AMC 10 · 2009 · #18
Grade 7 rate-ratioAt Jefferson Summer Camp, 60% of the children play soccer, 30% of the children swim, and 40% of the soccer players swim. To the nearest whole percent, what percent of the non-swimmers play soccer?
Pick an answer.
AMC 10 2009 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: 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.
Givens: 60% of the campers play soccer.; 30% of the campers swim.; 40% of the soccer players also swim.
Unknowns: Among the campers who do not swim, the percent who play soccer, rounded to the nearest whole percent.
Understand
Restated: 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.
Givens: 60% of the campers play soccer.; 30% of the campers swim.; 40% of the soccer players also swim.
Plan
Primary tool: #12 Draw a Venn Diagram
Secondary: #9 Solve an Easier Related Problem, #16 Change Focus / Count the Complement
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.
Execute — Answer: D
6.RP.A.3 Step 1 Assume 100 campers
- Pick a total that makes the percents into counts.
- If there are 100 campers, then 60 play soccer and 30 swim, because 60% and 30% of 100 are just 60 and 30.
💡 A percent is a count per 100, so choosing 100 campers turns every percent straight into a number of kids.
6.RP.A.3 Step 2 Count soccer players who swim
- 40% of the soccer players swim.
- There are 60 soccer players, and 40% of 60 is 24.
- So 24 campers play soccer and swim.
💡 The 40% is measured inside the soccer group, so take it of 60, not of 100.
7.NS.A.3 Step 3 Soccer players who don't swim
- Of the 60 soccer players, 24 also swim, so the rest do not.
- That leaves 60 minus 24, which is 36 campers who play soccer but do not swim.
💡 Every soccer player either swims or doesn't, so subtracting the swimmers leaves the non-swimmers.
6.RP.A.3 Step 4 Count all non-swimmers
30 campers swim, so the ones who do not swim are the rest of the camp: 100 minus 30, which is 70 campers.
💡 The non-swimmers are the complement of the swimmers, so subtract them from the whole camp.
7.RP.A.3 Step 5 Take the percent among non-swimmers
- Now look only at the 70 non-swimmers.
- Among them, 36 play soccer.
- As a percent that is 36 divided by 70, which is about 0.5143, or 51.43%.
- Rounded to the nearest whole percent, that is 51%.
- The answer is (D).
💡 The question changes the denominator from the whole camp to just the non-swimmers, so divide by 70, not 100.
6.RP.A.3 Pick a total that makes the percents into counts. If there are 100 campers, then 6.RP.A.3 40% of the soccer players swim. There are 60 soccer players, and 40% of 60 is 24 7.NS.A.3 Of the 60 soccer players, 24 also swim, so the rest do not. That leaves 60 minus 6.RP.A.3 30 campers swim, so the ones who do not swim are the rest of the camp: 100 minus 7.RP.A.3 Now look only at the 70 non-swimmers. Among them, 36 play soccer. As a percent t Review
Reasonableness: 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.
Alternative: Skip the diagram and work in percents of the whole camp directly: soccer-and-swim is 40% of 60% = 24%, so soccer-only is 60% - 24% = 36%, and non-swimmers are 100% - 30% = 70%. Then 36%/70% = 36/70 ≈ 51%.
CCSS standards used (min grade 7)
6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems (Treating each percent as a count per 100 and finding 40% of the 60 soccer players.)7.NS.A.3Solve real-world problems involving the four operations with rational numbers (Subtracting to find the soccer players who do not swim and the campers who do not swim.)7.RP.A.3Use proportional relationships to solve multi-step ratio and percent problems (Converting the ratio 36/70 into a percent and rounding to the nearest whole percent.)
⭐ When a question asks a percent of a smaller group, change the bottom of the fraction to that group's size before you divide.
⭐ When a question asks a percent of a smaller group, change the bottom of the fraction to that group's size before you divide.
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