A reliable method
Build a conversion as a fraction whose unwanted units cancel. In a recipe, compare the required number of servings with the original number before scaling an ingredient.
The common trap
Multiplying every number mentioned in a problem rarely gives a meaningful result. First name the desired unit, then choose the arithmetic that produces it.
Check and transfer
Check proportional reasoning by doubling a small case. If the situation includes a fixed setup charge, the total cost may not double even when the number of items does.
Worked comparison: why the alternative fails
A pack of 6 costs $18. Nine items cost (18/6)×9 = $27. Computing 18×9 treats the pack price as a unit price, while 9/(18/6) inverts the rate. Write dollars/item next to the intermediate result.
Transfer and explain
A recipe uses 3 cups for 8 people. For 12 people use 3×12/8 = 9/2 cups. The unit changes but the same scaling argument survives.
A pack of 5 replacement filters costs $15. At the same unit price, what do 8 filters cost in dollars?
- Divide pack cost by its filter count to find dollars per filter.
- Multiply the unit price by the new number of filters.
- The requested value is 24. Check its units and meaning against the original question.
Compare the other answer choices
Use the original pack cost without scaling.
Add the change in filter count as dollars.
Invert the unit price to one-third dollar per filter.
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.
GED Mathematical Reasoning overviewSiyavula Grade 10 mathematics foundation reference