Key takeaway: The constant of proportionality is the fixed number k that you multiply x by to get y in a proportional relationship, found by dividing any y-value by its x-value.
Grade 77.RP.A.2bRatios, Rates and Percents
Definition of Constant of Proportionality
The constant of proportionality is the number that stays the same across every pair of values in a proportional relationship. It is the multiplier that turns \(x\) into \(y\), and it is also the unit rate: the amount of \(y\) for one unit of \(x\).
Suppose a tablet uses 4.5 gigabytes of data in 6 hours of streaming. Dividing gives 0.75 gigabytes per hour, and that 0.75 is the constant. On the graph it is the height of the line above the point where \(x = 1\), which is also the slope of the line.
Formula and notation
The constant is usually named \(k\):
\[k = \frac{y}{x} \qquad\text{so}\qquad y = kx\]
Read the units as “y per x”. Choosing a different pair from the table gives the same \(k\); if it does not, the relationship is not proportional.
Key facts
- It equals y divided by x, never x divided by y.
- It is the y-value of the point where x equals 1.
- On the graph, it measures how steeply the line rises.
- A table of values must give the same quotient on every row.
Constant of Proportionality example
A tablet used 4.5 gigabytes in 6 hours of streaming. Find the constant of proportionality, write the equation, and find the data used in 10 hours.
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Divide data by time: \(k = \frac{4.5}{6} = 0.75\) gigabytes per hour.
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Write the rule: \(y = 0.75x\).
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Substitute \(x = 10\): \(y = 0.75 \times 10 = 7.5\).
The constant is 0.75 GB per hour, so 10 hours use 7.5 gigabytes.
Common mistake
Watch out
Ravi finds \(k = 6 : 4.5 \approx 1.33\) and writes \(y = 1.33x\) for gigabytes.
Why it is wrong
He divided in the wrong order. The constant is y divided by x, and here y is the gigabytes. His 1.33 means hours per gigabyte, which would suit the equation for hours, not gigabytes. Check: \(1.33 \times 6\) is nowhere near 4.5.
Related terms and lessons
- Grade 7 · Proportional RelationshipsLessonWorksheetQuiz
- Grade 7 · Equations of Proportional RelationshipsLessonWorksheetQuiz
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