- Write a proportion to express a rate or ratio
- Use units to determine which conversion factors are needed for dimensional analysis.
- Use a conversion factor to convert a rate
Proportions and Rates
Proportions are a powerful tool in expressing and solving problems involving rates and ratios. By setting two rates or ratios equal to each other, we can find unknown values and understand relationships between different quantities.
ratios and rates
- A ratio is a comparison between two quantities or measures. The ratio can be expressed in three ways: [latex]a[/latex] to [latex]b[/latex], [latex]a:b[/latex], or [latex]\frac{a}{b}[/latex].
- A rate is a specific kind of ratio in which two measurements with different units are related to each other. For example, if a car travels [latex]180[/latex] miles in [latex]3[/latex] hours, the rate of the car is [latex]60[/latex] miles per hour.
- A unit rate is a rate with a denominator of one.
Recall Reducing Fractions
The Equivalent Fractions Property states that:
If [latex]a,b,c[/latex] are numbers where [latex]b\ne 0,c\ne 0[/latex], then:
[latex]{\dfrac{a\cdot c}{b\cdot c}}={\dfrac{a}{b}}[/latex]
proportions
A proportion is an equation showing the equivalence of two rates or ratios.
Proportions have two key properties:
- The Cross-Product Property: The cross products of a proportion are always equal. That is, the product of the means equals the product of the extremes.
- The Reciprocal Property: If [latex]\frac{a}{b} = \frac{c}{d}[/latex], then [latex]\frac{b}{a} = \frac{d}{c}[/latex]. This property allows us to write the reciprocal of a proportion.
To express a rate or ratio as a proportion, we write two ratios or rates as fractions that are equal to each other. Here’s how to do it:
Let’s take the example of the car mentioned above traveling at a rate of [latex]60[/latex] miles per hour. If we want to know how far the car would travel in [latex]5[/latex] hours at the same rate, we would set up a proportion like this:
[latex]\frac{180 \text{ miles}}{3 \text{ hours}} = \frac{x \text{ miles}}{5 \text{ hours}}[/latex]
The proportion reads, “[latex]180[/latex] miles is to [latex]3[/latex] hours as [latex]x[/latex] miles is to [latex]5[/latex] hours.” We can solve for [latex]x[/latex] by cross-multiplying and find that [latex]x[/latex] equals [latex]300[/latex] miles. So, the car would travel [latex]300[/latex] miles in [latex]5[/latex] hours at the same rate.
Using Variables to represent unknowns
Recall that we can use letters we call variables to “stand in” for unknown quantities. Then we can use the properties of equality to isolate the variable on one side of the equation. Once we have accomplished that, we say that we have “solved the equation for the variable.”
You can view the example below to see how a proportion, which is an equality established between two fractions, is resolved to find the unknown value denoted by [latex]x[/latex].