AMC 8 · 2010 · #20

Grade 7 countinglogic

Archetype Overlap / Complement Counting └ Sub-archetype Two-Attribute Table: Fill or Optimize a Cell

fraction-arithmeticlcmset-partitioncomplementary-counting complementary-countingidentify-subproblems ↑ Prerequisites: fraction-arithmeticlcm

📏 Medium solution 💡 4 insights

Problem

In a room, exactly

\frac{2}{5}

of the people are wearing gloves and exactly

\frac{3}{4}

of the people are wearing hats. Both fractions must come out to whole numbers of people. Over all room sizes that work, what is the smallest possible number of people who are wearing both a hat and gloves?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Venn Diagram

The trigger word both points straight at Tool #12 (Venn Diagram): draw a gloves circle and a hats circle, and the answer lives in the overlap. Tool #7 (Identify Subproblems) splits the work into two clean halves — first pin down the smallest legal room size (an LCM question), then count the overlap inside that room. Tool #11 (Try Extreme Cases) is what "minimum" means here: to make the overlap as small as possible, push the two circles apart as far as the room will allow, i.e. fill the room with "glove-only" and "hat-only" people first and only force the leftover into the overlap.

1STEP 1

Find the smallest room size making both fractions whole: the LCM of 5 and 4 is 20.

T_min = lcm(5, 4) = 20

2STEP 2

With 20 people, gloves = $\frac{2}{5}$ × 20 = 8 and hats = $\frac{3}{4}$ × 20 = 15.

|G| =

\frac{2}{5}

× 20 = 8, |H| =

\frac{3}{4}

× 20 = 15

3STEP 3

Draw a gloves circle (8) and hats circle (15); label the overlap x, so the four regions sum to 20.

(8 - x) + x + (15 - x) + (neither) = 20

4STEP 4

To minimize the overlap, set "neither" = 0: 23 - x = 20, so x = 3 people must wear both.

(8 - x) + x + (15 - x) + 0 = 20 → 23 - x = 20 → x = 3

5STEP 5

The forced minimum matches (A); bigger rooms (multiples of 20) scale every region equally, so none beats x = 3.

x_min = 3 → (A)

Answer

Sanity check the Venn diagram with x = 3: glove-only = 8 - 3 = 5, both = 3, hat-only = 15 - 3 = 12, neither = 0. Sum = 5 + 3 + 12 + 0 = 20. Gloves total: 5 + 3 = 8 ✓. Hats total: 3 + 12 = 15 ✓. Trying x = 2 would force the union to be 8 + 15 - 2 = 21 ＞ 20, which is impossible — so x = 3 really is the smallest legal value.

💡Key takeaway

Once the Venn diagram is drawn, the minimum "both" overlap is just (gloves) + (hats) - (room total) — a Grade 7 multi-step reasoning move on top of Grade 5–6 fractions and LCM.

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