A piggy bank had 6 fewer quarters than dimes and twice as many dimes as nickels. If the total value was $21 and he only had nickels, dimes and quarters, how many of each coin was there?

Let the number of nickels be $ n $, the number of dimes be $ d $, and the number of quarters be $ q $.

From the problem, we have three key relationships:
1. There are 6 fewer quarters than dimes: 
   \[
   q = d - 6
   \]
2. There are twice as many dimes as nickels:
   \[
   d = 2n
   \]
3. The total value of the coins is $21. Since nickels are worth $0.05, dimes are worth $0.10, and quarters are worth $0.25, the value equation is:
   \[
   0.05n + 0.10d + 0.25q = 21
   \]

### Step 1: Substituting $ d = 2n $ and $ q = d - 6 $
From $ d = 2n $, substitute $ d $ into $ q = d - 6 $:
\[
q = 2n - 6
\]

Now substitute $ d = 2n $ and $ q = 2n - 6 $ into the value equation:
\[
0.05n + 0.10(2n) + 0.25(2n - 6) = 21
\]

### Step 2: Simplifying the equation
First, distribute the constants:
\[
0.05n + 0.20n + 0.25(2n - 6) = 21
\]
\[
0.05n + 0.20n + 0.50n - 1.50 = 21
\]
Combine like terms:
\[
0.75n - 1.50 = 21
\]

### Step 3: Solving for $ n $
Add 1.50 to both sides:
\[
0.75n = 22.50
\]
Now, divide by 0.75:
\[
n = \frac{22.50}{0.75} = 30
\]

### Step 4: Finding $ d $ and $ q $
Now that we know $ n = 30 $, substitute it into the equation for $ d $:
\[
d = 2n = 2(30) = 60
\]
Next, substitute $ n = 30 $ into the equation for $ q $:
\[
q = 2n - 6 = 2(30) - 6 = 60 - 6 = 54
\]

### Final Answer:
- Nickels: 30
- Dimes: 60
- Quarters: 54

The piggy bank contains 30 nickels, 60 dimes, and 54 quarters.

Would you like a detailed breakdown of the steps or have any questions?

Here are some related questions for further practice:
1. What if the total value was $25? How would that affect the number of coins?
2. What if the piggy bank also contained pennies, adding up to $22?
3. How would the problem change if there were 10 fewer quarters than dimes instead of 6?
4. Can you find the total number of coins in the piggy bank?
5. What is the total value of just the dimes and quarters in the bank?

**Tip:** When solving problems with relationships between variables, it's helpful to express everything in terms of one variable to simplify calculations.

Solve a piggy bank problem where there are 6 fewer quarters than dimes and twice as many dimes as nickels, with a total value of $21. Learn how to set up and solve the equations step-by-step. Suitable for Grades 8-10 algebra students.

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