AMC 8 · 2020 · #13

Grade 6 probabilityalgebra

Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Ratio Shift — Two State

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

📏 Medium solution 💡 3 insights

Problem

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?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Guess and Check

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 $\frac{18+x}{36+x}$ and compares to $\frac{3}{5}$ , while Tool #3 (Eliminate Possibilities) rules out anything that isn't $\frac{3}{5}$ . 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.

1STEP 1

Count the starting socks: 6 + 18 + 12 = 36, and rewrite 60% as the fraction $\frac{3}{5}$ .

6 + 18 + 12 = 36 socks, 60% =

\frac{60}{100}

\frac{3}{5}

2STEP 2

After adding x purple socks, the target ratio is $\frac{18+x}{36+x}$ = $\frac{3}{5}$ .

\frac{18+x}{36+x}

\frac{3}{5}

3STEP 3

Plug each choice into $\frac{18+x}{36+x}$ ; only x=9 lands on $\frac{27}{45}$ = $\frac{3}{5}$ , eliminating the rest.

\frac{27}{45}

\frac{27 ÷ 9}{45 ÷ 9}

\frac{3}{5}

✓ → (B)

4STEP 4

Cross-check: non-purple (6+12=18) never changes, so purple : non-purple = 3 : 2, purple = 27, x = 27-18 = 9.

non-purple = 6 + 12 = 18, purple =

\frac{3}{2}

× 18 = 27, x = 27 - 18 = 9

Answer

Plug x = 9 back in. Purple becomes 18 + 9 = 27; total becomes 36 + 9 = 45. The probability of purple is

\frac{27}{45}

\frac{3}{5}

= 60% — exactly what the problem demanded. The answer is also reasonable in magnitude: purple started as a

\frac{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%.

💡Key takeaway

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

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