1) The ratio of 
to
is the quotient of
divided by and
can be written either:
or 
The value of the quotient tells you the size of the number when
compared to .
For example, the value of 2/4 tells you that 2 is half the size of 4.
2) Two variables are said to be proportional when the
value of one variable is equal to a constant times the other variable. For
example, suppose the price of a watermelon can be figured using the
formula:
Where
is
the price of the watermelon and w is the weight of the watermelon in
pounds. We would say thatis
proportional to
sinceis
equal to the constant $0.50 times.
In this case the constant $0.50 is the price of the watermelon per pound.
3) A proportion is two ratios that are equal to each
other. An example would be:
Proportions are related to proportional variables. Consider the
watermelon example above. A $3.00 watermelon is to 6 pounds as a $4.00
watermelon is to 8 pounds. This relationship can be expressed with the
proportion above. Proportions are useful for solving problems. Suppose you
know the price of a watermelon is proportional to its weight and you are
given that a 6 pound watermelon cost $3.00. You can now figure out the
price of a 10 pound watermelon by setting up a proportion similar to the
one above and solving for the unknown variable.
Let x = the price of a 10 pound watermelon.
4) A rate is a ratio comparing two different quantities.
If you travel 200 miles in 4 hours then your average speed is the
following rate:
Another example would be a vehicle that travels 288 miles on 12 gallons
of gas. The gas mileage of the vehicle is a rate that compares miles to
gallons.
5) The basic formula for solving rate problems is:
For example, a car traveling at 60 mph for 2.5 hours will travel 
When given any two values in the formula it is always possible to solve
for the third value. For example, you can find out how long it would take
a car traveling at 50 mph to travel 187.5 miles. Let rate = 50, distance =
187.5, and time = t. Then
This basic formula can be generalized so that it can be used to solve
many different problems involving rates. For example, suppose an old pick–up
can travel 234 miles on 13 gallons of gas. How many miles per gallon does
the pick-up get? In this case we are looking for the rate, and instead of
comparing miles to hours the rate is comparing miles to gallons.
Generalizing the formula above, we get:
To solve the problem, substitute in the given values for gallons and
distance, and then solve for the rate.
6) A percent can be thought of as a rate. To take 75% of
a number, for example, involves the rate:
Here .75 is the decimal form of 75%. To convert a percent
into its decimal form, drop the percent sign and move the decimal point to
the left two places. Reverse this procedure to convert back to a percent.
For example, the decimal form of 5.6% is 0.056, and 0.455 is the decimal
form 45.5%.
To find the percent of a number use:
The percent in the formula is the decimal form of the percent. There
are three types of percent problems that can be solved with this formula.
For example,
i) What is 20% of 35?
ii) 60% of what number is 15?
iii) What percent of 80 is 60?
