AMC 10 · 2019 · #11

Grade 6 rate-ratio

Archetype Multiplicative Ratio with Integrality Constraints

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

📏 Medium solution 💡 2 insights

Problem

Two jars each contain the same number of marbles, and every marble is either blue or green. In Jar $1$ the ratio of blue to green marbles is $9:1$ , and the ratio of blue to green marbles in Jar $2$ is $8:1$ . There are $95$ green marbles in all. How many more blue marbles are in Jar $1$ than in Jar $2$ ?

$\textbf{(A) } 5\qquad\textbf{(B) } 10 \qquad\textbf{(C) }25 \qquad\textbf{(D) } 45 \qquad \textbf{(E) } 50$

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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Toolkit + CCSS Solution

Understand

Restated: 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?

Givens: Both jars contain the same total number of marbles, call it $N$; Jar 1 ratio blue:green $= 9{:}1$; Jar 2 ratio blue:green $= 8{:}1$; Total green marbles across both jars $= 95$; Answer choices: $5,\, 10,\, 25,\, 45,\, 50$

Unknowns: Blue in Jar 1 minus blue in Jar 2

Understand

Plan

Primary tool: #8 Analyze the Units

Secondary: #7 Identify Subproblems, #3 Eliminate Possibilities

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.

Execute — Answer: A

#8 Analyze the Units 6.RP.A.3 Step 1

Translate the ratios into the green fraction of each jar.
Jar 1 is $9$ blue $+ 1$ green per $10$ marbles, so green is $\tfrac{1}{10}$ of Jar 1.
Jar 2 is $8$ blue $+ 1$ green per $9$ marbles, so green is $\tfrac{1}{9}$ of Jar 2.

$$\text{Jar 1 green fraction} = \tfrac{1}{10}, \;\; \text{Jar 2 green fraction} = \tfrac{1}{9}$$

💡 Grade 6 ratios: an $a{:}b$ split makes $b/(a+b)$ of the whole the second color.

#7 Identify Subproblems 5.NF.A.1 Step 2

Let $N$ be the (same) number of marbles in each jar.
The total green count is the sum of green from each jar.

$$\tfrac{N}{10} + \tfrac{N}{9} = 95$$

💡 Grade 5 fractions: adding two parts of the same whole gives the total of that color.

#7 Identify Subproblems 6.EE.B.7 Step 3

Add the two fractions with common denominator $90$ and solve for $N$.

$$\tfrac{9N}{90} + \tfrac{10N}{90} = \tfrac{19N}{90} = 95 \;\Rightarrow\; N = \tfrac{95 \cdot 90}{19} = 5 \cdot 90 = 450$$

💡 Grade 6: a one-step equation $\tfrac{19}{90} N = 95$ gives the jar size at once.

#7 Identify Subproblems 5.NF.B.6 Step 4

Compute blue marbles in each jar.
Jar 1 is $\tfrac{9}{10}$ blue and Jar 2 is $\tfrac{8}{9}$ blue.

$$\text{Blue}_1 = \tfrac{9}{10} \cdot 450 = 405, \;\; \text{Blue}_2 = \tfrac{8}{9} \cdot 450 = 400$$

💡 Grade 5: take the same fraction-of-a-whole twice, once per jar.

#7 Identify Subproblems 4.NBT.B.4 Step 5

Subtract to get how many more blue marbles Jar 1 has.

$$405 - 400 = 5$$

💡 Grade 4: a small whole-number subtraction finishes the job.

#3 Eliminate Possibilities 4.NBT.A.2 Step 6

Match $5$ to the answer choices.

$$5 \;\Rightarrow\; \textbf{(A)}$$

💡 Grade 4: pick the matching whole number from the choice list.

[1] #8 6.RP.A.3 Translate the ratios into the green fraction of each jar. Jar 1 is $9$ blue $+ 1

[2] #7 5.NF.A.1 Let $N$ be the (same) number of marbles in each jar. The total green count is th

[3] #7 6.EE.B.7 Add the two fractions with common denominator $90$ and solve for $N$.

[4] #7 5.NF.B.6 Compute blue marbles in each jar. Jar 1 is $\tfrac{9}{10}$ blue and Jar 2 is $\t

[5] #7 4.NBT.B.4 Subtract to get how many more blue marbles Jar 1 has.

[6] #3 4.NBT.A.2 Match $5$ to the answer choices.

Review

Reasonableness: Check the green count: Jar 1 has $450/10 = 45$ green and Jar 2 has $450/9 = 50$ 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.

Alternative: Tool #6 (Guess & Check) on $N$: $N$ must be a multiple of both $10$ and $9$, so $N \in \{90, 180, 270, 360, 450, \dots\}$. Test green totals: at $N=90$ green $= 9+10 = 19$; at $N=450$ green $= 45+50 = 95$ — hit. Same answer, no equation needed.

CCSS standards used (min grade 6)

4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols (Matching the answer $5$ to the multiple-choice value.)
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers (Subtracting $405 - 400 = 5$ for the final difference.)
5.NF.A.1 Add and subtract fractions with unlike denominators (Combining $\tfrac{N}{10} + \tfrac{N}{9}$ over the common denominator $90$.)
5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers (Computing $\tfrac{9}{10} \cdot 450 = 405$ and $\tfrac{8}{9} \cdot 450 = 400$ for blue counts.)
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems (Reading $9{:}1$ and $8{:}1$ as fractions $\tfrac{1}{10}$ and $\tfrac{1}{9}$ green of each jar.)
6.EE.B.7 Solve real-world problems by writing and solving equations of the form px = q (Solving $\tfrac{19}{90} N = 95$ for the jar size $N = 450$.)

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

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: A

Review

CCSS standards used (min grade 6)

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