We can solve this problem by setting up a system of equations based on the information provided.

Step 1: Define the variables

Let:

• $d$ be the number of dimes
• $q$ be the number of quarters

We know the following:

1. Jennifer has 21 coins in total: $d + q = 21$

2. The value of the coins can be expressed as:

   • The value of the dimes: $0.10d$ dollars
   • The value of the quarters: $0.25q$ dollars

3. If the dimes were quarters and the quarters were dimes, she would have $1.05 less. This means the new total value would be: $0.25d + 0.10q$ The total value of her current coins is: $0.10d + 0.25q$ The difference in value is $1.05: $(0.25d + 0.10q) = (0.10d + 0.25q) - 1.05$

Step 2: Set up the equation for the difference in value

Simplifying the equation: $0.25d + 0.10q = 0.10d + 0.25q - 1.05$ Now, move the terms involving $d$ and $q$ to one side: $0.25d - 0.10d = 0.25q - 0.10q - 1.05$ Simplifying both sides: $0.15d = 0.15q - 1.05$

Step 3: Solve the system of equations

We now have the system of equations:

1. $d + q = 21$
2. $0.15d = 0.15q - 1.05$

We can solve the second equation by dividing through by 0.15: $d = q - 7$

Step 4: Substitute into the first equation

Substitute $d = q - 7$ into the first equation $d + q = 21$: $(q - 7) + q = 21$ Simplifying: $2q - 7 = 21$ $2q = 28$ $q = 14$

Step 5: Find $d$

Now that we know $q = 14$, substitute it into the equation $d + q = 21$: $d + 14 = 21$ $d = 7$

Step 6: Answer the question

The problem asks for the result of subtracting the number of dimes from the number of quarters: $q - d = 14 - 7 = 7$

Thus, the difference between the number of quarters and dimes is 7.

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Would you like further details on the solution? Or do you have any questions?

Here are some related questions:

1. How would the total value change if the number of dimes and quarters were reversed?
2. If Jennifer had 30 coins instead of 21, how would the equation change?
3. What happens if the amount by which the value changes is different, say $1.50?
4. How could you solve this problem if the coins involved were nickels and pennies?
5. How can you use this approach to solve similar word problems involving coins?

Tip: When setting up equations for word problems, always define clear variables and write down all given conditions as mathematical expressions.