জয়দীপের উঠানে তিনটা কল আছে: $A$,$B$ আর $C$। সে যদি তিনটা কলই একসাথে খুলে দেয়, তাহলে তার পুলটা $100$ মিনিটে পরিপূর্ণ হয়ে যায়। সে যদি খালি $A$ আর $B$ কল একসাথে খুলে দেয়, তাহলে তার পুলটা $150$ মিনিটে পরিপূর্ণ হয়ে যায়। সে যদি খালি $B$ আর $C$ কলটা একসাথে খুলে দেয়, তাহলে তার পুলটা $200$ মিনিটে পরিপূর্ণ হয়ে যায়। সে যদি খালি $A$ আর $C$ কলটা একসাথে খুলে দেয়, তাহলে তার পুলটা পরিপূর্ণ হতে কতো সময় লাগবে?
Joydip's backyard has three faucets: $A$,$B$ and $C$. If he turns on all three of them, it takes $100$ minutes to fill the pool. If he turns on only $A$ and $B$, it takes $150$ minutes, and if he turns on only $B$ and $C$, it takes $200$ minutes. If he turns on only $A$ and $C$, how long will it take (in minutes) to fill the pool?
BdMO National 2021 Primary Problem 2
- Anindya Biswas
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"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
— John von Neumann
— John von Neumann
- Anindya Biswas
- Posts:264
- Joined:Fri Oct 02, 2020 8:51 pm
- Location:Magura, Bangladesh
- Contact:
Re: BdMO National 2021 Primary Problem 2
Let $t_A,t_B,t_C$ be the time (in minutes) is takes to fill the pool using faucets $A,B,C$ respectively.
So, in $1$ minute, $A,B,C$ fills the fraction of $\frac1{t_A},\frac1{t_B},\frac1{t_C}$ of the whole pool respectively.
Therefore, we have,
$\frac1{t_A}+\frac1{t_B}+\frac1{t_C}=\frac1{100}\cdots\cdots\cdots(1)$
$\frac1{t_A}+\frac1{t_B}=\frac1{150}\cdots\cdots\cdots(2)$
$\frac1{t_B}+\frac1{t_C}=\frac1{200}\cdots\cdots\cdots(3)$
From $2\times(1)-(2)-(3)$, we get:
$\frac1{t_A}+\frac1{t_C}=\frac1{50}-\frac1{150}-\frac1{200}$
$\Rightarrow\frac1{t_A}+\frac1{t_C}=\frac1{120}$
So, it takes exactly $120$ minutes to fill the pool using only $A$ and $C$.
So, in $1$ minute, $A,B,C$ fills the fraction of $\frac1{t_A},\frac1{t_B},\frac1{t_C}$ of the whole pool respectively.
Therefore, we have,
$\frac1{t_A}+\frac1{t_B}+\frac1{t_C}=\frac1{100}\cdots\cdots\cdots(1)$
$\frac1{t_A}+\frac1{t_B}=\frac1{150}\cdots\cdots\cdots(2)$
$\frac1{t_B}+\frac1{t_C}=\frac1{200}\cdots\cdots\cdots(3)$
From $2\times(1)-(2)-(3)$, we get:
$\frac1{t_A}+\frac1{t_C}=\frac1{50}-\frac1{150}-\frac1{200}$
$\Rightarrow\frac1{t_A}+\frac1{t_C}=\frac1{120}$
So, it takes exactly $120$ minutes to fill the pool using only $A$ and $C$.
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
— John von Neumann
— John von Neumann