AMC 10 · 2009 · #18

Grade 7 rate-ratio

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

percentageset-partitionprinciple-of-inclusion-exclusion physical-representationcomplementary-counting ↑ Prerequisites: percentage

📏 Medium solution 💡 2 insights

Problem

At a summer camp, 60% of the campers play soccer and 30% of the campers swim. Also, 40% of the soccer players swim. Find, to the nearest whole percent, what percent of the campers who do not swim play soccer.

Pick an answer.

(A)

$\ 30\%$

(B)

$\ 40\%$

(C)

$\ 49\%$

(D)

$\ 51\%$

(E)

$\ 70\%$

AMC 10 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Venn Diagram

Two overlapping groups (soccer and swimming) split the camp into four regions: both, only one, or neither. A Venn diagram organizes those regions. To make the percents easy to handle, pretend the camp has exactly 100 campers so each percent becomes a headcount. The question asks about non-swimmers, so the last move is to change the denominator from the whole camp to just the non-swimmers.

1STEP 1

Assume 100 campers

Let the camp have 100 campers, so each percent becomes a count: 60 play soccer and 30 swim.

soccer = 60, swim = 30

2STEP 2

Count soccer players who swim

Take 40% of the 60 soccer players: 24 campers play soccer and swim.

0.40 × 60 = 24

3STEP 3

Soccer players who don't swim

Of the 60 soccer players, 24 also swim, so 36 play soccer but do not swim.

60 - 24 = 36

4STEP 4

Count all non-swimmers

30 campers swim, so the non-swimmers are the rest of the camp: 70.

100 - 30 = 70

5STEP 5

Take the percent among non-swimmers

Among the 70 non-swimmers, 36 play soccer, and $\frac{36}{70} \approx 51.43\%$ rounds to 51%, choice (D).

36/70 ≈ 0.5143 = 51.43% ≈ 51%

Answer

51%

The four Venn regions are 24 (soccer and swim), 36 (soccer only), 6 (swim only), and 34 (neither), and 24 + 36 + 6 + 34 = 100, so the counts fit the camp. Among the 70 non-swimmers, 36 out of 70 is just over half, so a value just above 50% like 51% makes sense; the near-miss choice 49% would be below half, which is wrong.

💡Key takeaway

When a question asks a percent of a smaller group, change the bottom of the fraction to that group's size before you divide.

Assume 100 campers
Count soccer players who swim
Soccer players who don't swim
Count all non-swimmers
Take the percent among non-swimmers

Assume 100 campers

Count soccer players who swim

Soccer players who don't swim

Count all non-swimmers

Take the percent among non-swimmers

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