Let's break down this problem step by step.

Given:

1. Relationship between coins:

   • For every 2 nickels, there are 3 dimes.
   • For every 2 dimes, there are 5 quarters.

2. Total coins: 500 coins.

Let's use these relationships to set up equations.

Step 1: Set Up Variables

Let:

• $N$ = Number of nickels
• $D$ = Number of dimes
• $Q$ = Number of quarters

Step 2: Set Up Ratios Based on Given Relationships

1. Nickel to Dime Ratio: For every 2 nickels, there are 3 dimes. This gives us: $\frac{N}{D} = \frac{2}{3} \Rightarrow N = \frac{2}{3}D$

2. Dime to Quarter Ratio: For every 2 dimes, there are 5 quarters. This gives us: $\frac{D}{Q} = \frac{2}{5} \Rightarrow D = \frac{2}{5}Q$

Step 3: Express All Variables in Terms of Quarters

From the second equation, we have: $D = \frac{2}{5}Q$ Now, substitute $D$ in the equation for $N$: $N = \frac{2}{3} \times \frac{2}{5}Q = \frac{4}{15}Q$

Step 4: Set Up the Total Coin Equation

Since the total number of coins is 500, we can write: $N + D + Q = 500$ Substitute $N$ and $D$ in terms of $Q$: $\frac{4}{15}Q + \frac{2}{5}Q + Q = 500$

Step 5: Simplify the Equation

To combine terms, find a common denominator. The denominators are 15 and 5, so convert them all to a denominator of 15: $\frac{4}{15}Q + \frac{6}{15}Q + \frac{15}{15}Q = 500$ Combine the terms: $\frac{25}{15}Q = 500$ Simplify the fraction: $\frac{5}{3}Q = 500$ Multiply both sides by 3 to solve for $Q$: $5Q = 1500$ Now, divide by 5: $Q = 300$

Step 6: Find $D$ and $N$

Now that we know $Q = 300$, substitute back to find $D$ and $N$:

1. Dimes: $D = \frac{2}{5}Q = \frac{2}{5} \times 300 = 120$
2. Nickels: $N = \frac{4}{15}Q = \frac{4}{15} \times 300 = 80$

Step 7: Calculate the Total Value

• Nickels: 80 nickels at $0.05 each = $80 \times 0.05 = 4$ dollars
• Dimes: 120 dimes at $0.10 each = $120 \times 0.10 = 12$ dollars
• Quarters: 300 quarters at $0.25 each = $300 \times 0.25 = 75$ dollars

Total value: $4 + 12 + 75 = 91 \text{ dollars}$

Final Answers

• Nickels: 80
• Dimes: 120
• Quarters: 300
• Total value: $91

Would you like a detailed breakdown of any specific step, or do you have further questions?

Here are five related questions to expand on this problem:

1. How would the total coin count change if there were an additional type of coin?
2. What if the ratio of nickels to dimes changed to 1:1? How would that affect the distribution?
3. How would the total value change if each coin type had a different value?
4. What is the algebraic approach to solving the problem without substituting for quarters first?
5. Can we use matrix methods to solve systems of equations for more complex coin problems?

Tip: For ratio-based problems, always try expressing all variables in terms of a single variable to simplify calculations.