We can solve this problem by setting up a system of equations based on the information provided. Let's define variables for the number of each type of coin:

• Let $n$ represent the number of nickels.
• Let $d$ represent the number of dimes.
• Let $q$ represent the number of quarters.

Step 1: Translate the problem into equations

1. The total value of the coins is $5.90, which we will convert to cents: $5.90 \times 100 = 590$ cents.

   • The value of each nickel is 5 cents, so the total value of the nickels is $5n$.
   • The value of each dime is 10 cents, so the total value of the dimes is $10d$.
   • The value of each quarter is 25 cents, so the total value of the quarters is $25q$.

   Thus, the equation for the total value is: $5n + 10d + 25q = 590$

2. The problem states that there are 3 times as many quarters as nickels: $q = 3n$

3. The problem also says there are 14 more dimes than nickels: $d = n + 14$

Step 2: Substitute the values from equations 2 and 3 into equation 1

Now, we substitute $q = 3n$ and $d = n + 14$ into the first equation: $5n + 10(n + 14) + 25(3n) = 590$

Step 3: Simplify and solve

Expand the equation: $5n + 10n + 140 + 75n = 590$ Combine like terms: $90n + 140 = 590$ Subtract 140 from both sides: $90n = 450$ Solve for $n$: $n = \frac{450}{90} = 5$

Step 4: Calculate the number of dimes and quarters

Now that we know $n = 5$ (there are 5 nickels), we can use the other equations to find $d$ and $q$.

• For the dimes: $d = n + 14 = 5 + 14 = 19$
• For the quarters: $q = 3n = 3 \times 5 = 15$

Final Answer:

• Nickels: 5
• Dimes: 19
• Quarters: 15

Verification:

• The value of 5 nickels is $5 \times 5 = 25$ cents.
• The value of 19 dimes is $19 \times 10 = 190$ cents.
• The value of 15 quarters is $15 \times 25 = 375$ cents.
• Total: $25 + 190 + 375 = 590$ cents, which is $5.90.

Thus, the solution is correct!

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Do you need more details or have any questions?

Here are some related questions:

1. How would the solution change if there were twice as many dimes as nickels?
2. What would the total value be if the number of quarters increased by 5?
3. How would the system of equations change if only the nickels and dimes totaled $5.90?
4. Can you determine the number of coins if the total amount was $6.50 instead of $5.90?
5. What would the impact be if there were equal numbers of nickels, dimes, and quarters?

Tip: Always verify the solution by recalculating the total value from each step.