A woman has 17 coins in her pocket, all of which are dimes and quarters. If the total value of the coins is $2.90, how many dimes and how many quarters does she have?

To solve this problem, let’s break it down step by step.

Let’s denote:

• D = number of dimes
• Q = number of quarters

We know two things:

1. The total number of coins is 17.
2. The total value of the coins is $2.90.

This gives us the following system of equations:

1. D + Q = 17
2. 0.10D + 0.25Q = 2.90

Let’s solve the first equation for D:

D = 17 – Q

Now, substitute this expression for D into the second equation:

0.10(17 – Q) + 0.25Q = 2.90

Simplify the equation:

1.70 – 0.10Q + 0.25Q = 2.90

Combine like terms:

1.70 + 0.15Q = 2.90

Subtract 1.70 from both sides:

0.15Q = 1.20

Divide both sides by 0.15:

Q = 8

Now, substitute Q = 8 back into the first equation to find D:

D + 8 = 17

D = 9

So, the woman has 9 dimes and 8 quarters.