Paula’s purse contains twice as many dimes as quarters and three fewer nickels than dimes. The total of these coins is $3.15. How many dimes are there?

Let's denote the number of quarters as $ q $, the number of dimes as $ d $, and the number of nickels as $ n $.

According to the problem, we have the following relationships:

1. **Twice as many dimes as quarters:**  
   \[
   d = 2q
   \]
   
2. **Three fewer nickels than dimes:**  
   \[
   n = d - 3
   \]

3. **Total value of the coins is $3.15:**  
   The value equation can be expressed in cents as:
   \[
   25q + 10d + 5n = 315
   \]

Now, let's substitute the expressions for $ d $ and $ n $ in terms of $ q $ into the value equation.

First, substitute $ d = 2q $ into $ n = d - 3 $:
\[
n = 2q - 3
\]

Now substitute $ d = 2q $ and $ n = 2q - 3 $ into the value equation:
\[
25q + 10(2q) + 5(2q - 3) = 315
\]

Simplify and solve for $ q $:
\[
25q + 20q + 10q - 15 = 315
\]
\[
55q - 15 = 315
\]
\[
55q = 330
\]
\[
q = 6
\]

Now that we know $ q = 6 $, we can find $ d $ and $ n $:

\[
d = 2q = 2(6) = 12
\]
\[
n = 2q - 3 = 2(6) - 3 = 12 - 3 = 9
\]

Thus, the number of dimes is **12**.

Would you like more details on this solution or have any other questions?

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Here are 5 related questions for further practice:

1. If Paula had 4 more dimes, what would be the new total value of the coins?
2. What is the total number of coins Paula has in her purse?
3. If Paula added 2 quarters, how much would the total value of her coins increase?
4. How many more dimes than nickels does Paula have?
5. If Paula had an equal number of nickels and quarters, what would be the new total value?

**Tip:** Always convert the total value into cents when working with coin problems to avoid confusion with decimals.

Learn how to solve Paula's purse problem involving quarters, dimes, and nickels. Follow a step-by-step solution to find the number of dimes, understanding key concepts such as linear equations and system of equations. Suitable for students in Grades 7-9.

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