Question

$A,B,C,D$ and $E$ play a game of cards $A$ says to $B$, "If you give me three cards, you will have as many as $E$ has and if I give you three cards, you will have as many as $D$ has. $A$ and $B$ together have $10$ cards more than what $D$ and $E$ together have If $B$ has two cards more than what $C$ has and the total number of cards be $133$, how many cards does $B$ have?

Answer 1

Finding the number of cards $B$ have:

Let $A,B,C,D$ and $E$ have $a,b,c,d$ and $e$ number of cards respectively.

Now, $A$ says to $B$, "If you give me three cards, you will have as many as $E$ has and if I give you three cards, you will have as many as $D$.

$b–3=e$……….$\left(1\right)$

$b+3=d$….. … $\left(2\right)$

$A$ and $B$ together have $10$ cards more than what $D$ and $E$ together,

$a+b=d+e+10$ …….… $\left(3\right)$

$B$ has two cards more than $C$,

$b=c+2$ ………………..… $\left(4\right)$

Total number of cards is $133$,

$a+b+c+d+e=133$……..… $\left(5\right)$

From $\left(1\right)$ and $\left(2\right)$, we get,

$\begin{array}{rcl}b-e& =& d-b\\ 2b& =& d+e\end{array}$

$2b=d+e$……….. .. $\left(6\right)$

From $\left(3\right)$ and $\left(6\right)$

$\begin{array}{rcl}a+b& =& 2b+10\\ b& =& a-10\end{array}$

$a=b+10$…………..$\left(7\right)$

From $\left(4\right)$, $\left(5\right)$, $\left(6\right)$ and $\left(7\right)$, we get,

$\begin{array}{rcl}a+b+c+d+e& =& 133\\ b+10+b+b-2+2b& =& 133\\ 5b+8& =& 133\\ 5b& =& 133-8\\ 5b& =& 125\\ b& =& 25\end{array}$

Hence, $B$ has $25$ cards.