Learning Outcomes
- Solve proportional equations
Previously when solving equations involving fractions, we cleared fractions by multiplying both sides of the equation by the lowest common denominator. For example, to solve the equation
[latex]\frac{x}{2} + \frac{x}{3} = 5[/latex]
we multiply both sides of the equation by the lowest common denominator, which is the least common multiple of the denominators in our equation. Since the least common multiple of 2 and 3 is 6, we’ll multiply both sides of the equation by 6.
[latex]6(\frac{x}{2})=6 \cdot 5[/latex]
[latex]\frac{6x}{2} + \frac{6x}{3} = 30[/latex]
[latex]3x+2x=30[/latex]
[latex]5x=30[/latex]
[latex]\frac{5x}{5} = \frac{30}{5}[/latex]
[latex]x=6[/latex]
This approach can be applied to equations in which a variable appears in a denominator.
Example
Solve [latex]\frac{12}{n}=3[/latex].
Proportion
A proportion is an equation of the form [latex]{\Large\frac{a}{b}}={\Large\frac{c}{d}}[/latex], where [latex]b\ne 0,d\ne 0[/latex].
The proportion states two ratios or rates are equal. The proportion is read “[latex]a[/latex] is to [latex]b[/latex], as [latex]c[/latex] is to [latex]d[/latex].”
In the next example, we will solve a proportion by multiplying by the Least Common Denominator (LCD) using the Multiplication Property of Equality.
Example
Solve [latex]\frac{x}{6} = \frac{2}{3}[/latex].
try it
In the next video we show another example of how to solve a proportion equation using the LCD.
Another approach to solving a proportion involves finding the cross products of the proportion. To find the cross products, we multiply each denominator with the opposite numerator (diagonally across the equal sign).
For any proportion of the form [latex]{\Large\frac{a}{b}}={\Large\frac{c}{d}}[/latex], where [latex]b\ne 0,d\ne 0[/latex], its cross products are equal.
Example
Solve [latex]\frac{x}{15}=\frac{2}{5}[/latex].
Let’s look at an example of a proportion involving a variable in the denominator.
Example
Solve [latex]\frac{48}{n}=\frac{4}{3}[/latex].
