karen saves her change and has dimes nickels and pennies totaling $4.52. If there is 1 more dime than three times the number of nickels and 10 more pennies than nickels, how many of each type of coin are there?
Let N=the number of nickels (5 cents).
Dimes (10 cents), D = 3N+1
Pennies (1 cent), P = N+10
Then
5N + 10(3N+1) + 1(N+10) = 452
36N + 20 = 452
Solve for N (=12) and the rest of the coins.
To solve this problem, we need to set up a system of equations and solve them simultaneously. Let's define some variables:
Let:
D = number of dimes
N = number of nickels
P = number of pennies
We are given the following information:
1. Karen saves her change, which totals $4.52. We can write this equation as:
0.10D + 0.05N + 0.01P = 4.52 (converting the coin values to dollars)
2. There is 1 more dime than three times the number of nickels. We can write this equation as:
D = 3N + 1
3. There are 10 more pennies than nickels. We can write this equation as:
P = N + 10
Now, we have a system of three equations. To solve it, we can use a method called substitution.
We'll start by using equation 2 to solve for D in terms of N:
D = 3N + 1
Next, we'll use equation 3 to solve for P in terms of N:
P = N + 10
Substitute these expressions into equation 1:
0.10(3N + 1) + 0.05N + 0.01(N + 10) = 4.52
Now, we can solve this equation for N by simplifying and combining like terms:
0.3N + 0.1 + 0.05N + 0.01N + 0.1 = 4.52
Combine like terms:
0.36N + 0.2 = 4.52
Subtract 0.2 from both sides:
0.36N = 4.52 - 0.2
Simplify:
0.36N = 4.32
Divide both sides by 0.36:
N = 4.32 / 0.36
N = 12
Now, substitute this value of N back into equation 2 to find D:
D = 3N + 1
D = 3(12) + 1
D = 36 + 1
D = 37
Finally, substitute N into equation 3 to find P:
P = N + 10
P = 12 + 10
P = 22
Therefore, Karen has 37 dimes, 12 nickels, and 22 pennies.