Diego is collecting dimes and nickels in a jar. he has collected $22.5 so far. the relationship between the numbers of dimes and nickels, and the amount of money in dollars is represented by the equation o.10d+0.05=22.25. select all the values (d,n) that could be solutions to the equation.

It seems there might be a small mistake in the equation you provided. The correct equation representing the relationship between dimes, nickels, and the total amount collected should be:

\[
0.10d + 0.05n = 22.50
\]

where:
- $ d $ is the number of dimes,
- $ n $ is the number of nickels,
- $ 0.10d $ represents the dollar amount from dimes,
- $ 0.05n $ represents the dollar amount from nickels.

Now, let's solve this equation and find possible values of $ d $ (dimes) and $ n $ (nickels).

First, since both $ d $ and $ n $ must be whole numbers, we'll look for combinations of $ d $ and $ n $ that satisfy the equation.

\[
0.10d + 0.05n = 22.50
\]

Multiplying the entire equation by 100 to eliminate decimals:

\[
10d + 5n = 2250
\]

This simplifies the problem to finding pairs of $ d $ and $ n $ that satisfy this equation. We can rewrite it as:

\[
2d + n = 450
\]

We now need integer solutions where $ d \geq 0 $ and $ n \geq 0 $. Let's explore possible values of $ d $ and solve for $ n $:

For each value of $ d $, we compute $ n = 450 - 2d $. Since $ n $ must be non-negative, $ d $ must satisfy $ 2d \leq 450 $, or $ d \leq 225 $.

We can now compute a few values of $ d $ and $ n $:

- $ d = 0 $, then $ n = 450 $
- $ d = 50 $, then $ n = 350 $
- $ d = 100 $, then $ n = 250 $
- $ d = 150 $, then $ n = 150 $
- $ d = 200 $, then $ n = 50 $
- $ d = 225 $, then $ n = 0 $

So, the valid pairs of $ (d, n) $ that satisfy the equation are:

\[
(0, 450), (50, 350), (100, 250), (150, 150), (200, 50), (225, 0)
\]

### Would you like more details, or do you have any questions?

Here are 5 related questions to deepen your understanding:
1. What if the total collected amount was $30.00? How would that change the equation and the solution?
2. How can we verify the solution pairs satisfy the original equation?
3. What if Diego collected only dimes? How many dimes would he need?
4. What would happen if Diego's total was an odd number, like $22.75? Could a solution exist?
5. Can we graph this equation? What would the graph look like?

**Tip:** Always check if simplifying an equation (like removing decimals) helps solve the problem more easily.

Diego is collecting dimes and nickels in a jar. He has collected $22.50 so far. The relationship between the numbers of dimes and nickels, and the amount of money in dollars is represented by the equation 0.10d + 0.05n = 22.50. Select all the values (d, n) that could be solutions to the equation.

Learn how to solve the linear equation representing Diego's collection of dimes and nickels, with the total amount of $22.50. This example covers solving systems of equations and determining valid (d, n) pairs for Grades 6-8.

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