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Mathematics › 🔄 Grade 7 › Unit Rates and the Constant of Proportionality
🔄 Grade 7 · Lesson 4 of 14

Unit Rates and the Constant of Proportionality

Every proportional relationship hides one number, k, that turns any input straight into its output — and it's the slope of a line through the origin.

Grade 7Middle school
Unit Rates and the Constant of Proportionality — illustration
💡
The big idea: Two quantities are proportional when one is always a fixed multiple of the other — double the input and the output doubles too. That fixed multiplier is called the constant of proportionality, or unit rate, and it is usually written k. Once you know k, the relationship becomes a simple rule, y = kx, and its graph is always a straight line that passes through the origin.
🎯 By the end, you'll be able to
  • Define a proportional relationship as y = kx for a constant k
  • Compute a unit rate from a table, graph, or word problem
  • Identify the constant of proportionality as the slope of a line through the origin
  • Use the constant of proportionality to predict an unknown value
📎 You should already know
  • Ratios and rates
  • Plotting points on the coordinate plane

One number that runs the whole relationship

Price per pound at the grocery store, miles per hour on the highway, cups of flour per batch of cookies — each of these is a rate that connects two quantities. When that rate stays exactly the same no matter how much you scale up or down, the two quantities are proportional.

Because the rate never changes, one single number captures the entire relationship. Find that number once, and you can predict any input-output pair.

🔑 y = kx
A relationship is proportional when it can be written as y = kx, where k is a fixed number called the constant of proportionality (also called the unit rate). Every time x increases by 1, y increases by exactly k.
\[ y = kx \]
y is proportional to x with constant of proportionality k.

Finding k

If you know one matching pair of values (x, y) from a proportional relationship, you can find k by dividing: k = y ÷ x. Do this with any pair from the relationship and you get the same k every time — that consistency is exactly what makes it “proportional.”

🎮 Constant-of-Proportionality Gauge LIVE
A proportional relationship is a line through the origin; its slope is the unit rate k.

The graph always passes through the origin

Because y = kx gives y = 0 whenever x = 0, every proportional relationship graphs as a straight line through (0, 0). The steepness of that line — how fast it rises — is exactly k. A steeper line means a bigger unit rate; a flatter line means a smaller one.

✨ If it doesn't start at the origin, it isn't proportional
A line that is shifted up or down — one that crosses the y-axis somewhere other than 0 — is not a proportional relationship, even if it is perfectly straight. A head start (a fixed starting fee, for example) breaks the y = kx pattern.
📝 Worked example: A car travels 150 miles in 3 hours at a constant speed. Find the unit rate (constant of proportionality) between distance and time.
  1. Identify the matching pair: x = 3 hours, y = 150 miles.
  2. Divide: k = y ÷ x = 150 ÷ 3.
  3. k = 50.
✓ The constant of proportionality is <strong>50 miles per hour</strong> &mdash; the car's speed.
📝 Worked example: Using the same car (k = 50 mph), how long will it take to travel 400 miles?
  1. Use y = kx, but solve for x this time: x = y ÷ k.
  2. x = 400 ÷ 50.
  3. x = 8.
✓ The trip will take <strong>8 hours</strong>.
⚠️ Check that k stays the same
Before assuming a relationship is proportional, check more than one pair of values. If y ÷ x gives a different number for different pairs, the relationship is not proportional — something else (like a starting fee) is going on.

Check your understanding

1. In the equation y = kx, what does k represent?
k is the fixed multiplier that connects x and y in a proportional relationship — the constant of proportionality.
2. A table shows: x = 2, y = 5; x = 4, y = 10; x = 6, y = 15. What is k?
k = y ÷ x = 5 ÷ 2 = 2.5, and it stays consistent across every pair in the table (10 ÷ 4 = 2.5, 15 ÷ 6 = 2.5).
3. Which graph represents a proportional relationship?
Proportional relationships always graph as a straight line passing through (0, 0).
4. If k = 4 and x = 6, what is y?
y = kx = 4 × 6 = 24.
5. A car travels at a constant rate with k = 60 miles per hour. How long does it take to travel 300 miles?
x = y ÷ k = 300 ÷ 60 = 5 hours.
✅ Key takeaways
  • A proportional relationship can always be written as y = kx, where k is a fixed constant.
  • k is called the constant of proportionality, or unit rate — divide y by x for any matching pair to find it.
  • Every proportional relationship graphs as a straight line through the origin, (0, 0).
  • The steepness of that line is exactly k: a bigger k means a steeper, faster-rising line.
  • If y ÷ x changes between different pairs, or the graph doesn't pass through the origin, the relationship is not proportional.