Competition · AMC preparation · step 4 of 4

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 (18+x)/(36+x) and compares to 3/5, while Tool #3 (Eliminate Possibilities) rules out anything that isn't 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 socks and convert the percent

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

6 + 18 + 12 = 36 socks, 60% = 60/100 = 3/5

2STEP 2

Set up the fraction equation

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

(18+x)/(36+x) = 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.

After Jamal adds x purple socks, the statement that a randomly drawn sock is purple with probability 60% is exactly the equation (18+x)/(36+x) = 3/5.

▸ Why?

The left side is the chance of drawing purple written out: the drawer now holds 18+x purple socks out of 36+x in all, and the chance of purple is that purple count over the total count.

▸ Why?

Every sock in the drawer is just as likely to be the one drawn, so the chance of purple is the number of purple socks divided by the total number of socks.

Probability of equally-likely outcomesWhen every outcome is equally likely, an event's probability is its count of favorable outcomes divided by the total number of outcomes.

▸ Why?

The purple count after the addition is 18 + x — the 18 purple socks already there together with the x that were put in.

Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.

▸ Why?

The total count after the addition is 36 + x, since only purple socks joined the original 36 while the other colors were left alone.

▸ Why?

The green count stays 6 and the orange count stays 12, because no socks of those colors were added and adding none leaves a count unchanged.

Additive identity (zero)Adding zero adds nothing, so the number stays exactly the same.

▸ Why?

The whole drawer is all its color piles combined with none left out and none counted twice, so it grows from 36 to 36 + x when x purple socks are added.

Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.

▸ Why?

The right side is the given 60% rewritten as the plain fraction it stands for, namely 3/5.

▸ Why?

60% means 60 out of every hundred, which is the fraction 60/100.

A percent is parts per hundredp% means p out of every hundred, so p% of a quantity is (p/100) times it.

▸ Why?

60/100 and 3/5 are the same value, because dividing both the top and the bottom by 20 leaves a fraction's value unchanged.

Equivalent fractionsMultiplying (or dividing) a fraction's top and bottom by the same nonzero number leaves its value unchanged.

6.RP.A.3Guess And Check

3STEP 3

Test each answer choice

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

27/45 = (27 ÷ 9)/(45 ÷ 9) = 3/5 ✓ → (B)

4STEP 4

Cross-check another way

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 = 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!

Count the socks and convert the percent
Set up the fraction equation
Test each answer choice
Cross-check another way

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Count the socks and convert the percent

Set up the fraction equation

Test each answer choice

Cross-check another way

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