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?

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Solution

## 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.

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