I have a total of $\ Rs. 300$ in coins of denominations $\ Rs. 1,\ Rs. 2$ and $\ Rs. 5$. The number of $\ Rs. 2$ coins is 3 times the number of $\ Rs. 5$ coins. The total number of coins is 160. How many coins of each denomination are with me?

Given :

The total number of coins $= 160$.

Sum total of all coins $= Rs. 300$.

Coins are in the denomination  $\ Rs. 1, \ Rs. 2,$ and $\ Rs. 5$. 

The number of $\ Rs. 2$ coins is 3 times the number of $\ Rs. 5$ coins.

To do :

We have to find the number of coins in each denomination.

Solution :

Let the number of Rs. 1 coins be x and the number of Rs. 5 coins be y.

The number of Rs. 2 coins is 3 times the number of Rs. 5 coins.

This implies,

The number of Rs. 2 coins $= 3y$.

Therefore,

$x+3y+y=160$

$x+4y=160$.................(i)

$[x(1)+3y(2)+y(5)] = Rs. 300$

$x+6y+5y=300$

$x+11y=300$..................(ii)

Subtract equation (i) from equation (ii).

$(x+11y)-(x+4y)=300-160$

$x-x+11y-4y=140$

$7y=140$

$y=\frac{140}{7}$

$y=20$

Substituting $y=20$ in equation (i),

$x+4y=160$

$x+4(20)=160$

$x+80=160$

$x=160-80$

$x=80$

Therefore,

The number of Rs. 1 coins is 80, Rs. 2 coins is 3(20)$=$60, and Rs. 5 coins is 20.