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Constant of Proportionality

The constant of proportionality is the fixed number kk that relates two quantities in a proportional relationship, written as y=kxy = kx. It tells you how much yy changes for every one unit of xx.

In a proportional relationship between two variables xx and yy, the constant of proportionality is the ratio k=yxk = \frac{y}{x} that remains the same for all corresponding pairs of values. When kk is positive and the relationship takes the form y=kxy = kx, any increase in xx produces a proportional increase in yy. The constant of proportionality is sometimes called the unit rate, since it represents the value of yy when x=1x = 1.

Key Formula

y=kxor equivalentlyk=yxy = kx \quad \text{or equivalently} \quad k = \frac{y}{x}
Where:
  • kk = the constant of proportionality
  • yy = the dependent variable (output)
  • xx = the independent variable (input)

Worked Example

Problem: A car travels at a constant speed. In 2 hours it covers 120 miles, and in 5 hours it covers 300 miles. Find the constant of proportionality and use it to predict how far the car travels in 7 hours.
Step 1: Check that the relationship is proportional by finding the ratio y/x for each data pair.
1202=60,3005=60\frac{120}{2} = 60, \quad \frac{300}{5} = 60
Step 2: Since both ratios are equal, the relationship is proportional. The constant of proportionality is that shared ratio.
k=60k = 60
Step 3: Write the equation using y = kx.
y=60xy = 60x
Step 4: Substitute x = 7 to predict the distance.
y=60×7=420y = 60 \times 7 = 420
Answer: The constant of proportionality is 60 miles per hour, and the car travels 420 miles in 7 hours.

Visualization

Why It Matters

The constant of proportionality shows up whenever two quantities grow together at a steady rate. Speed, price per item, and recipe scaling all depend on it. Recognizing kk helps you make predictions, compare rates, and solve problems quickly without building a full table of values.

Common Mistakes

Mistake: Dividing x by y instead of y by x when finding k.
Correction: Always compute k = y / x (output divided by input). In y = kx, k multiplies x to produce y, so you isolate k by dividing the y-value by the corresponding x-value.
Mistake: Assuming any linear equation is proportional.
Correction: The equation y = 3x + 2 is linear but NOT proportional because of the + 2. A proportional relationship must pass through the origin (0, 0), meaning y = kx with no added constant.

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