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.
What you need to know
- In inverse proportion, as one quantity goes up the other goes down: y = k over x.
- The product x y stays constant, so find k by multiplying a known pair.
- More workers means less time; go faster and the journey time falls.
- The graph is a reciprocal curve, not a straight line.
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
Practice
6 questions, marked instantly
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Practice
Now try these yourself.
Type your answer and check it. The worked solution appears once you have had a go.
It takes 4 people 6 hours to paint a fence. Working at the same rate, how long would it take 3 people?
6 machines fill an order in 10 days. How many days would 15 machines take?
y is inversely proportional to x. When x = 3, y = 8. Find y when x = 6.
y is inversely proportional to x. When x = 2, y = 15. Find y when x = 5.
y is inversely proportional to x. When x = 4, y = 9. Find x when y = 12.
The time for a journey is inversely proportional to the average speed. A trip takes 3 hours at 40 mph. How long does it take at 60 mph?
Avoid these
The mistakes students make most often
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 students and parents ask
Before this topic, make sure you know
What to learn next
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