AMC 8 · 2020 · #13
Grade 6 probabilityalgebraProblem
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?
Pick an answer.
AMC 8 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
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
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) $24$
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.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 $60\% = \tfrac{60}{100} = \tfrac{3}{5}$.
💡 Turning $60\%$ into the ratio $3 : 5$ is a Grade 6 percent-as-ratio move and sets up clean checking.
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 $\dfrac{18+x}{36+x} = \tfrac{3}{5}$.
💡 Writing the new purple fraction as a single expression in $x$ is just Grade 6 ratio writing — we have not solved any equation yet.
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) $x=24$: $\tfrac{42}{60} = \tfrac{7}{10} \neq \tfrac{3}{5}$.
💡 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.
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.
💡 Switching from purple : total to purple : non-purple uses the fact that the unchanged quantity is the safe anchor — pure Grade 6 ratio thinking.
6.RP.A.3 First, count the starting socks and translate the percentage into a fraction tha 6.RP.A.3 Set up the post-addition expression. After adding $x$ purple socks, purple count 4.NF.A.1 Now plug each answer choice into $\dfrac{18+x}{36+x}$ and simplify. The choice t 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.3Use 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.1Explain 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!
⭐ This AMC 8 problem only needs Grade 6 ratio reasoning — turning $60\%$ into the $3 : 5$ ratio you already know!
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