Difference between revisions of "2020 CIME II Problems/Problem 3"

(Created page with "In a jar there are blue jelly beans and green jelly beans. Then, <math>15\%</math> of the blue jelly beans are removed and <math>40\%</math> of the green jelly beans are remov...")
 
 
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==Solution 1==
 
==Solution 1==
Suppose there are <math>x</math> jelly beans total at the beginning. Suppose further that there are <math>b</math> blue jelly beans and <math>x-b</math> green jelly beans. Then, after the removal, there will be <math>0.85b</math> blue jelly beans and <math>0.6x-0.6b</math> green jelly beans. Because the total number of jelly beans at the end is <math>80\%</math> of the starting number, we can create an equation: <cmath>0.6x+0.25b=0.8x</cmath> <cmath>0.2x=0.25b</cmath> <cmath>0.8x=b</cmath> This tells us there were originally <cmath>0.8x</cmath> blue jelly beans and <cmath>0.2x</cmath> green jelly beans at the beginning, so now there must be <cmath>0.68x</cmath> blue and <cmath>0.12x</cmath> green. The percent of the remaining jelly beans that are blue is <cmath>\frac{0.68x}{0.68x+0.12x}=\frac{68}{80}=\frac{85}{100},</cmath> so the answer is <math>\boxed{085}</math>.
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Suppose there are <math>x</math> jelly beans total at the beginning. Suppose further that there are <math>b</math> blue jelly beans and <math>x-b</math> green jelly beans. Then, after the removal, there will be <math>0.85b</math> blue jelly beans and <math>0.6x-0.6b</math> green jelly beans. Because the total number of jelly beans at the end is <math>80\%</math> of the starting number, we can create an equation: <cmath>0.6x+0.25b=0.8x</cmath> <cmath>0.2x=0.25b</cmath> <cmath>0.8x=b</cmath> This tells us there were originally <math>0.8x</math> blue jelly beans and <math>0.2x</math> green jelly beans at the beginning, so now there must be <math>0.68x</math> blue and <math>0.12x</math> green. The percent of the remaining jelly beans that are blue is <cmath>\frac{0.68x}{0.68x+0.12x}=\frac{68}{80}=\frac{85}{100},</cmath> so the answer is <math>\boxed{085}</math>.
  
 
==See also==
 
==See also==

Latest revision as of 21:02, 5 September 2020

In a jar there are blue jelly beans and green jelly beans. Then, $15\%$ of the blue jelly beans are removed and $40\%$ of the green jelly beans are removed. If afterwards the total number of jelly beans is $80\%$ of the original number of jelly beans, then determine the percent of the remaining jelly beans that are blue.

Solution 1

Suppose there are $x$ jelly beans total at the beginning. Suppose further that there are $b$ blue jelly beans and $x-b$ green jelly beans. Then, after the removal, there will be $0.85b$ blue jelly beans and $0.6x-0.6b$ green jelly beans. Because the total number of jelly beans at the end is $80\%$ of the starting number, we can create an equation: \[0.6x+0.25b=0.8x\] \[0.2x=0.25b\] \[0.8x=b\] This tells us there were originally $0.8x$ blue jelly beans and $0.2x$ green jelly beans at the beginning, so now there must be $0.68x$ blue and $0.12x$ green. The percent of the remaining jelly beans that are blue is \[\frac{0.68x}{0.68x+0.12x}=\frac{68}{80}=\frac{85}{100},\] so the answer is $\boxed{085}$.

See also

2020 CIME II (ProblemsAnswer KeyResources)
Preceded by
Problem 2
Followed by
Problem 4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All CIME Problems and Solutions

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