Asked by Allen
Can anyone help with this question?
Your backpack contains 2010 chocolate bars, which you are going to divide between your best friend and yourself. You think you are a nice person, so you will give your best friend more than zero chocolate bars. However, you have been craving chocolate recently, and so you must have at least twice as many chocolate bars as your friend after you finish dividing them. What is the difference between the maximum and minimum number of chocolate bars you can have?
Your backpack contains 2010 chocolate bars, which you are going to divide between your best friend and yourself. You think you are a nice person, so you will give your best friend more than zero chocolate bars. However, you have been craving chocolate recently, and so you must have at least twice as many chocolate bars as your friend after you finish dividing them. What is the difference between the maximum and minimum number of chocolate bars you can have?
Answers
Answered by
Steve
let's say you have exactly twice as many bars as he. So you have 1340 and he has 670
So, as long as you have at least 1340, but fewer than 2009 bars, he has some, and you have at least twice as many.
So, as long as you have at least 1340, but fewer than 2009 bars, he has some, and you have at least twice as many.
Answered by
Bob
Sorry, but it's wrong...
Answered by
Anonymous
669
Answered by
324234
it is 670
Answered by
Kaitlyn
it is actually 669
Answered by
669
The answer is my name
Answered by
Leon Zhang
I bet you got it from AoPS
Answered by
Anonymous
i did
Answered by
The answer is 669
669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669
Answered by
The answer is 669
Since your friend receives at least $1$ chocolate bar, the maximum number of chocolate bars you can get is $2010 - 1 = 2009$. If you receive $n$ chocolate bars, then your friend gets $2010-n$ chocolate bars. Since you must have at least twice as many chocolate bars as your friend, we have the inequality\begin{align*}
n &\ge 2(2010-n) \\
\Rightarrow\qquad n &\ge 4020-2n \\
\Rightarrow\qquad 3n &\ge 4020 \\
\Rightarrow\qquad n &\ge 1340.
\end{align*}Therefore, you must keep a minimum of $1340$ chocolate bars. Therefore, the difference between the maximum and minimum number of chocolate bars you can have is $2009 - 1340 = \boxed{669}$.
n &\ge 2(2010-n) \\
\Rightarrow\qquad n &\ge 4020-2n \\
\Rightarrow\qquad 3n &\ge 4020 \\
\Rightarrow\qquad n &\ge 1340.
\end{align*}Therefore, you must keep a minimum of $1340$ chocolate bars. Therefore, the difference between the maximum and minimum number of chocolate bars you can have is $2009 - 1340 = \boxed{669}$.
Answered by
Responder
I got this from AoPs and thats why i searched it up