Constant of Proportionality- 7th Grade Math Guide
What Is the Constant of Proportionality?
The constant of proportionality is the fixed number you multiply by when two quantities are directly proportional. If y = kx, then k is your constant of proportionality.
That's it. That's the whole concept.
When two things are proportional, they grow or shrink at a steady rate. The constant tells you exactly how much one variable changes when the other changes by one unit.
How to Spot It in an Equation
Look for equations in the form y = kx. The coefficient of x is always your constant of proportionality.
Examples:
- y = 3x → k = 3
- d = 2.5t → k = 2.5
- c = 0.75p → k = 0.75
If you see y = kx + something extra, those quantities are not proportional. The relationship breaks the moment you add or subtract anything.
Finding k From a Table
When you have a table of values, divide y by x for each row. Every row should give you the same answer—that's k.
| x | y | y ÷ x |
|---|---|---|
| 2 | 10 | 5 |
| 5 | 25 | 5 |
| 9 | 45 | 5 |
In this table, k = 5. The equation is y = 5x.
⚠️ Warning: If your quotients aren't equal, the quantities aren't proportional. Check your work or the table itself.
Finding k From a Graph
On a graph, the constant of proportionality is the slope of the line through the origin.
Pick any point on the line. Read the x-value and y-value. Divide y by x.
Example: A point at (4, 28) gives you k = 28 ÷ 4 = 7.
The line must pass through (0, 0). If it doesn't, you're not looking at a proportional relationship.
Real-World Examples
You encounter this constantly without realizing it:
- Gas prices: If gas costs $3.50 per gallon, k = 3.50. Cost = 3.50 × gallons.
- Paychecks: If you earn $15/hour, k = 15. Pay = 15 × hours worked.
- Recipe scaling: A cookie recipe uses 2 cups flour per batch. k = 2. Flour = 2 × batches.
How to Solve Problems With k
Step 1: Find k from the information given.
Step 2: Plug k into y = kx.
Step 3: Solve for the missing variable.
Example problem: A car travels 180 miles in 3 hours at constant speed. How far does it travel in 7 hours?
Step 1: Find k. k = 180 ÷ 3 = 60 miles/hour
Step 2: Equation: distance = 60 × time
Step 3: Distance in 7 hours = 60 × 7 = 420 miles
Common Mistakes
- Confusing k with the total value—k is the rate, not the final answer.
- Forgetting that proportional relationships always pass through the origin.
- Mixing up direct and inverse proportionality. Direct: y = kx. Inverse: y = k/x.
- Not simplifying fractions when finding k from a table.
Quick Reference
| What You Have | How to Find k |
|---|---|
| Equation y = kx | Read the coefficient of x |
| Table of values | Divide any y by its matching x |
| Graph | Pick a point, divide y by x (slope) |
| Word problem | Divide total by units given |
The constant of proportionality is just the multiplier that connects your two variables. Find it once, plug it in, solve. No tricks.