Difference between revisions of "2009 AMC 10A Problems/Problem 18"
Williamhu888 (talk | contribs) m (→Solution 1) |
m (→Solution 3) |
||
| (10 intermediate revisions by 6 users not shown) | |||
| Line 1: | Line 1: | ||
== Problem == | == Problem == | ||
| − | |||
At Jefferson Summer Camp, <math>60\%</math> of the children play soccer, <math>30\%</math> of the children swim, and <math>40\%</math> of the soccer players swim. To the nearest whole percent, what percent of the non-swimmers play soccer? | At Jefferson Summer Camp, <math>60\%</math> of the children play soccer, <math>30\%</math> of the children swim, and <math>40\%</math> of the soccer players swim. To the nearest whole percent, what percent of the non-swimmers play soccer? | ||
| Line 15: | Line 14: | ||
</math> | </math> | ||
| + | == Solutions == | ||
=== Solution 1 === | === Solution 1 === | ||
| − | |||
Out of the soccer players, <math>40\%</math> swim. As the soccer players are <math>60\%</math> of the whole, the swimming soccer players are <math>0.4 \cdot 0.6 = 0.24 = 24\%</math> of all children. | Out of the soccer players, <math>40\%</math> swim. As the soccer players are <math>60\%</math> of the whole, the swimming soccer players are <math>0.4 \cdot 0.6 = 0.24 = 24\%</math> of all children. | ||
| Line 31: | Line 30: | ||
* <math>34\%</math>: swimming no, soccer no | * <math>34\%</math>: swimming no, soccer no | ||
| − | Now we can compute the answer. Out of all children, <math>70\%</math> are non-swimmers, and again out of all children <math>36\%</math> are non-swimmers that play soccer. Hence the | + | Now we can compute the answer. Out of all children, <math>70\%</math> are non-swimmers, and again out of all children <math>36\%</math> are non-swimmers that play soccer. Hence the percent of non-swimmers that play soccer is <math>\frac{36}{70} \approx 51\% \Rightarrow \boxed{D}</math>. |
=== Solution 2 === | === Solution 2 === | ||
| − | + | Let us set the total number of children as <math>100</math>. So <math>60</math> children play soccer, <math>30</math> swim, and <math>0.4\times60=24</math> play soccer and swim. | |
| − | Let us set | ||
Thus, <math>60-24=36</math> children only play soccer. | Thus, <math>60-24=36</math> children only play soccer. | ||
| Line 43: | Line 41: | ||
Our denominator is simply <math>100-\text{Swimmers}=100-30=70</math> | Our denominator is simply <math>100-\text{Swimmers}=100-30=70</math> | ||
| − | And so we get <math>\frac{36}{70}</math> which is roughly <math>51.4=\boxed{\ | + | And so we get <math>\frac{36}{70}</math> which is roughly <math>51.4\% \Rightarrow \boxed{\text{D}}</math> |
| + | |||
| + | ==Solution 3== | ||
| + | WLOG, let the total number of students be <math>100</math>. Draw a venn diagram with 2 circles encompassing these 4 regions: | ||
| + | |||
| + | |||
| + | Non-soccer players, non-swimmers: 34 people | ||
| + | |||
| + | Soccer players, non-swimmers: 36 people | ||
| + | |||
| + | Soccer players, swimmers: 24 people | ||
| + | |||
| + | Non-soccer players, swimmers: 6 people. | ||
| + | |||
| + | |||
| + | Hence the answer is <math>\frac{36}{70}=\frac{18}{35}</math>. We know this is a little bit larger than <math>\frac 12</math> because <math>\frac{17.5}{35}=\frac 12</math>. <math>\boxed{\textbf{(D) } 51\%}</math> | ||
| + | |||
| + | ~BakedPotato66 | ||
| + | |||
| + | ==Video Solution== | ||
| + | https://youtu.be/Ahei7_BKK9Q | ||
| + | |||
| + | ~savannahsolver | ||
== See Also == | == See Also == | ||
{{AMC10 box|year=2009|ab=A|num-b=17|num-a=19}} | {{AMC10 box|year=2009|ab=A|num-b=17|num-a=19}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 09:29, 2 August 2021
Contents
Problem
At Jefferson Summer Camp, 
of the children play soccer, 
of the children swim, and 
of the soccer players swim. To the nearest whole percent, what percent of the non-swimmers play soccer?
Solutions
Solution 1
Out of the soccer players, swim. As the soccer players are of the whole, the swimming soccer players are 
of all children.
The non-swimming soccer players then form 
of all the children.
Out of all the children, swim. We know that 
of all the children swim and play soccer, hence 
of all the children swim and don't play soccer.
Finally, we know that 
of all the children are non-swimmers. And as 
of all the children do not swim but play soccer, 
of all the children do not engage in any activity.
A quick summary of what we found out:
: swimming yes, soccer yes
: swimming no, soccer yes
: swimming yes, soccer no
: swimming no, soccer no
: swimming yes, soccer no
: swimming no, soccer noNow we can compute the answer. Out of all children, are non-swimmers, and again out of all children are non-swimmers that play soccer. Hence the percent of non-swimmers that play soccer is 
.
Solution 2
Let us set the total number of children as 
. So 
children play soccer, 
swim, and 
play soccer and swim.
Thus, 
children only play soccer.
So our numerator is 
.
Our denominator is simply 
And so we get 
which is roughly 
Solution 3
WLOG, let the total number of students be . Draw a venn diagram with 2 circles encompassing these 4 regions:
Non-soccer players, non-swimmers: 34 people
Soccer players, non-swimmers: 36 people
Soccer players, swimmers: 24 people
Non-soccer players, swimmers: 6 people.
Hence the answer is 
. We know this is a little bit larger than 
because 
. 
~BakedPotato66
Video Solution
~savannahsolver
See Also
| 2009 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 17 |
Followed by Problem 19 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
