Competition · AMC preparation · step 4 of 4

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)
20
(B)
$33\frac{1}{3}$
(C)
38
(D)
45
(E)
54

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 1/3, (C)38, (D)45, (E)54, only one matches the value we compute, so we can confirm the choice directly.

1STEP 1

Start with 100 marbles

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

Take away Pedro's 20 percent

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

Take away Ebony's 10 percent

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

Take away Jimmy's 25 percent

Stage 3 — Jimmy: the whole is 72. Since 25% = 14\frac{1}{4}, he gets 14\frac{1}{4} × 72 = 18, so Gilda keeps 72 - 18 = 54 marbles.

72 - 25% of 72 = 72 - 1/4 × 72 = 72 - 18 = 54
5STEP 5

Read the leftover as a percent

We started with 100 and Gilda kept 54, so she has 54100\frac{54}{100} = 54% of the original bag — only choice (E) matches.

54/100 = 54% → (E)
Answer
54
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 108200\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!

  • Start with 100 marbles
  • Take away Pedro's 20 percent
  • Take away Ebony's 10 percent
  • Take away Jimmy's 25 percent
  • Read the leftover as a percent

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