Decision and conditions
Read which variables are proportional and what stays fixed. A proportionality statement must be given; a physical story alone is insufficient.
Worked example
y is inversely proportional to x. When x=3, y=12. What is y when x=9? xy = 36. At x=9, divide the same product by 9. The resulting y is 4. Method choice: The product stays fixed under inverse proportionality.
Contrast: change the reasoning
z is directly proportional to x²y. If x doubles and y is divided by 3, by what factor is z multiplied? Doubling x multiplies x² by four. Dividing y by 3 multiplies it by 1/3. The combined multiplier is 4/3. Method choice: Apply the x multiplier twice before combining it with the y multiplier.
Transfer with solution
Quantity A is 13% of B. B is what percent of A? A/B = 13/100. Therefore B/A=100/13. Multiply by 100 to express the percentage: 10000/13. Method choice: Reverse the ratio before converting back to percent.
Boundary check
Positive quantities satisfy A:B=2:3 and B:C=4:5. If A+B+C=105, what is A? Scale A:B to 8:12 and B:C to 12:15, so A:B:C=8:12:15. The total has 35 ratio units, each worth 3. A has eight units and equals 24. Method choice: Match the shared B component before combining the two ratios.
Misconception to reject
Inverse proportionality reverses a multiplier instead of preserving it. Triples y as if the relationship were direct.
y is inversely proportional to x. When x=3, y=12. What is y when x=9?
- xy = 36.
- At x=9, divide the same product by 9.
- The resulting y is 4.
Compare the other answer choices
Subtracts the factor three instead of dividing by it.
Keeps y fixed despite the change in x.
Triples y as if the relationship were direct.
Takes the reciprocal of the correct new value.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
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