Let's set the following variables:
x= number of quarters
y= number of dimes
The question states there are 30 coins in total, thus:
x+y=30 [1]
Since each quarter is worth $0.25 and each dime is worth $0.10, the total money on the table is:
0.25x+0.10y=5.10 [2]
The equations [1] and [2] form a system that can be solved in several ways. Let's do it by the substitution method
From equation [1], we solve for x:
x=30-y
Now we substitute x into equation [2]:
0.25(30-y)+0.10y=5.10
Operate the parentheses:
7.5-0.25y+0.10y=5.10
Joining like terms:
-0.15y=5.10-7.5
Operating again:
-0.15y=-2.4
Solving:
y=(-2.4)/(-0.15)=16
Now we use this value to find x:
x=30-y=30-16=14
Thus, there are 14 quarters and 16 dimes