Mathematics: Proportions and Ratios

Proportions
A proportion is a relationship between two quantities that dictates how one changes when the other changes. A direct proportion describes a relationship in which a quantity increases by a set amount for every increase in the other quantity, or decreases by that same amount for every decrease in the other quantity.
Example: Assuming a constant driving speed, the time required for a car trip increases as the distance of the trip increases. The distance to be traveled and the time required to travel are directly proportional.

Inverse proportion is a relationship in which an increase in one quantity is accompanied by a decrease in the other, or vice versa. Example: the time required for a car trip decreases as the speed increases, and increases as the speed decreases, so the time required is inversely proportional to the speed of the car.

Ratios
A ratio is a comparison of two quantities in a particular order.

Example: If there are 14 computers in a lab, and the class has 20 students, there is a student to computer ratio of 20 to 14, commonly written as 20:14.

Ratios are normally reduced to their smallest whole number representation, so 20:14 would be reduced to 10:7 by dividing both sides by 2.

Constant of Proportionality
When two quantities have a proportional relationship, there exists a constant of proportionality between the quantities; the product of this constant and one of the quantities is equal to the other quantity. For example, if one lemon costs $0.25, two lemons cost $0.50, and three lemons cost $0.75, there is a proportional relationship between the total cost of lemons and the number of lemons purchased.

The constant of proportionality is the unit price, namely $0.25/lemon. Notice that the total price of lemons, t, can be found by multiplying the unit price of lemons, p, and the number of lemons, n:
.

Work/Unit Rate
Unit rate expresses a quantity of one thing in terms of one unit of another. For example, if you travel 30 miles every two hours, a unit rate expresses this comparison in terms of one hour: in one hour you travel 15 miles, so your unit rate is 15 miles per hour. Other examples are how much one ounce of food costs (price per ounce) or figuring out how much one egg costs out of the dozen (price per 1 egg, instead of price per 12 eggs). The denominator of a unit rate is always 1. Unit rates are used to compare different situations to solve problems.

For example, to make sure you get the best deal when deciding which kind of soda to buy, you can find the unit rate of each. If soda #1 costs $1.50 for a 1-liter bottle, and soda #2 costs $2.75 for a 2-liter bottle, it would be a better deal to buy soda
#2, because its unit rate is only $1.375 per 1-liter, which is cheaper than soda
#1. Unit rates can also help determine the length of time a given event will take. For example, if you can paint 2 rooms in 4.5 hours, you can determine how long it will take you to paint 5 rooms by solving for the unit rate per room and then multiplying that by 5.

Slope
On a graph with two points,
slope is found with the formula m=(y2-y1)/(x2-x1) , where

and m stands for slope.

If the value of the slope is positive, the line has an upward direction from left to right. If the value of the slope is negative, the line has a downward direction from left to right.

Consider the following example:
A new book goes on sale in bookstores and online stores. In the first month, 5,000 copies of the book are sold. Over time, the book continues to grow in popularity. The data for the number of copies sold is in the table below.

# of Months on Sale: 1 - 2 - 3 - 4 - 5
# of Copies Sold (In Thousands): 5 - 10 - 15 - 20 - 25

So, the number of copies that are sold and the time that the book is on sale is a proportional relationship.
In this example, an equation can be used to show the data:
, where x is the number of months that the book is on sale. Also, y is the number of copies sold. So, the slope of the corresponding line is
.

Finding an Unknown in Equivalent Expressions
It is often necessary to apply information given about a rate or proportion to a new scenario. For example, if you know that Jedha can run a marathon (26 miles) in 3 hours, how long would it take her to run 10 miles at the same pace? Start by setting up equivalent expressions:

Now, cross multiply and solve for
: 

So, at this pace, Jedha could run 10 miles in about 1.15 hours or about 1 hour and 9 minutes.