Applied Math: 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

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 10:7.

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,  and, the slope is found with the formula  ; 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.
 

 
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.

Practice
P1. Solve the following for .
(a)
(b)
(c)
P2. At a school, for every 20 female students there are 15 male students. This same student ratio happens to exist at another school. If there are 100 female students at the second school, how many male students are there?
P3. In a hospital emergency room, there are 4 nurses for every 12 patients. What is the ratio of nurses to patients? If the nurse-to-patient ratio remains constant, how many nurses must be present to care for 24 patients?
P4. In a bank, the banker-to-customer ratio is 1:2.
If seven bankers are on duty, how many customers are currently in the bank?
P5. Janice made $40 during the first 5 hours she spent babysitting. She will continue to earn money at this rate until she finishes babysitting in 3 more hours. Find how much money Janice earns per hour and the total she earned babysitting.
P6. The McDonalds are taking a family road trip, driving 300 miles to their cabin. It took them 2 hours to drive the first 120 miles. They will drive at the same speed all the way to their cabin. Find the speed at which the McDonalds are driving and how much longer it will take them to get to their cabin.
P7. It takes Andy 10 minutes to read 6 pages of his book. He has already read 150 pages in his book that is 210 pages long. Find how long it takes Andy to read 1 page and also find how long it will take him to finish his book if he continues to read at the same speed.

Practice Solutions
P1. First, cross multiply then solve for x:
(a)
(b)
(c)
 

P2. One way to find the number of male students is to set up and solve a proportion.

Represent the unknown number of male students as the variable x:

Cross multiply and then solve for x:

 
P3. The ratio of nurses to patients can be written as 4 to 12, 4:12, or . Because four and twelve have a common factor of four, the ratio should be reduced to 1:3, which means that there is one nurse present for every three patients. If this ratio remains constant, there must be eight nurses present to care for 24 patients.
 

P4. Use proportional reasoning or set up a proportion to solve. Because there are twice as many customers as bankers, there must be fourteen customers when seven bankers are on duty.

Setting up and solving a proportion gives the same result:

Represent the unknown number of patients as the variable x: .
To solve for x, cross multiply: , so .
 

P5. Janice earns $8 per hour. This can be found by taking her initial amount earned, $40, and dividing it by the number of hours worked, 5. Since , Janice makes $8 in one hour. This can also be found by finding the unit rate, money earned per hour: . Since cross multiplying yields , and division by 5 shows that , Janice earns $8 per hour.
Janice will earn $64 babysitting in her 8 total hours (adding the first 5 hours to the remaining 3 gives the 8-hour total).

Since Janice earns $8 per hour and she worked 8 hours, . This can also be found by setting up a proportion comparing money earned to babysitting hours.
Since she earns $40 for 5 hours and since the rate is constant, she will earn a proportional amount in 8 hours: .

Cross multiplying will yield , and division by 5 shows that .
 

P6. The McDonalds are driving 60 miles per hour. This can be found by setting up a proportion to find the unit rate, the number of miles they drive per one hour: . Cross multiplying yields  and division by 2 shows that .

Since the McDonalds will drive this same speed, it will take them another 3 hours to get to their cabin. This can be found by first finding how many miles the McDonalds have left to drive, which is .

The McDonalds are driving at 60 miles per hour, so a proportion can be set up to determine how many hours it will take them to drive 180 miles: . Cross multiplying yields  and division by 60 shows that . This can also be found by using the formula  (or ), where , and division by 60 shows that .
P7. It takes Andy 10 minutes to read 6 pages,  minutes, which is 1 minute and 40 seconds.
Next, determine how many pages Andy has left to read, . Since it is now known that it takes him  minutes to read each page, then that rate must be multiplied by however many pages he has left to read (60) to find the time he’ll need: , so it will take him 100 minutes, or 1 hour and 40 minutes, to read the rest of his book.