Lesson 3: Writing Proportions

Property

To solve a word problem using proportions, first identify the two quantities being compared and set up two equal ratios. Let a variable represent the unknown quantity. Ensure that the units are placed consistently in the numerators and denominators of the ratios.
$\frac{\text{unit A quantity 1}}{\text{unit B quantity 1}} = \frac{\text{unit A quantity 2}}{\text{unit B quantity 2}}$

Examples

• A doctor prescribes 5 ml of medicine for every 25 pounds of weight. For a child weighing 80 pounds, we set up $\frac{5 \text{ ml}}{25 \text{ lbs}} = \frac{a}{80 \text{ lbs}}$. Solving gives $25a = 400$, so $a = 16$ ml.
• A bag of popcorn with 3.5 servings has 120 calories per serving. To find total calories, set up $\frac{120 \text{ calories}}{1 \text{ serving}} = \frac{c}{3.5 \text{ servings}}$. Solving gives $c = 120 \cdot 3.5 = 420$ calories.
• If 1 U.S. dollar equals 12.54 Mexican pesos, to find how many pesos is 325 dollars, set up $\frac{1 \text{ dollar}}{12.54 \text{ pesos}} = \frac{325 \text{ dollars}}{p}$. Solving gives $p = 325 \cdot 12.54 = 4075.5$ pesos.

Explanation

Proportions are fantastic for real-world problems like scaling a recipe or converting units. The key is to set up your ratios carefully, making sure the corresponding units are in the same position on both sides of the equation.

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.