Basic Terms and Terminology Relating to Solving Problems Involving Ratios and Rates of Change

  • Ratio: The relationship of two numbers
  • Proportions: Two ratios that are equal to each other
  • Rate of change: The relationship between two numbers or quantities and how they change in relationship to each other

As more fully discussed and described in the section entitled "Ratios and Their Meaning", a ratio is the relationship of two numbers and proportions are two ratios that are equal to each other.

Rates of Change

Simply defined, a rate of change is the relationship between two numbers or quantities and how they change in relationship to each other. Similar to ratios, as discussed above, rates of change are expressed as ratios and fractions, but with some measure of change in addition to the numbers that are used in a ratio.

Some of the most commonly used rates of change relate to time, distance, speed, sales tax, and health care related rates of change such as respiratory rate, pulse rate, mortality rate and morbidity rates, for example.

For example, look at the table below ratios and their comparisons to some rate of change:

Ratio Rate of Change
15 : 1 OR 15 / 1 OR 15 is to 1 $15 : 1 hour OR $1 5/ 1 OR $15 is to 1 hour
12 : 6 OR 12 / 6 OR 12 is to 6 12 minutes : 6 miles OR 12 minutes / 6 miles OR 12 minutes is to 6 miles
3 : 6 OR 3 / 6 OR 3 is to 6 3 tsp : 6 cups OR 3 tsp / 6 cups OR 3 tsp is to 6 cups
16 : 4 OR 16 / 4 OR 16 is to 4 16 calculations : 4 minutes OR 16 calculations / 4 minutes OR 16 calculations is to 4 minutes
60 : 1 OR 60 / 1 OR 60 is to 1 60 words : 1 minute OR 60 words / 1 minute OR 60 words is to 1 minute

In everyday terms and everyday language, rates of change have meaning. For example, the rates of change for some of the examples above can be worded in this manner.

Example 1

Word Problem:

You are at an amusement theme park with your 10 year old child. You bought a ticket for 1/2 day, but your child wants to stay another 4 hours after the 1/2 day is used. The admissions office can accommodate your child's additional time but the rate is $ 15 an hour. How much will it cost to purchase the additional 4 hours at this rate?

Solution to the Word Problem:

$15 / 1 Hour = $x / 4 Hours

x= 4 x 15 = $60

Answer: It will cost you an additional $60 for 4 more hours of fun for your child.

Example 2

Word Problem:

You are following a recipe for a cake for your church group's get together. This recipe calls for 3 tsp of baking powder to every 6 cups of flour. How many tsp would you add to 24 cups of flour?

Solution to the Word Problem:

3 tsp : 6 cups OR 3 tsp / 6 cups OR 3 tsp is to 6 cups

3 tsp / 6 cups = x tsps / 24 cups

6x = 24 x 3 = 76

6x = 24 x 3 = 72 / 6 = 12 4/16 = 12 tsps

Answer: You will need 12 tsps. of baking powder for 24 cups of flour which is enough to make 4 cakes for the church group's get together.

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