bag contains 3 types of coins: nickels (worth 5 cents each), dimes (worth 10 cents each) and quarters (worth 25 cents each) in the ratio 5 : 3 : 2. If the total value of the dimes is 50 cents more than that of her nickels, then how many quarters does the bag contain?

Let's denote the number of nickels, dimes, and quarters in the bag as \( N \), \( D \), and \( Q \), respectively. According to the given problem, these coins are in the ratio 5:3:2.

### Step 1: Express the Number of Coins Using Ratios

Given the ratio of nickels to dimes to quarters is 5:3:2, we can introduce a variable \( x \) to represent the common multiplier for these ratios. Then:

\[
N = 5x, \quad D = 3x, \quad Q = 2x
\]

### Step 2: Set Up the Relationship Between the Values of Nickels and Dimes

The value of each nickel is 5 cents, and the value of each dime is 10 cents. Therefore, the total values of nickels and dimes are:

\[
\text{Value of nickels} = 5N = 5 \times 5x = 25x \quad \text{cents}
\]

\[
\text{Value of dimes} = 10D = 10 \times 3x = 30x \quad \text{cents}
\]

The problem states that the total value of the dimes is 50 cents more than that of the nickels. So, we have the equation:

\[
30x = 25x + 50
\]

### Step 3: Solve for \( x \)

Simplify the equation to solve for \( x \):

\[
30x - 25x = 50
\]

\[
5x = 50
\]

\[
x = 10
\]

### Step 4: Determine the Number of Quarters

Now that we have \( x = 10 \), we can find the number of quarters \( Q \):

\[
Q = 2x = 2 \times 10 = 20
\]

### Conclusion

The bag contains **20 quarters**.

Would you like further details, or do you have any questions?

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Here are 5 related questions to expand your understanding:

1. If the total value of the quarters was required, how would you calculate it?
2. How many nickels and dimes are in the bag?
3. What is the total value of all the coins in the bag?
4. How would the problem change if the value of the quarters was given instead?
5. What if the ratio of the coins were different, say 4:5:3? How would that affect the calculations?

**Tip:** Always start by identifying relationships and translating word problems into mathematical equations for clarity.

Learn how to solve a problem involving ratios and coin values with nickels, dimes, and quarters. Understand the relationship between coin counts and their respective values based on given ratios and conditions. Suitable for students in Grades 7-9.

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