Question

The quarters, buckled, and dimes totaled 20 and their value was $2.05. How many kind were there if there were 3 times as many dimes as quarters

Answer 1

We assume the question is intended to say there are a total of 20 quarters, nickels, and dimes, the value of which is $2.05. The number of dimes is 3 times the number of quarters.

Let q, n, d represent the numbers of quarters, nickels, and dimes, respectively. Then you have three equations in the unknowns:
  q +n +d = 20
  25q +5n +10d = 205
  -3q +d = 0

A calculator can give the solution to this matrix equation in short order. It is
  q = 3
  n = 8
  d = 9

There are 3 quarters, 8 nickels, and 9 dimes.

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If you use the third equation to write an expression for d, you can substitute that into the other two equations.
  q +n +3q = 20
  25q +5n +10(3q) = 205
Simplifying, these are
  4q +n = 20
  11q +n = 41
Subtracting the first from the second, we find
  7q = 21
  q = 3
From there, it is easy to find d=9, n=8.