We use the powerful mathematical concept of geometric similarity. You may have encountered this idea in geometry in connection with similar triangles (see Figure 18.2). Here we apply it to real objects.

Although similar objects need not be the same size, corresponding distances must be proportional, just as in similar triangles. For example, when a photo is enlarged, it is enlarged by the same factor in both the horizontal and vertical directions—in fact, in any direction (such as a diagonal). We call this magnification factor the linear scaling factor (or length scaling factor).

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In Figure 18.3, the linear scaling factor is 3; the enlargement is 3 times as wide and 3 times as high as the original. In fact, every pair of points becomes 3 times as far apart.

Objects can be scaled down as well as up; for example, the smaller photograph in Figure 18.3 is geometrically similar to the larger one. The linear scaling factor to get the smaller from the larger is 1/3.

How Area and Volume Scale

The enlargement can be divided into rectangles, each the size of the original. Hence, the enlargement has times the area of the original. More generally, if the linear scaling factor is some general number (not necessarily 3), the resulting enlargement has an area (“L squared”) times the area of the original.

We symbolize the relationship between the area and the linear scaling factor by

where the symbol is read as “is proportional to” or “scales as.”

What about enlarging three-dimensional objects? If we take a cube and enlarge it by a linear scaling factor of 3, it becomes 3 times as long, 3 times as high, and 3 times as deep as the original (see Figure 18.4).

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What about volume? How much bigger is the volume of the enlarged cube? The enlarged cube has three layers, each with little cubes, each the same size as the original. Thus, the total volume is times as much as the original cube. Denoting the volume by , we can write

Thus, for an object enlarged or shrunken by a linear scaling factor , the resulting volume is (“L cubed”) times the volume of the original. Like the relationship between area and , this relationship holds even for irregularly shaped objects, such as science-fiction monsters.

You can see, however, that the area of each face (side) of the enlarged cube is times as large as that of a face of the original cube, just as the area of the photo enlarged by a factor of 3 has 9 times the area of the original. The total surface area of the enlarged cube is 9 times the total surface area of the original cube.

More generally, for objects of any shape, the total surface area of a scaled-up object goes up with the square of the linear scaling factor. Thus, the surface area of an object scaled up by a linear scaling factor of is times the surface area of the original. This feature holds true even for irregular shapes.

Before we discuss scaling real three-dimensional objects, you should understand some language pitfalls in describing increases and decreases.

The Language of Growth, Enlargement, and Decrease

Let’s see if you can make sense of some incorrect and confusing claims you can find on the Internet:

1 out of 3 households earned 200 percent less than the poverty line in 2009.

—inthesetimes.com/working/entry/6793/since_great_recession_more_working_families_in_poverty

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Well, 50% less than would be half as much, 100% less than would be zero, and 200% less than simply makes no sense. But here it’s just that the word order is mixed up. What the writer meant to say is “earned less than 200% of the poverty line.”

Americans … bike 25 times less than the Dutch.

—cleantechnica.com/2014/01/03/americans-drive-twice-much-brits-bike-25-times-less-dutch/

You no doubt guess that “25 times less” really means “one twenty-fifth” as much. But the latter doesn’t sound as grand. Writing “25 times” does give the correct ratio, and “less” clues the reader about which is less. But mixing “times” with “less” or “more” offers potential for confusion, as you can see in the following example.

At a website giving information about federal student loans, you may find the following statement:

Under this plan, your monthly payments … won’t be more than three times greater than any other payment.

Let’s see. Suppose you have $1. If I have $1 greater than what you have, then I have $2. If I have $3 greater than what you have, then I have … $4.

The terms more, larger, and greater refer to adding to the original amount. The terms times and twice as much as refer to multiplication of the original amount. For instance, if you have $1 and I have “5 times as much,” then I have $5—which is $4 more than you have, hence “4 times more than” what you have (the original plus 4 times as much).

What we have in the loan information is a mixing of additive and multiplicative terms, and it is not clear what is meant. Further investigation reveals that what was intended was “no more than 3 times your lowest payment.” Using times together with more or greater than is, to use a football analogy, “piling on.” It is often designed to impress the audience by attempted exaggeration.

The relationship of an original amount to a larger amount can be expressed either in multiplicative terms (“one-fifth as much” or “20% as much”) or in subtractive terms (“four-fifths less than” or “80% less than”)—but don’t mix the two.

Texting while driving is the number one cause of death for American teenagers. What is wrong with the following claim, and how could you state it correctly?

Texting drivers looked at the road 400 percent less than drivers who did not text message while behind the wheel of a car.

—carlsbad.patch.com/groups/michael-bombergers-blog/p/bp-the-dangers-of-texting-while-driving

If we follow the same logic of the writer of the claim about American and Dutch cycling, we would conclude that texters look at the road one-fourth as much as non-texters. But we would be wrong; tracing this phrase to its original source (in Australia in 2006), we find:

Drivers spent up to 400 percent more time with their eyes off the road when text messaging, than when not text messaging…. The amount of time that drivers spent with their eyes off the road increased by up to 400% when retrieving and sending text messages.

—www.distraction.gov/downloads/pdfs/effects-of-text-Messaging.pdf

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This phrasing would mean that texters spent 5 times as much time with their eyes off the road as non-texters; or put another way, non-texters had their eyes off the road one-fifth as often as non-texters. But alas! The details of the report reveal that non-texters spent 10% of the time with their eyes off the road and texters spent 40% (aargh!). So the phrasing by the report writers and the “interpretation” of it by the blogger are both misleading. We could correctly talk about “4 times as much” or “300% more” or “one-fourth as much” or “25% as much” or “75% less”—as long as we don’t mix up additive and multiplicative terminology.

People often say “5 times more than” when they mean “5 times as much.” All you can do is be aware of the potential confusion, try to figure out what was meant, and be careful in your own expression.

Finally, we need to distinguish percent from percentage points: If support for the president decreased from 60% to 30%, it dropped 30 percentage points but decreased 50% (because the drop of 30 percentage points is 50% of the original 60 percentage points).

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