Lesson 4: Solving Ratio and Rate Problems

Property

A ratio table organizes equivalent ratios in rows or columns, where each ratio maintains the same proportional relationship: $\frac{a}{b} = \frac{2a}{2b} = \frac{3a}{3b} = \frac{ka}{kb}$ for any multiplier $k$.

Examples

Property

To solve problems involving proportional variables, we can use a build-up strategy. This involves finding a scale factor that relates a known quantity to a desired quantity. If we multiply one variable by this scale factor, we must multiply the other variable by the same scale factor to maintain the proportional relationship. This process can be organized in a ratio table.

Examples

• A recipe for soup requires 3 cups of broth to serve 4 people. To serve 12 people, you use a scale factor of 3 (since $4 \times 3 = 12$). Therefore, you need $3 \times 3 = 9$ cups of broth.
• A car travels 180 miles in 3 hours. To find how far it travels in 40 minutes, we use a scale factor of $\frac{2}{9}$ (since 40 minutes is $\frac{2}{3}$ of an hour, and we are starting from 3 hours, so $\frac{2/3}{3} = \frac{2}{9}$). The distance is $\frac{2}{9} \times 180 = 40$ miles.
• If 5 comic books cost 22 dollars, how much do 20 comic books cost? The number of books is multiplied by a scale factor of 4, so we multiply the cost by 4: $22 \times 4 = 88$ dollars.

Explanation

This is like resizing a photo. To keep the picture from looking stretched or squished, you have to scale the height and width by the same percentage. With proportions, you multiply both variables by the same scale factor to get the right answer.

Property

A ratio table organizes equivalent rates.
To find a unit rate, divide both quantities by the value of the quantity you want to be 1.
To find an equivalent rate, multiply both quantities in the unit rate row by the new desired quantity.

Examples