges, you can visually subtract and add time without much effort.

Take it one step further, and it’s even easier. The diameter of the circle splits it into two halves—six hours on one side and six hours on the other side. That means that numbers directly across from each other on the clock are exactly six hours apart.

Because 9 is across from 3, 9:00
P.M.
is six hours later than 3:00
P.M.
And because 11 is across from 5, 11:00
A.M.
is six hours before 5:00
A.M.

Once you can see the six-hour increments, it’s not too difficult to do more challenging subtraction and addition problems. For example, 8:00
P.M.
to 3:00
A.M.
is 6 hours plus 1 hour, or 7 hours.

There’s no need to reinvent the Thanksgiving dinner. If you’re not sure how much to buy for your dinner, consider these guidelines:

• A 1-pound bag of carrots makes 4 to 5 servings.

• A 12-ounce package of fresh cranberries makes about 2¼ cups of sauce.

• A serving of gravy is about 1∕3 cup.

• 1½ pounds of fresh green beans makes 6 to 8 servings.

• A 5-pound bag of potatoes makes 10 to 12 servings.

• A 14-ounce bag of stuffing makes about 11 servings.

And then there’s the almighty turkey. If you want leftovers, shoot for 1½ pounds per person. (Remember that the total weight includes bones, so you won’t be serving that much to everyone.) To do the math, multiply the number of guests by 1.5.

8 people • 1.5 pounds
=
12-pound turkey

10 people • 1.5 pounds
=
15-pound turkey

12 people • 1.5 pounds
=
18-pound turkey

14 people • 1.5 pounds
=
21-pound turkey

Is it just you and your cat? Why not order Chinese?

You can think of variables as the cats and dogs of the math world. Independent variables are the cats. They can get along just fine without others; they just need to have fresh bowls of water and food and a clean litter box.