Without actually solving the​ problem, choose the correct solution by deciding which choice satisfies the given conditions.

Kesha has a total of 

170170 

​coins, all of which are either dimes or quarters. The total value of the coins is 

​$29.0029.00. 

Find the number of each type of coin.

To find the correct solution without solving the problem, let's define the variables and create the equations based on the conditions given:

### Step 1: Define Variables
- Let $ d $ be the number of dimes.
- Let $ q $ be the number of quarters.

### Step 2: Set Up Equations
We know that:
1. The total number of coins is 170, so:
   \[
   d + q = 170
   \]
2. The total value of the coins is $29.00. Since dimes are worth $0.10 and quarters are worth $0.25, the value equation will be:
   \[
   0.10d + 0.25q = 29.00
   \]

### Step 3: Check the Setup
The solution must satisfy both of these equations:
- The number of coins must add up to 170.
- The total value of the coins must be $29.00.

Without solving, the correct solution would involve finding the values of $ d $ and $ q $ that satisfy both equations simultaneously.

### Final Note:
To find the correct number of dimes and quarters, you would substitute different possible values for $ d $ or $ q $ that fulfill the conditions above. The pair of values that makes both the total number of coins and the total value correct would be the solution.

Would you like to explore this further or have any questions?

### Relative Questions:
1. How would the problem change if the total value of the coins was $35.00 instead?
2. What would the number of each type of coin be if there were only 150 coins in total?
3. How could this method be used to solve for a mixture of three types of coins (e.g., pennies, nickels, and dimes)?
4. If Kesha had 170 coins with a total value of $34.00, how would that affect the number of dimes and quarters?
5. What method can we use to efficiently solve systems of equations like this?

### Tip:
When dealing with coin problems, setting up equations based on both the number of coins and the total value is key to finding the solution.

Explore how to solve the problem of determining the number of dimes and quarters totaling $29.00 with a total of 170 coins. Learn the step-by-step approach involving algebraic concepts like linear equations and systems of equations. Suitable for students in Grades 7-9.

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