Lesson 2: Proportion

New Concept

This lesson introduces proportional variables, where the ratio between two quantities remains constant. You'll learn to identify proportional relationships, solve problems using scaling and unit rates, and recognize their distinct straight-line graphs through the origin.

What’s next

Next, you'll put this concept into action with interactive examples, practice cards on scaling, and challenges involving graphs of proportional relationships.

Property

Two variables are said to be proportional if their ratio is constant, or always the same. This means one variable is a constant multiple of the other. To check if two variables are proportional, you can identify several pairs of corresponding values for the variables, and then compute their ratios to see if they are equal.

Examples

• A baker uses 3 cups of sugar for every 2 dozen muffins. The amount of sugar is proportional to the number of dozens of muffins because the ratio $\frac{3}{2}$ is constant.
• A taxi fare includes a 3 dollars flat fee plus 2 dollars per mile. The total cost is not proportional to the miles driven because the ratio of cost to miles changes. For 2 miles, the ratio is $\frac{2 \times 2 + 3}{2} = 3.5$, but for 5 miles it is $\frac{2 \times 5 + 3}{5} = 2.6$.
• The perimeter of a regular octagon is given by the formula $P = 8s$, where $s$ is the side length. The perimeter is proportional to the side length because the ratio $\frac{P}{s} = 8$ is always constant.

Explanation

Think of it like this: if two variables are proportional, they are partners that always move together at a steady pace. If you double one variable, the other one doubles too. Their relationship is perfectly predictable and consistent.

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

An effective method for solving proportion problems is the unit rate strategy. This involves two steps: first, simplify a known ratio to find the value for a single unit (the unit rate). Second, multiply this unit rate by the desired quantity to find the final answer. This strategy is particularly useful when the build-up scale factor is not a simple number.

Examples

• If 4 gallons of paint cover 1680 square feet, how much area will 7 gallons cover? The unit rate is $\frac{1680}{4} = 420$ square feet per gallon. Thus, 7 gallons will cover $7 \times 420 = 2940$ square feet.
• Maria drove 224 miles on 8 gallons of gasoline. How far can she travel on a full tank of 15 gallons? Her car's efficiency is $\frac{224}{8} = 28$ miles per gallon. On a full tank, she can drive $15 \times 28 = 420$ miles.
• A 12-ounce coffee costs 3.00 dollars. To find the cost of a 20-ounce coffee at the same rate, first find the unit cost: $\frac{3.00}{12} = 0.25$ dollars per ounce. The 20-ounce coffee costs $20 \times 0.25 = 5.00$ dollars.

Explanation

This strategy is all about finding the 'price for one.' If you know one can of soda costs 1.50 dollars, it's easy to find the cost of five cans. You find the value of a single unit first, then multiply up to what you need.

Property

When graphed, the relationship between two proportional variables has two key characteristics:

1. The graph is a straight line.
2. The graph passes through the origin, which is the point $(0, 0)$.

These features occur because the rate of change is constant and because if one variable is zero, the other must also be zero.

Examples

• A graph shows the cost of bulk almonds. The point $(4, 24)$ is on the line, meaning 4 pounds cost 24 dollars. Since the graph is a line through the origin, the unit price is constant: $\frac{24}{4} = 6$ dollars per pound.
• The graph of a monthly bus pass cost is a horizontal line at $y=50$. This is not proportional to the number of rides because it does not pass through $(0,0)$ and the cost is constant regardless of the number of rides.
• A caterer's fee is shown on a graph that is a straight line through $(0,0)$ and $(10, 150)$. The relationship is proportional. The unit rate (cost per person) is $\frac{150}{10} = 15$ dollars per person. For 30 people, the cost would be $30 \times 15 = 450$ dollars.

Explanation

Think of a proportional graph as a perfectly straight ramp that starts right at the ground. It is straight because the steepness (the rate) never changes, and it starts at $(0,0)$ because zero input means zero output, like working 0 hours earns 0 dollars.