AMC 10 · 2018 · #2

Grade 7 arithmetic
Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Percent / Fraction Recovery
percentageratio-proportionfraction-arithmetic convert-to-algebra ↑ Prerequisites: percentage
📏 Short solution 💡 1 insight
Problem
Liliane has 50% more soda than Jacqueline, and Alice has 25% more soda than Jacqueline. Decide how Liliane's amount compares to Alice's amount — expressed as a percent more.

Pick an answer.

(A)
Liliane has 20% more soda than Alice.
(B)
Liliane has 25% more soda than Alice.
(C)
Liliane has 45% more soda than Alice.
(D)
Liliane has 75% more soda than Alice.
(E)
Liliane has 100% more soda than Alice.

AMC 10 2018 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

Jacqueline's amount is the hidden anchor that both percents lean on, so Tool #4 (Introduce a Variable) leads — name Jacqueline's soda and write Liliane and Alice in terms of it. Because every quantity scales with Jacqueline, Tool #6 (Guess and Check) lets us pick one convenient value (100) to make the percents concrete; the answer cannot depend on which value we pick, which the review confirms. The real work is noticing the comparison's base switches from Jacqueline to Alice.

1STEP 1

Name Jacqueline's soda

Every amount is a percent of Jacqueline, so call her amount 100 — a percent answer doesn't depend on the actual size.

Jacqueline = 100
2STEP 2

Write Liliane and Alice

Liliane is 50% more than Jacqueline, so 150; Alice is 25% more, so 125.

Liliane = 100 + 50 = 150, Alice = 100 + 25 = 125
3STEP 3

Compare Liliane to Alice

Switch the base to Alice: the gap 150-125=25 measured against Alice's 125 gives 25/125 = 20%, choice (A).

(150 - 125)/125 = 25/125 = 1/5 = 20% → (A)
Answer
Liliane has 20% more soda than Alice.
The tempting wrong move is to subtract the percents (50% - 25% = 25%) and pick (B); that fails because those percents are measured against Jacqueline, while the question measures against Alice. Re-test with a different base to be sure the choice of 100 did not matter: let Jacqueline = 4, so Liliane = 6 and Alice = 5; then 6 is (6-5)/5 = 1/5 = 20% more than 5. Same answer, so 20% — choice (A) — is solid.
💡Key takeaway

"Percent more" always measures against the thing you compare to — here that is Alice, not Jacqueline, so divide the gap by Alice's amount.

  • Name Jacqueline's soda
  • Write Liliane and Alice
  • Compare Liliane to Alice

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