There are some $25$ paise coins and some $50$ paise coins in a bag. The total number of coins is $150$ and the amount in the bag is Rs. $55$. Find the number of coins of each value in the bag.

(A) Suppose there are $x$ coins of $50$ paise and $y$ coins of $25$ paise in the bag.
The total number of coins is $x + y = 150$ ....... $(1)$
The value of $x$ coins of $50$ paise is $50x$ paise and the value of $y$ coins of $25$ paise is $25y$ paise.
The total amount is $50x + 25y$ paise.
Since the total amount is Rs. $55$,which is $5500$ paise,we have:
$50x + 25y = 5500$
Dividing the equation by $25$,we get:
$2x + y = 220$ ....... $(2)$
Subtracting equation $(1)$ from equation $(2)$:
$(2x + y) - (x + y) = 220 - 150$
$x = 70$
Substituting $x = 70$ in equation $(1)$:
$70 + y = 150$
$y = 80$
Thus,the number of $50$ paise coins is $70$ and the number of $25$ paise coins is $80$.