AMC 8 · 2020 · #13

Grade 6 probabilityalgebra

Archetype Multiplicative Ratio with Integrality Constraints

probability-basiclinear-equations-one-varfraction-arithmeticpercentage convert-to-algebra ↑ Prerequisites: linear-equations-one-varfraction-arithmetic

📏 Medium solution 💡 3 insights

Problem

Jamal has a drawer containing $6$ green socks, $18$ purple socks, and $12$ orange socks. After adding more purple socks, Jamal noticed that there is now a $60\%$ chance that a sock randomly selected from the drawer is purple. How many purple socks did Jamal add?

$\textbf{(A) }6 \qquad \textbf{(B) }9 \qquad \textbf{(C) }12 \qquad \textbf{(D) }18 \qquad \textbf{(E) }24$

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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Toolkit + CCSS Solution

Understand

Restated: Jamal's drawer starts with $6$ green, $18$ purple, and $12$ orange socks. He adds some number of purple socks (and nothing else). After the addition, a randomly chosen sock has a $60\%$ chance of being purple. How many purple socks did he add?

Givens: Starting counts: $6$ green, $18$ purple, $12$ orange (total $36$); Only purple socks are added; green and orange counts stay the same; After the addition, $\Pr(\text{purple}) = 60\% = \tfrac{3}{5}$; Answer choices: (A) $6$, (B) $9$, (C) $12$, (D) $18$, (E) $24$

Unknowns: The number of purple socks added, call it $x$

Understand

Plan

Primary tool: #6 Guess and Check

Secondary: #3 Eliminate Possibilities, #15 Reorganize Information

This is a multiple-choice problem with only $5$ candidate values for $x$, so the toolkit's best AMC strategy applies: Tool #6 (Guess and Check) plugs each choice into $\dfrac{18+x}{36+x}$ and compares to $\tfrac{3}{5}$, while Tool #3 (Eliminate Possibilities) rules out anything that isn't $\tfrac{3}{5}$. As a sanity check, Tool #15 (Reorganize Information) notices that the non-purple socks $(6+12=18)$ never change, so purple : non-purple must equal $3 : 2$ — which forces purple $= 27$ and confirms $x = 9$ without any algebra.

Execute — Answer: B

#6 Guess and Check 6.RP.A.3 Step 1

First, count the starting socks and translate the percentage into a fraction that is easier to compare with.
The drawer holds $6 + 18 + 12 = 36$ socks, and $60\% = \tfrac{60}{100} = \tfrac{3}{5}$.

$$6 + 18 + 12 = 36 \text{ socks}, \qquad 60\% = \tfrac{60}{100} = \tfrac{3}{5}$$

💡 Turning $60\%$ into the ratio $3 : 5$ is a Grade 6 percent-as-ratio move and sets up clean checking.

#6 Guess and Check 6.RP.A.3 Step 2

Set up the post-addition expression.
After adding $x$ purple socks, purple count is $18 + x$ and total is $36 + x$.
The target is $\dfrac{18+x}{36+x} = \tfrac{3}{5}$.

$$\dfrac{18+x}{36+x} = \tfrac{3}{5}$$

💡 Writing the new purple fraction as a single expression in $x$ is just Grade 6 ratio writing — we have not solved any equation yet.

#3 Eliminate Possibilities 4.NF.A.1 Step 3

Now plug each answer choice into $\dfrac{18+x}{36+x}$ and simplify.
The choice that lands on $\tfrac{3}{5}$ wins; the rest get eliminated.
(A) $x=6$: $\tfrac{24}{42} = \tfrac{4}{7} \neq \tfrac{3}{5}$.
(B) $x=9$: $\tfrac{27}{45} = \tfrac{3}{5}$.
(C) $x=12$: $\tfrac{30}{48} = \tfrac{5}{8} \neq \tfrac{3}{5}$.
(D) $x=18$: $\tfrac{36}{54} = \tfrac{2}{3} \neq \tfrac{3}{5}$.
(E) $x=24$: $\tfrac{42}{60} = \tfrac{7}{10} \neq \tfrac{3}{5}$.

$$\tfrac{27}{45} = \tfrac{27 \div 9}{45 \div 9} = \tfrac{3}{5} \;\checkmark \;\Rightarrow\; \textbf{(B)}$$

💡 Recognizing $\tfrac{27}{45}$ and $\tfrac{3}{5}$ as equivalent fractions (both shrink by dividing top and bottom by $9$) is a Grade 4 equivalence skill.

#15 Reorganize Information 6.RP.A.3 Step 4

Cross-check with the reorganization trick.
The non-purple socks $(6 + 12 = 18)$ never change.
If the final ratio purple : total is $3 : 5$, then purple : non-purple is $3 : 2$.
Since non-purple $= 18$, purple must equal $\tfrac{3}{2} \times 18 = 27$.
So $x = 27 - 18 = 9$ — same answer.

$$\text{non-purple} = 6 + 12 = 18, \quad \text{purple} = \tfrac{3}{2} \times 18 = 27, \quad x = 27 - 18 = 9$$

💡 Switching from purple : total to purple : non-purple uses the fact that the unchanged quantity is the safe anchor — pure Grade 6 ratio thinking.

[1] #6 6.RP.A.3 First, count the starting socks and translate the percentage into a fraction tha

[2] #6 6.RP.A.3 Set up the post-addition expression. After adding $x$ purple socks, purple count

[3] #3 4.NF.A.1 Now plug each answer choice into $\dfrac{18+x}{36+x}$ and simplify. The choice t

[4] #15 6.RP.A.3 Cross-check with the reorganization trick. The non-purple socks $(6 + 12 = 18)$

Review

Reasonableness: Plug $x = 9$ back in. Purple becomes $18 + 9 = 27$; total becomes $36 + 9 = 45$. The probability of purple is $\tfrac{27}{45} = \tfrac{3}{5} = 60\%$ — exactly what the problem demanded. The answer is also reasonable in magnitude: purple started as a $\tfrac{18}{36} = 50\%$ share, and we only need to push it up to $60\%$, so adding a moderate number like $9$ (smaller than the original $18$) feels right. Choices like (D) $18$ or (E) $24$ would push the share well past $60\%$.

Alternative: Tool #13 (Convert to Algebra) gives the same answer with a single equation: $\dfrac{18+x}{36+x} = \tfrac{3}{5}$ cross-multiplies to $5(18+x) = 3(36+x)$, i.e. $90 + 5x = 108 + 3x$, so $2x = 18$ and $x = 9$. This is fine but less revealing for a young solver than the Guess and Check + Reorganize path used above.

CCSS standards used (min grade 6)

6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems (Treating $60\%$ as the ratio $3 : 5$, building the post-addition expression $\dfrac{18+x}{36+x}$, and using the unchanging non-purple count to force purple : non-purple $= 3 : 2$.)
4.NF.A.1 Explain why a fraction is equivalent to another fraction (Recognizing that $\tfrac{27}{45}$ reduces to $\tfrac{3}{5}$ when checking the candidate $x = 9$, and that the other candidates' fractions do not.)

⭐ This AMC 8 problem only needs Grade 6 ratio reasoning — turning $60\%$ into the $3 : 5$ ratio you already know!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Review

CCSS standards used (min grade 6)

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