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
X dimes.
(15-X)Qtrs.
0.10x + 0.25(15-x) = $2.85
10x + 25(15-x) = 285
10x + 375-25x = 285
-15x = 285-375 = -90
X = 6
15-X = 15-6 = 9
Let's assume Rachel has x number of dimes and y number of quarters.
The value of x dimes is 10x cents.
The value of y quarters is 25y cents.
According to the given information, Rachel has a total of 15 coins, so x + y = 15 ------(equation 1)
Also, the total value of the coins is $2.85, which is equal to 285 cents. So, the equation representing the total value is 10x + 25y = 285 ------(equation 2)
We can solve this system of equations by substitution or elimination method.
Let's solve it using the elimination method:
Multiply equation 1 by 10 to make the x coefficients the same:
10x + 10y = 150 ------(equation 3)
Subtract equation 3 from equation 2 to eliminate x:
(10x + 25y) - (10x + 10y) = 285 - 150
15y = 135
y = 9
Substitute the value of y into equation 1:
x + 9 = 15
x = 15 - 9
x = 6
Therefore, Rachel has 6 dimes and 9 quarters.
To solve this problem using a system of equations, let's use the variables "d" for the number of dimes and "q" for the number of quarters Rachel has. We need to set up two equations based on the given information.
1. The total number of coins:
d + q = 15
2. The total value of the coins:
0.10d + 0.25q = 2.85
Now we have a system of two equations. To solve it, we can use the method of substitution or elimination.
Let's solve it using the method of substitution:
From the first equation, we have:
d = 15 - q
We can substitute this value of d into the second equation:
0.10(15 - q) + 0.25q = 2.85
1.5 - 0.10q + 0.25q = 2.85
Combine like terms:
0.15q + 1.5 = 2.85
Subtract 1.5 from both sides:
0.15q = 2.85 - 1.5
0.15q = 1.35
Now divide both sides by 0.15:
q = 1.35 / 0.15
q = 9
So, Rachel has 9 quarters.
To find the number of dimes, we can substitute the value of q back into the first equation:
d + 9 = 15
Subtract 9 from both sides:
d = 15 - 9
d = 6
Therefore, Rachel has 6 dimes.
In summary, Rachel has 6 dimes and 9 quarters.