ACT Math: Write and Solve Equations for Directly and Inversely Proportional Quantities

Direct proportionality: y=kx (as x increases, y increases at a constant rate). Example: Distance traveled is directly proportional to time at constant speed. If you travel 60 miles per hour, distance=60*time. Inverse proportionality: y=k/x (as x increases, y decreases). Example: Time to complete a task is inversely proportional to the number of workers. If 10 workers take 5 days, the relationship is days=50/workers. To identify which type, check: Does one variable increase while the other increases (direct) or decreases (inverse)? Once you identify the relationship type and find the constant k, you can write the equation and answer any question about the relationship.

Example: The cost of a group trip is inversely proportional to the number of participants. If 4 people cost $100 each, the total cost is $400=k (constant). If 10 people participate, cost per person=400/10=$40. This inverse relationship (more people, lower cost per person) is common in real-world problems.

Trap 1: Confusing direct and inverse relationships. "As attendance increases, revenue per person increases" is direct. "As attendance increases, revenue per person decreases" is inverse. Read carefully to identify the relationship direction. Trap 2: Forgetting to find the constant k before solving. You must identify the type (y=kx or y=k/x), then use given values to find k, then use k to answer the question. Skipping the k-finding step leads to wrong answers. Always write the relationship formula first (y=kx or y=k/x), find k using given information, then use k to answer questions about other values.

When you see a proportionality problem, pause and ask: Do both variables increase (direct) or does one increase while the other decreases (inverse)? Write the formula. Find k. Use k to answer. This process prevents errors.

Problem 1: The distance a spring stretches is directly proportional to the weight hung on it. A 5-pound weight stretches the spring 2 inches. How much will a 12-pound weight stretch it? Relationship: distance=k*weight. Find k: 2=k*5, k=0.4. For 12 pounds: distance=0.4*12=4.8 inches. Problem 2: The intensity of light is inversely proportional to the square of the distance from the source. At 1 meter, intensity is 100 units. What is the intensity at 2 meters? Relationship: intensity=k/distance^2. Find k: 100=k/1^2, k=100. At 2 meters: intensity=100/2^2=100/4=25 units. Problem 3: The number of days to complete a job is inversely proportional to the number of workers. 5 workers take 20 days. How many days do 8 workers take? Relationship: days=k/workers. Find k: 20=k/5, k=100. For 8 workers: days=100/8=12.5 days. All three follow the same pattern: identify relationship, find k, use k to answer.

Do ten more proportionality problems daily for one week. By test day, you'll set up and solve proportionality problems in seconds.

Proportionality problems appear regularly on ACT Math and test whether you can translate word descriptions into mathematical relationships. Once you master identifying and setting up proportionality relationships, you'll solve these problems faster than peers and gain time for harder questions.

This week, practice identifying direct vs. inverse relationships and finding constants. By test day, proportionality problems will feel straightforward and you'll solve them with speed and confidence.