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# Bansi has 3 times as many two-rupee coins as he has five-rupee coins. He has in all a sum of rupees 77, how many coins of each denomination does he have?

Last updated date: 19th Sep 2024
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Hint: We shall find the number of coins of each denomination by assuming the number of coins as two different variables. Then we will form a linear equation in two variables by equating the sum of all the money he has. Further, we will use the relation given between the number of coins of each denomination to convert it into one variable only and find our solution.

Complete step by step solution:
Let the number of two-rupee coins Bansi has be $x$ and let the number of five-rupee coins Bansi has be $y$.
Given that he has a sum of total rupees 77. Adding up the total value of coins, we can write:
$2x+5y=77$ …………. (1)
Now, we have formed a linear equation in 2-variables.
According to the problem, number of two-rupee coins $=3\times$ number of five-rupee coins
$\Rightarrow x=3y$
Substituting this value in (1), we have
\begin{align} & 2\left( 3y \right)+5y=77 \\ & \Rightarrow 6y+5y=77 \\ & \Rightarrow 11y=77 \\ \end{align}
Dividing both sides by 11, we get
$\Rightarrow \dfrac{11y}{11}=\dfrac{77}{11}$
$\Rightarrow y=7$
Thus, the total number of five-rupee coins is 7.
Using $x=3y$, we will find a number of two-rupee coins.
\begin{align} & \Rightarrow x=3\left( 7 \right) \\ & \Rightarrow x=21 \\ \end{align}
Thus, the number of two-rupee coins is 21.

Therefore, Bansi has 21 two-rupee coins and 7 five-rupee coins.

Note: In order to find the solution of a linear equation in two variables, we prefer sketching the graph of the equation so that the value of one variable at a particular value of the second variable can be easily marked and determined. For plotting the graph, we find two points lying on the line by putting the values of both the variables equal to zero one by one. Doing so, we get two points on the two axes which can be further joined to obtain the graph.