Relationships Between Quantities
Mathematics often becomes more useful when we stop looking at quantities separately and begin asking how they are related. A class may contain 12 students wearing blue shirts and 8 wearing red shirts. A car may travel 180 miles using 6 gallons of fuel. A recipe may use 3 cups of flour for every 2 cups of milk.
In each situation, the individual numbers matter, but so does the relationship between them. Ratios and rates give us ways to describe those relationships.
These ideas are the foundation of proportion. Before we can understand when two ratios are equivalent, solve proportions, or study direct and inverse relationships, we need to understand clearly what a ratio is and what information it represents.
What Is a Ratio?
A ratio compares two quantities by division. It tells us how much of one quantity there is relative to another.
Suppose a basket contains 6 apples and 4 oranges. The ratio of apples to oranges can be written in several equivalent forms:
\[ 6:4 \] \[ \frac{6}{4} \]or in words:
6 to 4.
All three forms describe the same comparison.
Because a ratio represents division, the ratio \(\frac{6}{4}\) can also be simplified:
\[ \frac{6}{4}=\frac{3}{2}. \]This tells us that the quantities are related in the ratio \(3:2\). For every 3 apples, there are 2 oranges.
Order Matters
Ratios must be written in the same order as the quantities being compared. Reversing the order changes the meaning.
If there are 6 apples and 4 oranges, then the ratio of apples to oranges is
\[ \frac{6}{4}=\frac{3}{2}. \]But the ratio of oranges to apples is
\[ \frac{4}{6}=\frac{2}{3}. \]These ratios contain the same two quantities, but they answer different questions.
A useful habit is to write the names of the quantities before inserting their numbers:
\[ \frac{\text{apples}}{\text{oranges}} = \frac{6}{4}. \]This becomes especially helpful later when proportions contain several different measurements or units.
Part-to-Part and Part-to-Whole Ratios
Ratios can compare two parts of the same collection, or they can compare one part with the entire collection.
Return to the basket containing 6 apples and 4 oranges. There are 10 pieces of fruit altogether.
The ratio of apples to oranges is a part-to-part ratio:
\[ \frac{6}{4}=\frac{3}{2}. \]The ratio of apples to all fruit is a part-to-whole ratio:
\[ \frac{6}{10}=\frac{3}{5}. \]The ratio of oranges to all fruit is
\[ \frac{4}{10}=\frac{2}{5}. \]These comparisons describe different relationships, so identifying exactly what is being compared is an important part of proportional reasoning.
Ratios with the Same Units
Some ratios compare quantities measured in the same units.
Suppose one board is 8 feet long and another is 12 feet long. The ratio of the first length to the second is
\[ \frac{8\text{ ft}}{12\text{ ft}} = \frac{2}{3}. \]Because both quantities are measured in feet, the units cancel. The final ratio is simply
\[ \frac{2}{3}. \]This means the first board is two-thirds as long as the second.
When comparing quantities of the same kind, make sure they are expressed in compatible units before forming the ratio. For example, comparing 2 feet with 18 inches requires converting one measurement first.
\[ 2\text{ ft}=24\text{ in}. \]Therefore,
\[ \frac{24\text{ in}}{18\text{ in}} = \frac{4}{3}. \]What Is a Rate?
A rate is a ratio that compares quantities measured in different units.
If a car travels 180 miles in 3 hours, its distance-to-time rate is
\[ \frac{180\text{ miles}}{3\text{ hours}}. \]Unlike the earlier board example, the units do not cancel because miles and hours measure different things.
Rates commonly appear throughout everyday life:
- miles per hour;
- dollars per pound;
- words per minute;
- miles per gallon;
- dollars per hour;
- people per square mile; and
- items produced per day.
The word per is especially useful. It usually signals division and can often be read as “for each.”
Ratio versus Rate
Every rate is a ratio, but not every ratio is usually described as a rate.
A ratio can compare quantities of the same kind:
\[ \frac{8\text{ students}}{12\text{ students}}. \]A rate typically compares quantities with different units:
\[ \frac{180\text{ miles}}{3\text{ hours}}. \]The distinction is useful because rates carry information about how one type of quantity changes relative to another type.
