Proportional Relationships
Proportions
True Proportions
- A proportion is an expression of the equality of two ratios or rates.
- A proportion is true if the fractions are equal when written in lowest terms.
Example
- Any units expressed in a proportion must be consistent. (When units are used, the units of the numerators are the same, and the units of the denominators are the same.)
- In any true proportion, the "cross products" are equal.
Solving Proportions
Sometimes one of the numbers in a proportion is unknown. In this case, it is necessary to solve the proportion.
To solve the proportion, find a number to replace the unknown so that the proportion is true. You can use cross-products to solve a proportion.
Example
Proportions are helpful when solving problems that require for a scale or rate to be followed.
When writing a proportion to solve a problem, be sure to use follow a consistency of units.
Example:
A bakery makes two kinds of pies: cherry and apple. The ratio of cherry pies to apple pies is 5 to 9. If there are 70 pies in all, how many are cherry?
Strategy:
Step 1: Determine the ratio of cherry pies to total pies.
Step 2: Set up a proportion of cherry pies to total pies to find the number of cherry pies at the bakery.
Solution:
Step 1:
For every 14 pies, 5 are cherry.
Step 2:
5 x 70 = 14 c Cross products property.
Divide each side by 14.
25 = c Simplify.
There are 25 cherry pies at the bakery.
Practice
1.) Use cross products to determine whether each proportion given below is a true proportion.
a.)
b.)
c.)
d.)
2.) Solve each proportion. Round to the nearest hundredth if needed.
a.)
b.)
c.)
d.)
e.)
3.) Solve the following problems using a proportion.
a.) The dosage of a certain medication is 2 ounces for every 50 pounds of body weight. How many ounces of medication are required for a person who weighs 175 pounds?
b.) Three tablespoons of fertilizer are added to every 4 gallons of water. How many tablespoons of fertilizer are required for 10 gallons of water?
c.) The scale on a map is 1.25 inches equals 10 miles. How far apart in miles are two towns that are 2 inches apart on the map?
d.) A contractor charges $9.25 per square foot for stucco work. A homeowner was given the estimate of $5,550 for stucco work on her home. What was the square footage used for the estimate?
Answer Key
1.)
a.) not true
b.) not true
c.) not true
d.) true
2.)
a.) n = 8
b.) w = 9
c.) t = 16
d.) a = 6
e.) d = 7
3.)
a.) 7
b.) 7.5
c.) 16 miles
d.) 600