Competition · AMC preparation · step 4 of 4

AMC 10 · 2019B · #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 1/10 of the jar is green; an 8{:}1 ratio means 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 the ratios into fractions

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 = 1/10, Jar 2 green fraction = 1/9

2STEP 2

Set up the marble equation

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

N/10 + N/9 = 95

3STEP 3

Solve for the jar size

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

9N/90 + 10N/90 = 19N/90 = 95 → N = (95 · 90)/19 = 5 · 90 = 450

4STEP 4

Count blue in each jar

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 = 9/10 · 450 = 405, Blue₂ = 8/9 · 450 = 400

5STEP 5

Subtract the two counts

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

405 - 400 = 5

6STEP 6

Match against the choices

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).

Turn the ratios into fractions
Set up the marble equation
Solve for the jar size
Count blue in each jar
Subtract the two counts
Match against the choices

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