Translating Slowly at a Glance

Don't you hate it when someone's translating something from another language and they go so quickly that you can't keep up? We hate that, too. We also hate it when we need to go from English to math in a hurry. We prefer going slowly, piecing together a little bit at a time. We don't need to go from a paragraph straight to a symbolic expression, and it's often easier not to. Ne comprenez-vous?

Sample Problem

What is the surface area of a rectangular box with a lid?

Let's think about this with common sense. Fortunately, we kept some in reserve for exactly this moment. To find the surface area, we need to add the surface area of the top, the surface area of the bottom, and the surface areas of the four sides. Writing this partially in symbols, we get:

(surface area of top) + (surface area of bottom) + (surface areas of sides)

To find the surface areas of the top, bottom, and sides, we'll need variables for the dimensions of the box. Let's use h for height, w for width, and l for length. It's a good idea to label these in the picture, too. Don't worry that this will mess up the box. We weren't planning to enter it into any art contests anyway.

The surface area of the top and the surface area of the bottom are each lw, which comes less from common sense and more from the memorization of an uber-useful formula. We can now translate a bit more into symbols:

lw + lw + (surface areas of sides)

All we have left to worry about are the surface areas of the sides. Worry we will, until we have it figured out. We're perfectionists like that. Also, we're Yoda. You had no idea.

The two sides on the left and right ends each have surface area wh, and the front and back sides each have surface area lh. Now we can finish translating to get:

lw + lw + wh + wh + lh + lh

Nice. We didn't even need to use Google Translate. Finally, we tidy up a little, because this equation is a mess. Where was it raised, a barn?

The surface area of the box is:

2lw + 2wh + 2lh

Translating from English to math a bit at a time can make the work take a little longer, but if it helps you find the right answer consistently, it's probably worth it. When we say "probably," we mean "definitely." We were being sarcastic. There's a first time for everything.

Examples
Exercises

Example 1

Liana has 7 times as many chocolate bars as her friend Beth. Liana may be happy about that right now, but we'll come back to her after her third heart attack. Express the number of chocolate bars Liana has in terms of the number of chocolate bars Beth has.

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Example 2

Felicity has some cookies, and Liana has four times as many cookies. Apparently, Liana has polished off all those candy bars and still isn't full. Goodness gracious, Liana, eat an apple.

Express the number of cookies Liana has in terms of the number of cookies Felicity has.

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Example 3

The length of a bed is two feet longer than its width. In terms of its width, how long is the bed?
Let w be the width of the bed. Then the length of the bed is w + 2. Hey, as long as our feet aren't hanging out over the edge, we'll be happy.

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Example 4

Jonathan is three inches shorter than Jules. Jules is two inches shorter than Justin. How tall is Jonathan in relation to Justin?

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Example 5

What is the total area covered by the shape shown below (in terms of s)?

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Example 6

Linda divides a pie equally between herself, her two kids, and three of their friends. She won't even come close to dividing the ice cream evenly, but that's another story for another day. What fraction of the pie does each person get?
Divide 1 pie between Linda, 2 kids, and 3 friends to get , or any equivalent expression.