Competition · AMC preparation · step 4 of 4

AMC 8 · 2008 · #9

Grade 7 arithmetic
Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps
percentagefraction-multiplicationmulti-digit-arithmetic identify-subproblems ↑ Prerequisites: percentagefraction-multiplication
📏 Short solution 💡 3 insights
Problem
Tycoon Tammy starts with $100. In the first year her investment drops by 15%. In the second year, the leftover amount grows by 20%. What is the overall percent change after two years?

Pick an answer.

(A)
$5\%\text{ loss}$
(B)
$2\%\text{ loss}$
(C)
$1\%\text{ gain}$
(D)
$2\% \text{ gain}$
(E)
$5\%\text{ gain}$

AMC 8 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Identify Subproblems

Two percent changes happen in order, so Tool #7 (Identify Subproblems) splits the problem into two clean steps: first apply the 15% loss, then apply the 20% gain to the result. Tool #3 (Write an Equation) gives the right form for each step — multiply by 0.85 for a 15% loss, multiply by 1.20 for a 20% gain — and then one more equation turns the final dollar amount into a percent change. The trap to avoid is adding -15% and +20% to get +5%; that ignores the fact that the second percent is taken on a smaller base.

1STEP 1

Apply the first-year loss

A 15% loss keeps 85% of the money, so $100 × 0.85 leaves $85 after Year 1.

100×0.85=100 × 0.85 =85
2STEP 2

Apply the second-year gain

Apply the 20% gain to the reduced $85, not the original $100, so $85 × 1.20 = $102.

85×1.20=85 × 1.20 =102
3STEP 3

Compare to the starting price

The change is $102 - $100 = $2, and $2 on the $100 base is a 2% gain → (D).

(102−102 -100)/100×100100 × 100% =2/$100 × 100% = 2% gain → (D)
Answer
2% gain
Quick sanity check on the combined factor: a 15% loss followed by a 20% gain multiplies the original by 0.85 × 1.20 = 1.02, which is exactly a 2% gain on $100, giving $102. That matches the step-by-step answer. It also explains why the naive -15% + 20% = +5% is wrong: the order and the changing base matter.
💡Key takeaway

Sequential percent changes do not add up the way they look. A 15% loss then a 20% gain becomes 0.85 × 1.20 = 1.02 — a 2% gain — because the second percent is taken on a smaller amount.

  • Apply the first-year loss
  • Apply the second-year gain
  • Compare to the starting price

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