AMC 8 · 2005 · #22

Grade 7 rate-ratio

Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Multiplicative Rate or Series

percentageratio-proportionratefraction-arithmetic identify-subproblemsratio-proportion ↑ Prerequisites: percentageratio-proportion

📏 Medium solution 💡 3 insights

Problem

A detergent comes in small (S), medium (M), and large (L) boxes. The medium costs 50% more than the small and holds 20% less detergent than the large. The large holds twice as much as the small and costs 30% more than the medium. Rank the three sizes from best buy to worst buy.

Pick an answer.

(A)

SML

(B)

LMS

(C)

MSL

(D)

LSM

(E)

MLS

AMC 8 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Try a Simpler Case (WLOG)

The question asks only for a ranking, so the actual prices and sizes do not matter — only their ratios do. Tool #9 (Try a Simpler Case) lets us pick convenient round numbers for the small box without loss of generality, then derive the other two boxes from the percentage rules. Tool #4 (Introduce a Variable) backs this up: if you set the small box at cost c and size s, every other quantity is a fixed multiple of c or s, so the ratios cost/size only depend on those multiples. Picking c = $1 and s = 5 oz keeps the arithmetic clean.

1STEP 1

Give the small box easy round numbers — cost $1, size 5 oz; any baseline gives the same ranking.

cost(S) = $1, size(S) = 5 oz

2STEP 2

The large holds twice the small, so size(L) = 10 oz; its cost is set later via the medium.

size(L) = 2 · 5 = 10 oz

3STEP 3

Medium: cost is 50% more (· 1.5 = $1.50), size is 20% less than large (· 0.8 = 8 oz).

cost(M) = 1.5 ·

1 =

1.50, size(M) = 0.8 · 10 = 8 oz

4STEP 4

Large costs 30% more than medium: 1.3 · $1.50 = <span class="mk-a"><span class="hl-ask">$ 1.95.

cost(L) = 1.3 ·

1.50 =

1.95

5STEP 5

Unit price = cost per oz: S = $0.200, <span class="mk-a"><span class="hl-ask">M =$ 0.1875, L = $0.195 — lower is better.

1/(5 oz) =

0.200/oz,

1.50/(8 oz) =

0.1875/oz,

1.95/(10 oz) =

0.195/oz

6STEP 6

Order the unit prices low to high: M ＜ L ＜ S, so the best-to-worst buy is (E).

0.1875 ＜ 0.195 ＜ 0.200 → M ＜ L ＜ S → (E) MLS

Answer

MLS

Try a different baseline to confirm the ranking is invariant. Let the small cost

2 and hold 10 oz. Then large = 20 oz and costs 1.3 · (1.5 ·

2) =

3.90; medium costs

3.00 and holds 0.8 · 20 = 16 oz. Unit prices: S =

0.200, M =

\frac{3.00}{16}

=

0.1875, L =

\frac{3.90}{20}

= $0.195. Same order: M ＜ L ＜ S. Also, M being the cheapest per ounce passes the sanity check that prices grew by 50% but size grew by 60% (5 → 8 oz) from S to M.

💡Key takeaway

When the question is "which is the best buy?", only cost per ounce matters — pick easy numbers for the small box, follow the percentages to the medium and large, then divide.

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