Difference between revisions of "2022 AIME II Problems/Problem 1"

Revision as of 11:44, 9 March 2022

Problem

Adults made up $\frac5{12}$ of the crowd of people at a concert. After a bus carrying $50$ more people arrived, adults made up $\frac{11}{25}$ 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 $x$ be the number of people at the party before the bus arrives. We know that $x\equiv 0\pmod {12}$ , as $\frac{5}{12}$ of people at the party before the bus arrives are adults. Similarly, we know that $x + 50 \equiv 0 \pmod{25}$ , as $\frac{11}{25}$ of the people at the party are adults after the bus arrives. $x + 50 \equiv 0 \pmod{25}$ can be reduced to $x \equiv 0 \pmod{25}$ , and since we are looking for the minimum amount of people, $x$ is $300$ . That means there are $350$ people at the party after the bus arrives, and thus there are $350 \cdot \frac{11}{25} = \boxed{154}$ adults at the party.

~eamo

Solution 2 (Kind of lame)

Since at the beginning, adults make up $\frac{5}{12}$ of the concert, the amount of adults must be a multiple of 12.

Call the amount of people in the beginning $x$ .Then $x$ must be divisible by 12, in other words: $x$ must be a multiple of 12. Since after 50 more people arrived, adults make up $\frac{11}{25}$ of the concert, $x+50$ is a multiple of 25. This means $x+50$ must be a multiple of 5.

Notice that if a number is divisible by 5, it must end with a 0 or 5. Since 5 is impossible (obviously, since multiples of 12 end in 2, 4, 6, 8, 0,...), $x$ must end in 0.

Notice that the multiples of 12 that end in 0 are: 60, 120, 180, etc.. By trying out, you can clearly see that $x=300$ is the minimum number of people at the concert.

So therefore, after 50 more people arrive, there are $300+50=350$ people at the concert, and the number of adults is $350*\frac{11}{25}=154$ . Therefore the answer is $\boxed{154}$ .

I know this solution is kind of lame, but this is still pretty straightforward.

~hastapasta

Video Solution (Mathematical Dexterity)

https://www.youtube.com/watch?v=gBIxZ6SUr_w

@@ Line 11: / Line 11: @@
 Since at the beginning, adults make up <math>\frac{5}{12}</math> of the concert, the amount of adults must be a multiple of 12.
-Call the amount of people in the beginning <math>x</math>.Then <math>x</math> must be divisible by 12, in other words -- <math>x</math> must be a multiple of 12.
+Call the amount of people in the beginning <math>x</math>.Then <math>x</math> must be divisible by 12, in other words: <math>x</math> must be a multiple of 12.
 Since after 50 more people arrived, adults make up <math>\frac{11}{25}</math> of the concert, <math>x+50</math> is a multiple of 25. This means <math>x+50</math> must be a multiple of 5.
@@ Line 23: / Line 23: @@
 ~hastapasta
 ==Video Solution (Mathematical Dexterity)==
 https://www.youtube.com/watch?v=gBIxZ6SUr_w

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