. Rachel has 15 coins with a value of $2.85. If the coins are either dimes or quarters, how many of each coin does she have? Write and solve using a system of equations.
d = dimes
q = quarters
Equations:
10d + 25q = 285
d + q = 15
Solve by substitution or elimination
Substitution:
10d + 25q = 285
d = 15 - q
Elimination:
10d + 25q = 285
-10(d + q) = -10(15)
To solve this problem using a system of equations, let's start by assigning variables to the unknown quantities. Let's say that Rachel has "d" dimes and "q" quarters.
We can then form two equations based on the given information:
1. The first equation represents the number of coins:
d + q = 15 (since Rachel has a total of 15 coins)
2. The second equation represents the total value of the coins:
0.10d + 0.25q = 2.85 (since each dime has a value of $0.10 and each quarter has a value of $0.25)
Now, we have a system of equations that we can solve to find the values of "d" and "q."
To solve this system, we can use the method of substitution or elimination. Let's use the method of substitution:
1. Rearrange equation 1 to solve for "d":
d = 15 - q
2. Substitute this value for "d" in equation 2:
0.10(15 - q) + 0.25q = 2.85
3. Simplify the equation:
1.50 - 0.10q + 0.25q = 2.85
4. Combine like terms:
0.15q = 1.35
5. Divide both sides of the equation by 0.15:
q = 9
6. Substitute the value of "q" back into equation 1 to find "d":
d + 9 = 15
d = 15 - 9
d = 6
Therefore, Rachel has 6 dimes (d = 6) and 9 quarters (q = 9).