A store increases the price of a $40 item by 30%, then later decreases the new price by 30%. What is the final price?

The final price equals $36.40 when the 30% increase is first applied to $40 yielding $52 (40 × 1.30 = 52), then the 30% decrease is applied to this new price yielding $36.40 (52 × 0.70 = 36.40), demonstrating that percentage increases and decreases are not symmetric when applied to different base amounts the decrease operates on a larger base than the increase, producing a net reduction despite equal percentage magnitudes.

A. $36.40

This value correctly represents the final price through sequential percentage application with comprehensive verification. Increase calculation: 30% of $40 = $12; new price = $40 + $12 = $52. Decrease calculation: 30% of $52 = $15.60; final price = $52 - $15.60 = $36.40. Multiplicative method: $40 × 1.30 × 0.70 = $40 × 0.91 = $36.40, revealing the net multiplier 0.91 < 1. Percentage analysis: net change = (1.30)(0.70) - 1 = 0.91 - 1 = -0.09 or -9% overall reduction. Verification pathway: $36.40 ÷ $40 = 0.91, confirming 9% net decrease. This solution demonstrates comprehensive understanding of sequential percentage changes including base-dependent calculation (each percentage applies to current value, not original), multiplicative combination revealing net effect, recognition of asymmetry between equal-magnitude increases/decreases on different bases, and verification through division to confirm net percentage change integrated competencies essential for finance (compound interest with withdrawals), retail mathematics (markdown after markup), economics (inflation-deflation sequences), and personal finance (investment growth with fees) where accurate sequential percentage determination prevents costly misconceptions about symmetry in percentage operations with direct financial implications for pricing strategies, investment returns, and purchasing power calculations.

B. $40.00

This amount suggests students incorrectly assumed percentage increases and decreases of equal magnitude cancel exactly regardless of base values a pervasive misconception rooted in additive intuition misapplied to multiplicative contexts. Students might execute 30% - 30% = 0% net change then apply to $40 to get $40, revealing fundamental misunderstanding that percentages operate multiplicatively on changing bases rather than additively on a fixed base. Another plausible pathway involves calculating increase to $52 correctly but then decreasing by 30% of original $40 ($12) rather than 30% of new price ($15.60), yielding $52 - $12 = $40 demonstrating base misidentification during the decrease operation. The psychological appeal of returning to original price creates cognitive bias favoring this answer despite mathematical invalidity a dangerous heuristic requiring explicit counterexamples with concrete calculations to dismantle the symmetry misconception. This distractor effectively identifies learners with additive percentage models who treat percentages as absolute adjustments rather than relative proportions a foundational deficiency requiring explicit reinforcement of percentage as "per hundred of current value" with sequential base tracking protocols to build multiplicative reasoning preventing symmetry assumptions that produce systematically optimistic financial projections in contexts like investment returns with fees or retail pricing with sequential markups/markdowns.

C. $42.80

This figure exceeds the original price by $2.80, suggesting students likely applied both percentages to the original base: 30% increase = $12, 30% decrease = $12, net change $0 but then added $2.80 through unstructured adjustment. More plausibly, students executed $40 × 1.30 = $52 correctly but then decreased by 20% instead of 30% ($52 × 0.80 = $41.60) then adjusted upward to $42.80. Another error pathway involves calculating net multiplier as 1.30 - 0.30 = 1.00 then adding 7% for tax or other adjustment. Students might solve $40 × (1 + 0.30 - 0.30) = $40 then recognize error and guess $42.80 as "reasonable" increase. The value $42.80 also equals $40 × 1.07 suggesting possible confusion with 7% net increase through miscalculation of (1.30)(0.70) = 0.91 misread as 1.07. This distractor effectively identifies learners with fragmented sequential percentage understanding who recognize multiplicative combination is required but execute arithmetic errors during multiplier calculation a transitional deficiency requiring explicit net multiplier computation protocols with verification through sequential step execution to prevent mental calculation errors that distort financial outcomes in pricing and investment contexts where precise sequential percentage determination affects profit margins and return calculations.

D. $44.00

This amount equals precisely 10% above original price ($40 × 1.10 = $44), suggesting students likely calculated net change as 30% - 20% = 10% through arbitrary adjustment or executed $40 × 1.30 = $52 correctly but then decreased by 15.38% ($8) to reach $44 without justification. Another plausible pathway involves solving $40 × 1.30 × x = $44 → x = 44/52 ≈ 0.846 then misinterpreting as 15.4% decrease but selecting $44 as answer without completing calculation. Students might recognize the asymmetry produces net loss but underestimate magnitude, guessing $44 as "close to original." The value $44 also equals $40 + $4 suggesting possible addition of fixed $4 after percentage operations without geometric rationale. This distractor functions primarily as an optimistic distractor testing whether students recognize net loss occurs despite equal percentage magnitudes a critical conceptual hurdle where learners must overcome additive intuition to accept multiplicative reality producing net reduction. Its inclusion proves educationally valuable for emphasizing the mathematical necessity of sequential base-dependent calculation rather than net percentage subtraction, with concrete demonstration showing why 30% of larger base exceeds 30% of smaller base, producing irreversible net loss that cannot be overcome without exceeding original percentage magnitude during recovery phase a principle with profound implications for investment recovery requirements and retail pricing strategies where sequential percentage operations require careful base tracking to avoid financial miscalculations.

Conclusion

The final price of $36.40 emerges through rigorous sequential percentage application recognizing each operation applies to the current price base rather than the original amount, verified through multiplicative combination revealing net multiplier 0.91 corresponding to 9% overall reduction. This problem reinforces critical percentage reasoning competencies essential across financial domains: understanding that percentage operations are base-dependent with sequential changes compounding multiplicatively rather than additively, recognizing the asymmetry between equal-magnitude increases and decreases (recovery requires larger percentage than loss to return to original), executing precise base tracking during multi-step percentage sequences, and verifying results through net multiplier calculation to detect symmetry misconceptions. Mastery of these integrated skills proves indispensable for personal finance (investment return calculations with fees), retail management (pricing strategy development with sequential markups/markdowns), economics (inflation-adjusted value determination), and business analytics (growth rate analysis with periodic fluctuations) where accurate sequential percentage determination prevents costly financial misconceptions with direct implications for wealth accumulation, pricing profitability, and economic forecasting accuracy. The distractors strategically target pervasive misconceptions including percentage symmetry assumptions, base misidentification during sequential operations, and net percentage subtraction errors highlighting the necessity of explicit sequential base tracking protocols with intermediate value recording and mandatory net multiplier verification to prevent intuitive but mathematically invalid assumptions that compromise financial decision accuracy in increasingly complex percentage operation sequences with real-world economic consequences.