Hello. I'm Professor Von Schmohawk and welcome to Why U.
In the last few lectures we studied several forms of linear equations
and saw how they can be applied to various types of real-world situations.
In this chapter we will lay the groundwork for solving real-world problems
which involve percentages.
The word "percent"
is derived from the Latin word "per centum"
meaning "per hundred".
Percentages are one method of describing a fraction of a quantity.
This fraction is always given in hundredths
and the percentage specifies the number of hundredths in the fraction.
In other words, a percentage is the numerator of a fraction
whose denominator is understood to be 100.
Instead of writing the word "percent"
percent is usually denoted by the percent symbol.
To understand percentages visually let's start with a quantity, x
which can be divided into 100 equal divisions.
We can visualize one percent of x
as one hundredth of x or one division.
Likewise, 25 percent of x
is 25 hundredths or one-quarter of x.
50 percent of x
is 50 hundredths or one-half of x.
And 100 percent of x
is 100 hundredths or the entire quantity, x.
Percentages can also have values greater than 100
and thus represent values greater than the original quantity x.
For example, 150 percent of x
is 150 hundredths or three-halves x.
So 150 percent represents a 50 percent increase over the original quantity.
Likewise, 200 percent of x
is 200 hundredths or twice x
which represents a 100 percent increase over the original quantity.
In these examples we converted percentages
into equivalent fractions with denominators of 100
and reduced those fractions to their simplest form.
Percentages can also be converted to equivalent decimal numbers.
Instead of converting a percentage to a fraction with a denominator of 100
we can convert the percentage to a decimal number which is equivalent to that fraction
by dividing the fraction's numerator by 100.
In this example 15 divided by 100 is 0.15.
Of course we could produce the same result of dividing by 100
by simply dropping the percent sign
and shifting the decimal point two digits to the left.
Decimal equivalents of percentages are sometimes easier to work with.
For instance, instead of writing "15 percent of x"
we can simply write "0.15x".
Let's say for example that out of a class of 40 students
15 percent received an A on a test.
We can find the number of students who received an A
by calculating 15 percent of 40.
To do this, we multiply the decimal equivalent of 15 percent, 0.15
times the size of the class, 40.
The result is six.
So six students received an A.
The decimal equivalent of a percentage can be thought of
as a ratio of the percentage amount to the entire amount
in other words the amount after the percentage is taken
to the amount before the percentage is taken.
In this example, this is the ratio of the six students who made an A
to the class of 40 students
or 6 over 40.
6 divided by 40 is 0.15
which is the decimal equivalent to 15 percent.
Determining the ratio of quantities is an easy way to calculate percentage.
For example, to find what percentage six apples is out of a total of 75 apples
we compute the ratio "6 to 75".
6 divided by 75 is 0.08
which is the decimal equivalent to eight percent.
In this chapter, we have introduced the concept of percentages
and explored the principles on which percentages are based.
In the next chapter we will use Algebra to solve real-world problems which involve percentages.