Learning Outcomes
- Write a proportion to express a rate or ratio
- Solve a proportion for an unknown
If you wanted to power the city of Lincoln, Nebraska using wind power, how many wind turbines would you need to install? Questions like these can be answered using rates and proportions.
RATES
A rate is the ratio (fraction) of two quantities.
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].
Ex. [latex]\dfrac{500}{20}=\dfrac{25\cdot 20}{1\cdot 20}=\dfrac{25}{1}=25[/latex]
Example
Your car can drive 300 miles on a tank of 15 gallons. Express this as a rate.
Proportion Equation
A proportion equation is an equation showing the equivalence of two rates or ratios.
An example of a proportion would be: [latex]\dfrac{7}{3}=\dfrac{35}{15}[/latex]
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.”
In the example below, you are asked to solve the proportion (an equality given between two fractions) for the unknown value [latex]x[/latex].
Ex. Solve the proportion [latex]\dfrac{7}{3}=\dfrac{x}{15}[/latex]
We see that the variable we wish to isolate is being divided by 15. We can reverse that by multiplying on both sides by 15.
[latex]\dfrac{7}{3}=\dfrac{x}{15}[/latex]
[latex]15\cdot \dfrac{7}{3}=x[/latex], giving [latex]x=35[/latex].
Example
Solve the proportion [latex]\displaystyle\frac{5}{3}=\frac{x}{6}[/latex] for the unknown value x.
