Inverse Proportion
In an inverse proportion, the product xy stays constant; if x grows by a factor, y falls by the same factor.
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Formula
x and y use their context units; k has units of x times y.
Variables & units
| Symbol | Meaning | Unit |
|---|---|---|
| Dependent quantity | context-dependent | |
| Nonzero independent quantity | context-dependent | |
| Constant product x·y | context-dependent |
LaTeX
y=\frac{k}{x},\quad k=xy
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Derivation & background
The product form xy=k is a direct way to test inverse proportionality. The graph y=k/x is not a straight line. The model requires fixed conditions: more workers shorten a job only if the workload and productivity per worker remain unchanged.
Exam blueprint
Validity range
x cannot be zero; interpreting a real-world relationship as inverse proportion also requires fixed conditions.
Derivation steps
If the product xy is constant, division gives y=k/x.
- 1Compute k=x·y from a known pair of values.
- 2Divide k by the new x value to obtain y.
Rearrangements
Find the constant product
The product stays the same for every valid pair.
Task variant
A job takes 30 hours with 4 people. Ideally, how long does it take with 10 equally productive people?
k=4·30=120 person-hours.
For 10 people, t=120/10=12 hours.
This assumes the same workload and productivity per person.
Common mistakes
Failing to keep x·y constant or changing the task conditions while scaling.
Check product constancy and state the model assumptions.
Exam context
- Work time for a fixed job, speed for a fixed distance, and inverse-ratio word problems.
Formula cluster
Inverse relationships
The product stays constant while one quantity changes inversely with the other.
Worked example
A job takes 30 hours for 4 equally productive people, so k=4·30=120 person-hours. For 10 people it ideally takes 120/10=12 hours. Check: 4·30=10·12=120.
Applications
Unit-rate problems for fixed work, flow questions, and comparing inversely varying quantities.
How do you calculate with Inverse Proportion?
Here is how to work through a typical Inverse Proportion (y = k/x) task step by step:
- 1
Task
A job takes 30 hours with 4 people. Ideally, how long does it take with 10 equally productive people?
Solution path
k=4·30=120 person-hours.
For 10 people, t=120/10=12 hours.
This assumes the same workload and productivity per person.
Frequently asked questions about Inverse Proportion
How can I recognize inverse proportion?+
In an inverse proportion, the product x·y stays constant. If x doubles, y halves; if x is multiplied by four, y falls to one quarter. Four people taking 30 hours represent 120 person-hours. With 10 equally productive people, the same job ideally takes 120/10=12 hours. Check the product for several pairs. One quantity increasing while another decreases is not enough: the change must follow the reciprocal exactly. The constant k carries the unit of the product, such as person-hours; it is not generally dimensionless.
Why is work time only ideally inversely proportional to the number of workers?+
The model assumes a fixed amount of work and equal productivity per person. In reality, coordination, limited space, different skills, or material shortages can prevent twice as many people from halving the time. Thus y=k/x is a model assumption to assess, not automatically a law for every team. State what stays constant in a word problem. If the assumptions are plausible, calculate k from people times time and divide by the new team size. If material, distance, or workload grows with the number of people, the product need not stay constant.
How do I solve an inverse-proportion word problem?+
For an inverse-proportion problem, keep the product constant. Four equally powerful pumps empty a tank in 30 minutes: k=4·30=120 pump-minutes. With six pumps it takes 120/6=20 minutes. Alternatively, scale from four pumps to one and then from one to six. Check the product: 4·30=6·20. This works only if each pump’s capacity and the amount of water stay unchanged. Do not multiply both quantities by the same factor, because that would be direct proportion. Inverse scaling divides y by the same factor when x increases instead of multiplying both quantities together.
What does the graph of y=k/x look like?+
For positive k, the graph has two separate branches in quadrants I and III; for negative k they lie in quadrants II and IV. The branches approach but do not reach either axis: x=0 is excluded, and y becomes small for very large x. Unlike a straight line, the graph is curved. Doubling x halves y. In school applications, quantities are often positive, so only one branch may be relevant. The axes are only approached; x=0 is excluded and the curve reaches neither axis at finite values.
How is inverse proportion different from negative correlation?+
Inverse proportion is a precise mathematical form: y=k/x and x·y=k. Negative correlation only means that larger x values tend to accompany smaller y values; it does not require a constant product. Data may be negatively correlated without being inversely proportional. Check whether the products of the given pairs stay approximately constant and whether a plausible mechanism explains the form. For statistical data, a model with uncertainty is usually more informative than simply labeling the relation an inverse unit-rate problem. Check the model assumptions before calculating; a decreasing trend alone does not prove inverse proportion.
Quanta exam set
Curated exam set for "Inverse Proportion":
Question (front)
Which formula describes Inverse Proportion?
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Question (front)
How do you rearrange Inverse Proportion for Find the constant product?
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Question (front)
Which common mistake happens with Inverse Proportion?
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Subject references and further reading
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More formulas: Functions, sequences and series
Retain Inverse Proportion for exams
Create a curated FSRS exam set for y = k/x: formula recall, variables, derivation, rearrangement, worked example, common mistakes and exam context.
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