Unit 12.2 Ratios and Proportions
Ratios
A ratio is a way to compare two quantities by showing the relative size of one quantity to another. Ratios are often written in the form a:b, where a and b are the quantities being compared. Ratios can also be expressed as fractions or as a division, [latex]\frac{a}{b}[/latex]
Ratios
If there are 4 apples and 6 oranges in a basket, what is the ratio of apples to oranges?
Apples : Oranges
4 : 6
Simplify
2 : 3
Proportions
A proportion is an equation that states that two ratios are equal. Proportions can be used to solve problems involving equivalent ratios.
[latex]\frac{a}{b} = \frac{c}{d}[/latex]
Common proportion examples:
Wheatstone Bridge: [latex]\frac{R_X}{R_S}\ =\ \frac{R_1}{R_2}[/latex]
Transformers: [latex]\frac{V_P}{V_S}\ =\ \frac{N_P}{N_S}[/latex]
Charles’s Law: [latex]\frac{V_1}{T_1}\ =\ \frac{V_2}{T_2}[/latex]
Proportions: Example 1
If a recipe requires 2 cups of sugar for 3 cups of flour, but you only have 1 cup of sugar, how much flour should you use?
Set up the proportion. If the original recipe calls for 2 cups of sugar for every 3 cups of flour, that is the fraction on the left. Since we only have 1 cup of sugar, and sugar is in the numerator of the other fraction, it goes into the numerator of this fraction.
We are wondering how much flour to use for our 1 cup sugar, so it will be represented by x
[latex]\frac{2}{3} = \frac{1}{x}[/latex]
Cross Multiply
2 ⋅ x = 3 ⋅ 1
2x = 3
Solve for x
x = [latex]\frac{3}{2}[/latex]
x = 1.5 cups