For example, a speed does not merely compare two numbers. It connects distance with time. A price per pound connects money with weight. A wage connects money with time worked.
Unit Rates
A unit rate is a rate whose second quantity is one unit. Unit rates make comparisons especially easy because everything is placed on the same basis.
Suppose 3 notebooks cost $9. The price rate is
\[ \frac{\$9}{3\text{ notebooks}}. \]Dividing both quantities by 3 gives
\[ \frac{\$3}{1\text{ notebook}}. \]The unit rate is therefore $3 per notebook.
Likewise, if a car travels 240 miles in 4 hours,
\[ \frac{240\text{ miles}}{4\text{ hours}} = \frac{60\text{ miles}}{1\text{ hour}}. \]The unit rate is 60 miles per hour.
Unit rates are one of the most practical tools in proportional reasoning. They allow us to compare different package sizes, speeds, wages, efficiencies, and many other quantities using a common reference of one.
Comparing with Unit Rates
Suppose one store sells 8 bottles of water for $6.40, while another sells 12 bottles for $9.00.
The first store charges
\[ \frac{\$6.40}{8} = \$0.80 \]per bottle.
The second store charges
\[ \frac{\$9.00}{12} = \$0.75 \]per bottle.
Although the second package has the higher total price, it has the lower unit price. Reducing both offers to the same unit makes the comparison straightforward.
This same strategy works far beyond shopping. Two vehicles can be compared by miles per gallon, two workers by items produced per hour, and two internet plans by cost per unit of service.
From Ratios to Proportions
A ratio describes one relationship. A proportion compares two ratios and states that they represent the same relationship.
For example,
\[ \frac{2}{3} = \frac{4}{6}. \]The two ratios use different numbers, but they describe the same relative comparison.
This distinction is fundamental:
- A ratio compares two quantities.
- A rate is a ratio comparing quantities with different units.
- A unit rate expresses a rate per one unit.
- A proportion states that two ratios are equal.
The next lesson examines that final idea more closely: how we determine whether ratios are equivalent and why equal ratios have important algebraic properties.
Let the Units Guide You
Units are not decorations attached to numbers. They are part of the mathematical meaning of a quantity.
Consider
\[ \frac{150\text{ miles}}{3\text{ hours}}. \]The numerator measures distance and the denominator measures time. The resulting rate is therefore measured in miles per hour:
\[ 50\frac{\text{ miles}}{\text{ hour}}. \]Keeping track of units helps prevent many common mistakes. Later, when setting up proportions, matching the same kinds of quantities in the same positions will become an important strategy for building correct equations.
Check Your Understanding
Before moving on, see whether you can identify the mathematical idea in each situation.
- A class has 15 boys and 10 girls. What is the ratio of boys to girls?
- A worker earns $120 for 8 hours of work. What is the hourly unit rate?
- A vehicle travels 210 miles on 7 gallons of fuel. What rate compares distance with fuel?
- Are \(\frac{3}{5}\) and \(\frac{9}{15}\) ratios, rates, or both?
- What additional statement would be needed to turn two ratios into a proportion?
Show Answers
- \[ \frac{15}{10} = \frac{3}{2}, \] so the ratio of boys to girls is \(3:2\).
- \[ \frac{\$120}{8\text{ hours}} = \$15\text{ per hour}. \]
- \[ \frac{210\text{ miles}}{7\text{ gallons}} = 30\text{ miles per gallon}. \]
- They are ratios. Because they contain no different measurement units, they would not ordinarily be described as rates.
- We must state that the two ratios are equal: \[ \frac{3}{5}=\frac{9}{15}. \] That equality forms a proportion.
Key Ideas
- A ratio compares two quantities through division.
- The order of quantities in a ratio matters.
- Ratios may compare parts with parts or parts with a whole.
- A rate compares quantities measured in different units.
- A unit rate expresses a comparison for one unit.
- Units help explain what a rate means and how it should be interpreted.
- A proportion is an equation stating that two ratios are equal.