Ratio, Proportion & Rates of Change
Grade 6-7Inverse Proportion
In an inverse proportion, as one quantity increases, the other decreases in a matching way, so their product always stays the same. This lesson covers setting up an inverse proportion equation, finding the constant of proportionality, and working with relationships where one quantity is inversely proportional to the square of another.
Written by Asad, Co-Founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Setting up an inverse proportion equation
If y is inversely proportional to x, write y = k/x, where k is the constant of proportionality. Substitute a known pair of values to find k, then use the equation to find any missing value.
If y is inversely proportional to x, and y = 15 when x = 4, then k = 4 × 15 = 60, so y = 60/x. When x = 10, y = 60 ÷ 10 = 6.
Using the constant product method
For real-world contexts like workers completing a job, multiply the two starting quantities to find a constant total, then divide by the new value of one quantity to find the other.
6 workers can complete a job in 8 days, giving a constant of 6 × 8 = 48 worker-days. With 4 workers, the job takes 48 ÷ 4 = 12 days.
Working with inverse-square proportion
If y is inversely proportional to x², write y = k/x². Substitute a known pair of values, remembering to square x first, to find k, then use the equation as before.
If y is inversely proportional to x², and y = 18 when x = 2, then k = 2² × 18 = 72, so y = 72/x². When x = 3, y = 72 ÷ 9 = 8.
Finding x when y is known
Once the equation is set up, it can also be rearranged to find x from a given value of y, by substituting into y = k/x and solving for x.
If y is inversely proportional to x, with k = 72, and y = 6, then 6 = 72/x, so x = 72 ÷ 6 = 12.
Worked Examples
Three exam-style questions, fully solved.
6 workers can complete a job in 8 days. How many days would it take 4 workers to complete the same job?
Easy- 1.Find the constant total: 6 × 8 = 48 worker-days
- 2.Divide by the new number of workers: 48 ÷ 4
Answer: 12 days
y is inversely proportional to x. When x = 4, y = 15. Find the value of y when x = 10.
Medium- 1.Find the constant of proportionality: k = 4 × 15 = 60
- 2.Write the equation: y = 60/x
- 3.Substitute x = 10: y = 60 ÷ 10
Answer: y = 6
The intensity of light, I lux, from a bulb is inversely proportional to the square of the distance, d metres, from the bulb. At a distance of 2 m, the intensity is 20 lux. Find the intensity at a distance of 5 m.
Hard- 1.Find the constant of proportionality: k = 2² × 20 = 80
- 2.Write the equation: I = 80/d²
- 3.Substitute d = 5: I = 80 ÷ 25
Answer: I = 3.2 lux
Avoid These
The most common mistakes students make.
Treating inverse proportion like direct proportion, multiplying by a scale factor instead of dividing, or dividing instead of multiplying.
Forgetting to square x when finding k or when substituting into an inverse-square relationship.
Making an arithmetic slip when finding the constant of proportionality, especially when the numbers are large.
In worker-and-days style problems, forgetting to find the constant total first, and instead trying to scale the number of days directly.
When rearranging the equation to find x from a given value of y, making an error solving the resulting equation.
FAQ
Questions parents and students ask.
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