Understanding Algebra

James W. Brennan
 

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Percents

Percent means “per hundred”, so

or x hundredths.

  • A percent is just a fraction

However, it is a fraction with a denominator of 100, not just any fraction. When we write the percent, we are just writing the numerator of the fraction. The denominator of 100 is expressed by the percent symbol “ %.” Remembering that the percent symbol means “over one-hundred” can prevent a lot of confusion.

“ % ” means “ /100 ”

To convert a percent to a decimal

  • Divide the percentage by 100 (or move the decimal point two places to the left).

Since , the decimal equivalent is just the percentage divided by 100. But dividing by 100 just causes the decimal point to shift two places to the left:

To convert a decimal to a percent

  • Multiply the decimal number by 100 (or move the decimal point two places to the right).

Since x% = x/100, it is also true that 100 x% = x. Another way to look at is to consider that in order to convert a number into a percent, you have to express it in hundredths. Recall that the hundredths place is the second place to the right of the decimal, so this is the digit that gives the units digit of the percent. Of course, all this means is that you move the decimal point two places to the right.

WARNING: If you just remember these rules as “move the decimal two places to the left” and “move the decimal two places to the right,” you are very likely to get them confused. If you accidentally move the decimal in the wrong direction it will end up four places off from where it should be, which means that your answer will be either ten-thousand times too big or ten-thousand times too small. This is generally not an acceptable range of error. It is much better to remember these rules by simply remembering the meaning of the percent sign, namely that “ % “ means “ /100.” If you just write the problem that way, you should be able to see what you need to do in order to solve it.

Converting between percents and their decimal equivalents is so simple that it is usually best to express all percents in decimal form when you are working percent problems. The decimal numbers are what you will need to put in your calculator, and you can always express the result as a percent if you need to.

  Calculator  note: Some calculators have a percent key that essentially just divides by 100, but it can do other useful things that might save you a few keystrokes. For instance, if you need to add 5% to a number (perhaps to include the sales tax on a purchase), on most calculators you can enter the original number and then press “ + 5 % = “. Just make sure you understand what it does before you blindly trust it. What it is doing in this example is multiplying the original number by 0.05 and then adding the result onto the original number. You should be able to work any percent problem without using this key, but once you understand what is going on it can be a convenient short-cut.

To convert a percent to a fraction

·        Put the percentage over a denominator of 100 and reduce

Writing a percent as a fraction is very simple if you remember that the percent is the numerator of a fraction with a denominator equal to 100.

Examples: 

In this last example, the first fraction has a decimal in it, which is not a proper way to represent a fraction. To clear the decimal, just multiply both the numerator and the denominator by 10 to produce an equivalent fraction written with whole numbers.

To convert a fraction to a percent

·         Divide the numerator by the denominator and multiply by 100

To write a fraction as a percent you need to convert the fraction into hundredths. Sometimes this is easy to do without a calculator. For example, if you saw the fraction

,

you should notice that doubling the numerator and the denominator will produce an equivalent fraction that has a denominator of 100. Then the numerator will be the percent that you are seeking:

With other fractions, though, it is not always so easy. It is not at all obvious how to convert a fraction like 5/7 into something over 100. In this case, the best thing to do is to convert the fraction into its decimal form, and then convert the decimal into a percent. To convert the fraction to a decimal, remember that the fraction bar indicates division:

The “approximately equal to” sign () is used because the decimal parts have been rounded off. Because it is understood that approximate numbers are rounded, we will not continue to use the approximately equal sign. It is more conventional to just use the standard equal sign with approximate numbers, even though it is not entirely accurate.

Working percent problems

In percent problems, just as in fraction problems, the word “of” implies multiplication:

x percent of a number” means “x% times a number”

Example:  What is 12% of 345?

12% is 12/100, which we can express in decimal form as 0.12. 12% of 345 means 12% times 345, or

Notice how it is easier to just move the decimal over two places instead of explicitly dividing by 100.

We solve a problem like this by translating the question into mathematical symbols, using x to stand for the unknown “what” and that the “of” means “times”:

Example:  What percent of 2342 is 319?

Once again we translate this into mathematical symbols:

Solving this equation involves a little bit of algebra. To isolate the x% on one side of the equation we must divide both sides by 2342:

The calculator tells us that

x% = 0.1362

Now the right-hand side of this equation is the decimal equivalent that is equal to x%, which means that = 13.62, or

319 is 13.62% of 2342

If that last step confused you, remember that the percent symbol means “over 100”, so the equation

x% = 0.1362

really says

or

x = 100(0.1362)
x = 13.62

Example:  2.4 is what percent of 19.7?

Translating into math symbols:

Solving for x:

2.4 = x% (19.7)

x% = 0.1218

x = 12.18

So we can say that 2.4 is 12% of 19.7 (rounding to 2 significant figures)

Example:  46 is 3.2% of what?

Translating into math symbols:

Solving for x:

46 = 3.2% (x)

46 = 0.032x

x = 1437.5

Therefore, we can say that 46 is 3.2% of 1400 (rounding to 2 significant figures). Notice that in the second step the percentage (3.2%) is converted into its decimal form (0.032).

 
      

 


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James W. Brennan