AMC 10 · 2019 · #11

Grade 6 rate-ratio

Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Chained Ratio Linking

ratio-proportionfraction-arithmeticlinear-equations-one-var ratio-proportionidentify-subproblems ↑ Prerequisites: ratio-proportionfraction-arithmetic

📏 Medium solution 💡 2 insights

Problem

Two jars hold the same number of marbles, each marble blue or green. Jar 1 has blue:green = 9/:1 and Jar 2 has blue:green = 8/:1. Across both jars there are 95 green marbles. How many more blue marbles are in Jar 1 than in Jar 2?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2019 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Analyze the Units

Tool #8 (ratio reasoning as a unit check): a 9{:}1 ratio means $\frac{1}{10}$ of the jar is green; an 8{:}1 ratio means $\frac{1}{9}$ of the jar is green. Tool #7 (Subproblems): first find the common jar size N from the green-count equation, then read off blue counts. Tool #3 verifies the difference 5 matches choice (A). Algebra (#13) would also work but a fraction-of-the-jar view keeps the numbers small and elementary.

1STEP 1

Turn each ratio into a green fraction: Jar 1 is $\frac{1}{10}$ green, Jar 2 is $\frac{1}{9}$ green.

Jar 1 green fraction =

\frac{1}{10}

, Jar 2 green fraction =

\frac{1}{9}

2STEP 2

With N marbles in each jar, the total green is the sum from both: $\frac{N}{10}$ + $\frac{N}{9}$ = 95.

\frac{N}{10}

\frac{N}{9}

= 95

3STEP 3

Combine over denominator 90: $\frac{19N}{90}$ = 95, so N = 450 marbles per jar.

\frac{9N}{90}

\frac{10N}{90}

\frac{19N}{90}

= 95 → N =

\frac{95 · 90}{19}

= 5 · 90 = 450

4STEP 4

Blue is $\frac{9}{10}$ of Jar 1 and $\frac{8}{9}$ of Jar 2: $\frac{9}{10}$ ·450 = 405 and $\frac{8}{9}$ ·450 = 400.

Blue₁ =

\frac{9}{10}

· 450 = 405, Blue₂ =

\frac{8}{9}

· 450 = 400

5STEP 5

Subtract the blue counts: 405 - 400 = 5 more in Jar 1.

405 - 400 = 5

6STEP 6

Match the 5 to the list: it is (A).

5 → (A)

Answer

Check the green count: Jar 1 has

\frac{450}{10}

= 45 green and Jar 2 has

\frac{450}{9}

= 50 green, total 95 — matches the given 95. Blue counts are 405 and 400, totals 450 each — both jars equal. The difference 5 is small, which fits intuition: the two ratios 9/:1 and 8/:1 are close, so the blue counts should also be close.

💡Key takeaway

This AMC 10 problem only needs Grade 6 ratio thinking you already know! A 9/:1 jar is $\frac{1}{10}$ green and an 8/:1 jar is $\frac{1}{9}$ green, so $\frac{N}{10}$ + $\frac{N}{9}$ = 95 gives N = 450 marbles per jar. Then blue counts are 405 and 400, differing by 5, answer (A).

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