Choose a representation
Use multiplication factors for sequential changes, rates for simultaneous work, and upward rounding for capacity. Compare consecutive differences to distinguish linear from quadratic sequences.
Worked example: follow the quantity
A price increases by 20% and is then discounted by 25%, ending at $27. What was the original price? Multiply the sequential factors 1.20 and 0.75 to obtain 0.90. Divide the final price by that combined factor. Substitute the given numbers: 27 ÷ (1.2 × 0.75) = 30. Therefore the requested value is 30.
Contrast a plausible mistake
For that example, the tempting answer 243/10 follows this error: Multiply the final price by 0.90 instead of reversing the changes. The requested quantity is 30. Explain why the wrong operation changes the mathematical object, not merely the arithmetic.
A second representation
Each van holds 3 passengers. What is the minimum number of vans needed for 16 passengers? Divide passengers by van capacity and identify the remainder. A positive remainder needs an additional whole van. Substitute the given numbers: 5+1 = 6. Therefore the requested value is 6.
Transfer before checking
A price increases by 20% and is then discounted by 25%, ending at $36. What was the original price? Try without viewing the answer. Check: 40. Multiply the sequential factors 1.20 and 0.75 to obtain 0.90. Divide the final price by that combined factor. Substitute the given numbers: 36 ÷ (1.2 × 0.75) = 40.
Scope and conditions
Do not average completion times to combine workers. A noninteger capacity calculation is a lower bound when an additional whole vehicle is necessary. These original practice tasks sample the official topic; they do not certify complete coverage or a predicted ACT score.
A price increases by 20% and is then discounted by 25%, ending at $27. What was the original price?
- Multiply the sequential factors 1.20 and 0.75 to obtain 0.90.
- Divide the final price by that combined factor.
- Substitute the given numbers: 27 ÷ (1.2 × 0.75) = 30.
- Therefore the requested value is 30.
Compare the other answer choices
Undo only the 25% discount.
Multiply the final price by 0.90 instead of reversing the changes.
Subtract percentage points and divide by 0.95 instead of multiplying factors.
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.
ACT Mathematics content