Difference between revisions of "2022 AIME II Problems/Problem 1"
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==Solution 1== | ==Solution 1== | ||
Let <math>x</math> be the number of people at the party before the bus arrives. We know that <math>x\equiv 0\pmod {12}</math>, as <math>\frac{5}{12}</math> of people at the party before the bus arrives are adults. Similarly, we know that <math>x + 50 \equiv 0 \pmod{25}</math>, as <math>\frac{11}{25}</math> of the people at the party are adults after the bus arrives. <math>x + 50 \equiv 0 \pmod{25}</math> can be reduced to <math>x \equiv 0 \pmod{25}</math>, and since we are looking for the minimum amount of people, <math>x</math> is <math>300</math>. That means there are <math>350</math> people at the party after the bus arrives, and thus there are <math>350 \cdot \frac{11}{25} = \boxed{154}</math> adults at the party. | Let <math>x</math> be the number of people at the party before the bus arrives. We know that <math>x\equiv 0\pmod {12}</math>, as <math>\frac{5}{12}</math> of people at the party before the bus arrives are adults. Similarly, we know that <math>x + 50 \equiv 0 \pmod{25}</math>, as <math>\frac{11}{25}</math> of the people at the party are adults after the bus arrives. <math>x + 50 \equiv 0 \pmod{25}</math> can be reduced to <math>x \equiv 0 \pmod{25}</math>, and since we are looking for the minimum amount of people, <math>x</math> is <math>300</math>. That means there are <math>350</math> people at the party after the bus arrives, and thus there are <math>350 \cdot \frac{11}{25} = \boxed{154}</math> adults at the party. | ||
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| + | ~eamo | ||
==See Also== | ==See Also== | ||
{{AIME box|year=2022|n=II|before=First Problem|num-a=2}} | {{AIME box|year=2022|n=II|before=First Problem|num-a=2}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 00:40, 18 February 2022
Problem
Adults made up 
of the crowd of people at a concert. After a bus carrying 
more people arrived, adults made up 
of the people at the concert. Find the minimum number of adults who could have been at the concert after the bus arrived.
Solution 1
Let 
be the number of people at the party before the bus arrives. We know that 
, as 
of people at the party before the bus arrives are adults. Similarly, we know that 
, as of the people at the party are adults after the bus arrives. can be reduced to 
, and since we are looking for the minimum amount of people, is 
. That means there are 
people at the party after the bus arrives, and thus there are 
adults at the party.
~eamo
See Also
| 2022 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by First Problem |
Followed by Problem 2 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
