Question 1132084
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This problem, and similar ones, are NOT useless.  Solving them requires logical reasoning, organized thinking, and useful math skills.<br>
Both of the responses from other tutors assumed the problem was to use only quarters, dimes, and nickels.  There is no mention in the statement of the problem about not using pennies, or half dollars.  But adding pennies and/or half dollars to the problem just makes the solution more tedious; the same problem solving skills are used if we just use quarters, dimes, and nickels.  So I will address the same problem.<br>
There is no need to list out all the combinations, as one tutor did successfully and the other tutor did unsuccessfully.  And if the problem were the harder problem with pennies and/or half dollars, you would not want to try to write out all the solutions.<br>
Instead, with only quarters, dimes, and nickels, you only need to look at each possible number of quarters and determine through logical analysis and simple arithmetic how many solutions there are with each number of quarters.<br>
For example, if you use 2 quarters, that is 50 cents; you have 30 cents more to make using dimes and nickels.  You can have any number of dimes from 0 to a maximum of 3; the nickels will make up anything that remains.  So with 2 quarters there are 4 solutions.<br>
The complete solution can then look like this:<br>
0 quarters  -->  80 cents left  -->  0 to 8 dimes  -->  9 solutions with 0 quarters
1 quarter  -->  55 cents left  -->  0 to 5 dimes  -->  6 solutions with 1 quarter
2 quarters  -->  30 cents left  -->  0 to 3 dimes  -->  4 solutions with 2 quarters
3 quarters  -->  5 cents left  --> 0 dimes  -->  1 solution with 3 quarters<br>
Total number of solutions: 9+6+4+1 = 20