Proportional Reasoning

Proportional reasoning is the ability to compare ratios and understand the relationship between quantities. It involves recognizing that two ratios are equivalent and can be used to solve problems involving scaling, rates, and percentages.

To calculate a percentage of a total, multiply the total by the percentage (in decimal form). For example, to find 75% of 60, calculate 60 * 0.75 = 45.

To find the actual distance for a different scale, set up a proportion. For example, if 2 inches = 70 miles, then 4.8 inches = x miles can be solved using the equation: $$\frac{2}{70} = \frac{4.8}{x}$$ .

The formula to calculate sales tax is: Sales Tax = Price × Tax Rate. For example, if a bike costs $738 and the tax rate is 5.5%, the sales tax is $$738 \times 0.055 = 40.59$$ .

Two ratios are equivalent if they represent the same relationship between quantities. For example, the ratios $$\frac{1}{2}$$ and $$\frac{2}{4}$$ are equivalent.

To convert a fraction to a percentage, divide the numerator by the denominator and multiply by 100. For example, $$\frac{3}{4} = 0.75 \times 100 = 75$$ .

A scale drawing is a representation of an object where the dimensions are proportional to the actual dimensions. For example, a scale drawing of the Eiffel Tower might use a scale of 1 cm = 40 m.

To find the height of a scale drawing, use the scale factor. If the actual height is 330 meters and the scale is 1 cm = 40 m, then the height in the drawing is $$\frac{330}{40} = 8.25 cm$$ .

Ratios compare two quantities, while proportions state that two ratios are equal. For example, if $$\frac{a}{b} = \frac{c}{d}$$ , then a, b, c, and d are in proportion.

To solve for an unknown in a proportion, cross-multiply and then divide. For example, if $$\frac{a}{b} = \frac{c}{x}$$ , then solve for x by calculating $$x = \frac{b \cdot c}{a}$$ .