Math for Grownups (17 page)

You’ll need 1¾ cups of dark chocolate to go with your 1∕3 cup of pistachios.

Whew! Those pistachios had better be worth it. That was a lot of work!

When you alter a recipe, you often find yourself having to multiply or divide fractions. Here’s a quick refresher.

To multiply fractions, just multiply the numerators together and then multiply the denominators. If you want to halve the amount of olive oil you need in Mom’s spaghetti sauce, multiply the ¼ cup by ½:

1 • 1 (the numerators)
=
1 and 4 • 2 (the denominators)
=
8.

¼ • ½
=
1/8;

To halve the recipe, then, you need to use 1/8 cup olive oil.

Dividing fractions is like multiplying, but you have to do one step first: Invert the second fraction. Say you have a punch recipe that serves 100 people, but you need to make a much smaller batch—reducing it to 1/9 its original size. If the original recipe calls for ¾ gallon of orange juice, how much would you need for your smaller batch?

In this case, you want to divide ¾ by 9.

First invert the second fraction—that is, turn it upside down. Then multiply.

You already know that to multiply fractions, you just multiply the numerators and then the denominators.

3 • 1
=
3 and 4 • 9
=
36

So, you’ll need 1∕12 gallon of orange juice for your smaller batch of punch.

And now you can alter any recipe you know!

In the Yard: If a Train Leaves Omaha at 8 A.M., How Much Lawn Edging Do You Need?

You may like the idea of a well-landscaped yard with room for entertaining and for the kids to play. Fresh flowers from a cutting garden look great on the dining room table, and how about cooking with herbs from your own backyard!

You can almost hear the bees buzzing in agreement. This is a wonderful plan.

But if you can’t really afford the services of a landscape architect, horticulturist, or gardener, you’ll be doing much of this work yourself. And you will probably require a donation of blood, sweat, and brain cells—and a little bit of help from your good friend, mathematics.

If you have a yard at all, you’ve probably considered digging up part of the lawn and installing a flowerbed or two. Whether you fill them with multicolored blossoms or evergreens, the first step is to define the shape of those flowerbeds.

Addison’s yard looks great—except for that giant dead spot in the back left corner. The neighbor’s live oak shades the area so densely that grass won’t grow. But Addison’s been reading up on shade-loving plants. She figures a little flowerbed of hostas and impatiens will work great there.

Heading out to the backyard, Addison decides to define the space with a garden hose before she starts digging. This will give her some idea of the quantity of materials she needs to buy.

The flowerbed is in the corner, at the angle where two fences meet. She decides on a wedge shape (think of a slice of pizza), so the flowerbed will have two straight sides, where the fences are. But the third side, she decides, will be curved (also like the slice of pizza). That’s pretty.

Addison whips out her measuring tape. She jots down the measurements as she goes. One straight side is 3' 10" long. The other is 3' even.

The straight sides are easy. But what about that curve? Addison decides to leave that until later; first, she needs to think about what she has to buy.

She knows that the garden center can help her figure out how many flowers she needs, so that’s not a problem. But she also knows, from experience, that weeds will take over her flowerbed if she’s not careful. Therefore, she decides that she needs to buy a barrier—a product known as weed guard—to put down before the topsoil and mulch. Her neighbor has offered to give her his leftover topsoil and mulch, so that’s taken care of. But because she wants to keep the grass from creeping into the bed, landscape edging is a must.

Addison needs to know how much weed guard and landscape edging to buy. That, of course, depends on the size of the flower bed. If she buys too little weed guard, she won’t be able to cover the entire bed. If she buys too much border, she’ll have wasted her money.

But will one measurement tell her how much weed guard and landscape edging to buy? The weed guard will cover the entire flowerbed. The border will go around it. In other words, she needs enough weed guard to cover the entire
area
of the flowerbed, whereas she needs enough landscape edging to go around the entire
perimeter
of the flowerbed (that is, the distance all the way around it). Looks like she will have to make two different measurements.

Addison starts by figuring out the perimeter. As long as she measures all the way around the flowerbed, she’ll know how much landscape edging she needs. No fancy formula involved!

