Daniel takes pleasure in experimenting with various kinds of toys. He bought three distinct types of toys for $25 in total. Furthermore, he possesses a toy that costs $20 more than three times the combined price of the other two toys. It's revealed that the total cost of the three toys multiplied by five, amounts to $125.
Find how much is the cost of each toy. Explain the answer clearly.

Let's solve this problem step by step by setting up equations based on the information provided.

### Step 1: Define Variables
Let the costs of the three distinct types of toys be $ x $, $ y $, and $ z $.

- $ x $: Cost of the first toy
- $ y $: Cost of the second toy
- $ z $: Cost of the third toy

### Step 2: Set Up the Equations
Based on the problem, we have the following information:

1. **Total cost of the three toys:**
   \[
   x + y + z = 25
   \]

2. **The cost of the third toy:**
   $ z $ is $20 more than three times the combined price of the other two toys:
   \[
   z = 20 + 3(x + y)
   \]

3. **Total cost multiplied by five:**
   The total cost of the three toys multiplied by five is $125:
   \[
   5(x + y + z) = 125
   \]
   Simplifying this equation:
   \[
   x + y + z = \frac{125}{5} = 25
   \]
   This is consistent with our first equation.

### Step 3: Substitute and Solve
We now have two key equations:
1. $ x + y + z = 25 $
2. $ z = 20 + 3(x + y) $

Let's substitute the expression for $ z $ from the second equation into the first equation:

\[
x + y + (20 + 3(x + y)) = 25
\]

Simplify the equation:
\[
x + y + 20 + 3x + 3y = 25
\]
\[
4x + 4y + 20 = 25
\]
\[
4x + 4y = 5
\]

### Step 4: Simplify Further
Divide the equation by 4:
\[
x + y = \frac{5}{4} = 1.25
\]

Now, substitute $ x + y = 1.25 $ back into the equation for $ z $:
\[
z = 20 + 3(1.25)
\]
\[
z = 20 + 3.75 = 23.75
\]

### Step 5: Conclusion
So, the costs of the toys are:

- $ x + y = 1.25 $
- $ z = 23.75 $

Since $ x + y = 1.25 $, there are infinitely many possibilities for $ x $ and $ y $ as long as their sum equals 1.25. For example:

- If $ x = 0.25 $, then $ y = 1.00 $
- If $ x = 0.75 $, then $ y = 0.50 $

Thus, the costs of the toys could be $ 0.25 $, $ 1.00 $, and $ 23.75 $ or any other combination of $ x $ and $ y $ that satisfies $ x + y = 1.25 $.

### Summary:
- $ x $ and $ y $ are any values that sum to $ 1.25 $.
- $ z = 23.75 $.

Would you like more details or have any questions? 

Here are some related questions:
1. What if the sum of $ x + y $ were different?
2. How would the solution change if $ z $ were only $15 more than twice the sum of $ x $ and $ y $?
3. Can you find a specific solution for $ x $ and $ y $ if $ x = 0.5 $?
4. What happens if $ x + y + z $ had to be $30 instead of $25?
5. How would the equations change if $ z $ were dependent on the difference rather than the sum of $ x $ and $ y $?
6. What if there were four toys instead of three?
7. Could the solution be represented graphically? How?
8. How would rounding affect the solutions for $ x $ and $ y $?

**Tip:** When dealing with equations involving sums, always check if the equation can be simplified early on to reduce complexity.

Learn how to solve a problem involving the costs of different toys. This problem covers algebraic concepts such as solving systems of linear equations. Suitable for students in Grades 7-9.

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