Note that, while the values below do not refer to money, the procedures used to solve these problems are otherwise identical to the markup - markdown examples on the previous page. While markup and markdown almost always refer only to increases and decreases in terms of prices and costs, the same general setup and solution process will apply to anything that is seeing increases or decreases. It's only the terminology that has changed; the method remains the same.

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- Growing up, you lived in a tiny country village. When you left for college, the population was 840. You recently heard that the population has grown by 5%. What is the present population?

First, I'll find the actual amount of the increase. Since the increase is five percent of the original, then the increase is:

(0.05)(840) = 42

The new population is the old population plus the increase, or:

840 + 42 = 882

The population is now 882.

- Your friend diets and goes from 125 pounds to 110 pounds. What was her percentage weight loss?

First, I'll find the absolute weight loss:

125 – 110 = 15

This fifteen-pound decrease is some percentage of the original, since the rate of change is always with respect to the original value. So the percentage is "change over original", or:

15 = (*x*)(125)

15 ÷ 125 = *x*

(See? The change, 15, is over the original, 125.)

15 ÷ 125 = 0.12

The change is a percentage, so I need to convert this decimal to percentage form:

She lowered her weight by 12%.

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- Your boss says that his wife has put an 18 × 51 foot garden in along the whole back end of their back yard. He says that this has reduced the back-yard lawn area by 24%. What are the total dimensions of his back yard? What are the dimensions of the remaining lawn area?

Since no suburban lot is going to be only eighteen feet wide (because then the house couldn't fit along the street frontage), the width of the lot must be the 51-foot dimension. Now I need to figure out the length of the back yard. The area of the garden is:

(18)(51) = 918

This represents 24% of the total yard area; that is, 24% of the original lawn area. This says that (918 square feet) is (twenty-four percent) of (the original), so:

918 = 0.24*x*

918 ÷ 0.24 = *x* = 3825

The total back yard area is 3825 square feet. Since the width is 51 feet, then:

3825 ÷ 51 = 75

The length then is 75 feet. Since 18 feet are taken up by the garden, then the lawn area is 75 − 18 = 57feet deep.

The back yard measures 51' × 75' and the lawn measures 51' × 57'.

(What follows is an aside.)

You may have wondered about the repeated contrast in these pages between "relative" numbers and "absolute" numbers. You should be aware of the difference between the two, because they can be used by those with an ax to grind to take *exactly the same* numbers and come to *exactly the opposite* conclusions.

For instance, consider the politics of welfare in the United States. I would posit that any family on welfare is a tragedy, and that the legal structure of the economy should be changed to allow more opportunities for advancement. But others with a vested interest in the status quo just squabble over how the present system, and thus their power base, should be expanded. One of their techniques is to argue for more benefits for "their" group, while claiming that "their" group shouldn't be blamed for the problem. How can they do this, when they're using the same statistics each time? They do it by picking and choosing when to use relative numbers versus absolute numbers:

In the United States, those classified as "black" comprise about 12% of the population, while those classified as "white" comprise about 75% of the population. Since "whites" outnumber "blacks" by more than six to one, it is only reasonable that there would be more "whites" on welfare than "blacks" — *in absolute numbers*. That is, when you count them up, there are more "whites" receiving welfare than "blacks", as one would reasonably expect. When certain "leaders" claim that "blacks" aren't the majority of welfare recipients, they are using the absolute numbers.

However, of all welfare recipients, only 50% are "white" (you would expect 75% to be "white") and 35% are "black" (you would expect only 12% to be). In other words, "blacks" are three times more likely to be on welfare than they *ought* to be, if you just look at their percentage of the population. When certain "leaders" claim that cutbacks to welfare programs are somehow "aimed" at "blacks", because "blacks" will be the ones allegedly most harmed, they are using these relative numbers.

Politicians and other charlatans often operate this way: riding both sides of the fence, and changing their interpretations of the numbers to suit the latest money-making opportunities. You should be aware of this two-facedness, and should listen to their arguments while carrying this fact in the back of your head. It'll make you a lot harder to fool. Each interpretation above is correct — *as far as it goes* — but you should always wonder what the actual numbers are, what the speaker's motivations are, and whether you have enough information to come to your own independent conclusions.

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