To find the perimeter of her flowerbed, she can simply add together the lengths of all sides—including the curve. To measure the curve, she lays her garden hose around it and then marks on the hose, with a crayon, where the curve begins and where it ends. Then she straightens out the hose and measures between the lines she marked. (Again, nothing fancy: Straightening out the curve doesn’t change the measurements.) The curved edge measures 5'. She then just adds all the sides together:

3' 10"
+
3'
+
5'
=
11' 10"

Addison rounds up and sees she needs 12' of plastic border.

Now she just needs to figure out the area of the flowerbed. If she were dealing with a rectangle, she’d just multiply the length by the width. But Addison doesn’t have a rectangle. She’s going to have to get creative.

To be honest, Addison doesn’t need to know the
exact
area. If she has a little too much weed barrier, that’s no big deal. But having too little would be a problem. And she also doesn’t want to buy double the amount she needs.

But what’s the easiest way to estimate the area? She needs a familiar shape that is a little larger than her flowerbed.

Passing by her sketch, her son Mario asks, “Mommy, why are you making a triangle flowerbed?”

A light bulb goes on over Addison’s head. Her flowerbed is not a triangle, of course, but it
looks
like one—at least to 4-year-old Mario. She doesn’t remember the formula for the area of a triangle, but that would be easy to find out, right?

She looks at her sketch again. If she somehow straightened the curve of her flowerbed, she could make a triangle. Experimenting, she realizes that if she extends the straight lines of her wedge a bit, then she can connect them with a straight line to make a triangle that is slightly bigger than her flowerbed. If she uses that measure, she’ll have some leftover weed guard, but that’s better than not having enough.

Addison goes back out to the garden and moves the garden hose so that it’s a straight line, with each end of the line meeting one of the fences. Now she has a perfect triangle. She measures the new line. It’s 6' long.

Now it’s time for some more math. What is the formula for the area of a triangle? Addison knows just the person to ask for help: her 14-year-old daughter, Grace.

“Duh,” Grace says, rolling her eyes. “The area of a triangle is one-half of the base times height.” And she turns back to her cell phone to finish the text she was writing when she was so
rudely
interrupted.

Addison looks at her sketch again. She knows the base of the triangle is 6', but what is the height? It’s tempting to just measure one of the sides, but Addison has the nagging feeling that’s not right. Looking at her sketch again, she realizes she needs to measure from the top of the triangle (the vertex opposite the base) to the base. Back out in the yard, Addison measures the height of the triangle-shaped flowerbed: 4'.

Now Addison can find the area. She just does what Grace told her. She takes half of the base of the triangle and multiplies that by the triangle’s height.

½ • 6 feet • 4 feet

3 feet • 4 feet

12 square feet

Because the answer is the area of the triangle, she uses
square feet
as her unit. The area of the triangle is 12 ft
²
.

Now Addison knows that she needs to buy 12 ft of plastic border and 12 ft
²
of weed guard.

If only she could get Mario and Grace to do the digging for her.

Addison doesn’t need to be an artist to solve her flowerbed problem. But it does help that she can visualize the situation and draw a sketch.

And then there’s that organization thing again. Updating her sketch and being careful with her measurements helped her be certain not to omit any of the steps in solving the problem—and to perform them in the right order.

Note that Addison didn’t really know how to solve her problem at first. She created a plan while she was doing it. And this is the sign of a confident problem solver. She didn’t wig out when she was unsure of something. She didn’t question her abilities. She simply plodded along—using what she knew and asking someone else when she didn’t know.

Having smart kids was a plus, too. (We should all be so lucky.)

You don’t have a yard, you say? But the thought of fresh tomatoes makes your mouth water? And your green thumb has been itching to create some horticultural magic?

Try container gardening!

Along with water and sunshine, all you need is a container, soil, and plants, and you’re good to go.

Let’s say you have four containers that are shaped like rectangular prisms (in other words, they are solid or three-dimensional rectangles). You’d like to grow four different varieties of tomato plants—one per container. How much potting soil will you need?