AMC 8 · 2019 · #8

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

percentagefraction-multiplication identify-subproblemsdimensional-analysis ↑ Prerequisites: percentagefraction-multiplication

📏 Medium solution 💡 2 insights

Problem

Gilda starts with a full bag of marbles. She gives 20% of the bag to Pedro, then 10% of what is left to Ebony, then 25% of what is still left to Jimmy. We want the percentage of the original bag that Gilda still holds for herself after all three gifts.

Pick an answer.

(A)

(B)

$33\frac{1}{3}$

(C)

(D)

(E)

AMC 8 2019 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Solve an Easier Related Problem

The trap in this problem is that each percentage applies to a different total (the current bag, not the original). Tool #9 (Easier Related Problem) makes this concrete by pretending Gilda starts with exactly 100 marbles — then every percentage becomes a plain whole number we can track. Tool #7 (Identify Subproblems) splits the journey into three clean one-step "give and keep" calculations, one per friend. Tool #3 (Eliminate Possibilities) is a nice safety net at the end: among (A)20, (B)33 $\frac{1}{3}$ , (C)38, (D)45, (E)54, only one matches the value we compute, so we can confirm the choice directly.

1STEP 1

Pretend Gilda starts with 100 marbles: the answer is a percent of her own bag, so size can't change it, and 100 makes every percent whole.

start = 100 marbles

2STEP 2

Stage 1 — Pedro: he gets 20% of 100 = 20, so Gilda keeps 100 - 20 = 80 marbles (that's 80% of what she had).

100 - 20% of 100 = 100 - 20 = 80

3STEP 3

Stage 2 — Ebony: the whole is now 80, not the original 100. She gets 10% of 80 = 8, so Gilda keeps 80 - 8 = 72 marbles.

80 - 10% of 80 = 80 - 8 = 72

4STEP 4

Stage 3 — Jimmy: the whole is 72. Since 25% = $\frac{1}{4}$ , he gets $\frac{1}{4}$ × 72 = 18, so Gilda keeps 72 - 18 = 54 marbles.

72 - 25% of 72 = 72 -

\frac{1}{4}

× 72 = 72 - 18 = 54

5STEP 5

We started with 100 and Gilda kept 54, so she has $\frac{54}{100}$ = 54% of the original bag — only choice (E) matches.

\frac{54}{100}

= 54% → (E)

Answer

A quick sanity check: Gilda gave away three chunks, but each chunk was at most a quarter of the current bag, so she should still have more than half left. 54% is just over half — exactly the right neighborhood. The answer is also independent of starting count: with 200 marbles the same logic gives 200 → 160 → 144 → 108, and

\frac{108}{200}

= 54%. Same percentage, confirming the result.

💡Key takeaway

This AMC 8 problem only needs Grade 6 percent reasoning — taking a percent of whatever is left at each step — that you already know!

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