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

A 15% loss keeps 85% of the money, so $100 × 0.85 leaves <span class="mk-a"><span class="hl-ask">$ 85 after Year 1.

100 × 0.85 =

2STEP 2

Apply the 20% gain to the reduced $85, not the original$ 100, so $85 × 1.20 = <span class="mk-a"><span class="hl-ask">$ 102.

85 × 1.20 =

102

3STEP 3

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

\frac{\$102 - \$100}{\$100}

× 100% =

\frac{\$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.

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