A coin bank contains only nickels,dimes, and quarters. the value of the 19 coins in the bank is $2.00. There are twice as many nickels as dimes. Find the number of each type of coin in the bank.
How do I solve this?
To solve this problem, you can set up a system of equations based on the given information.
Let's create variables for the number of nickels, dimes, and quarters in the bank.
Let N be the number of nickels, D be the number of dimes, and Q be the number of quarters.
1. The total number of coins is 19:
N + D + Q = 19 (Equation 1)
2. The value of all the coins is $2.00:
0.05N + 0.10D + 0.25Q = 2.00 (Equation 2)
3. There are twice as many nickels as dimes:
N = 2D (Equation 3)
Now you can solve this system of equations using substitution or elimination method.
Substitution Method:
Substitute Equation 3 into Equation 1:
2D + D + Q = 19
3D + Q = 19 (Equation 4)
Substitute Equation 3 into Equation 2:
0.05(2D) + 0.10D + 0.25Q = 2.00
0.10D + 0.10D + 0.25Q = 2.00
0.20D + 0.25Q = 2.00 (Equation 5)
Now you have two equations with two variables (Equations 4 and 5). You can solve this system of equations to find the values of D (number of dimes) and Q (number of quarters).
Elimination Method:
Multiply Equation 4 by 0.20 and Equation 5 by 3 to eliminate D:
0.60D + 0.20Q = 3.80 (Equation 6)
0.60D + 0.75Q = 6.00 (Equation 7)
Now subtract Equation 6 from Equation 7 to solve for Q:
0.60D + 0.75Q - 0.60D - 0.20Q = 6.00 - 3.80
0.55Q = 2.20
Q = 2.20 / 0.55
Q = 4
Substitute the value of Q (4) into Equation 4 to find D:
3D + 4 = 19
3D = 19 - 4
3D = 15
D = 15 / 3
D = 5
Finally, substitute the values of D (5) and Q (4) into Equation 3 to find N:
N = 2D
N = 2(5)
N = 10
Therefore, the number of nickels, dimes, and quarters in the bank are N = 10, D = 5, and Q = 4, respectively.
To solve this problem, we can set up a system of equations to represent the given information.
Let's represent the number of nickels as 'n', the number of dimes as 'd', and the number of quarters as 'q'.
1) The first equation represents the total number of coins: n + d + q = 19
2) The second equation represents the total value of the coins: 0.05n + 0.10d + 0.25q = 2.00
3) The third equation represents the relationship between the number of nickels and dimes: n = 2d
Now, we can solve these equations to find the values for 'n', 'd', and 'q'.
First, let's substitute n in terms of d into equations 1 and 2:
n + d + q = 19 --> 2d + d + q = 19 --> 3d + q = 19 (Equation 4)
0.05n + 0.10d + 0.25q = 2.00 --> 0.05(2d) + 0.10d + 0.25q = 2.00 --> 0.10d + 0.10d + 0.25q = 2.00 --> 0.20d + 0.25q = 2.00 (Equation 5)
Now we have two equations with two variables: Equation 4 and Equation 5. We can solve them simultaneously using substitution or elimination.
Let's use elimination.
Multiply Equation 4 by 0.20:
(3d + q) * 0.20 = 19 * 0.20
0.60d + 0.20q = 3.80 (Equation 6)
Subtract Equation 6 from Equation 5:
(0.20d + 0.25q) - (0.60d + 0.20q) = 2.00 - 3.80
0.20d - 0.60d + 0.25q - 0.20q = -1.80
-0.40d + 0.05q = -1.80 (Equation 7)
Now we have two linear equations:
-0.40d + 0.05q = -1.80 (Equation 7)
0.20d + 0.25q = 2.00 (Equation 5)
To eliminate the decimals, we can multiply both sides of Equation 7 by 4 and Equation 5 by 20:
(-0.40d + 0.05q) * 4 = (-1.80) * 4
-1.60d + 0.20q = -7.20 (Equation 8)
(0.20d + 0.25q) * 20 = 2.00 * 20
4.00d + 5.00q = 40.00 (Equation 9)
Now, let's multiply Equation 8 by -5 and Equation 9 by 2 to eliminate 'q':
(-1.60d + 0.20q) * -5 = (-7.20) * -5
8.00d - 1.00q = 36.00 (Equation 10)
(4.00d + 5.00q) * 2 = 40.00 * 2
8.00d + 10.00q = 80.00 (Equation 11)
Now, let's subtract Equation 10 from Equation 11:
(8.00d + 10.00q) - (8.00d - 1.00q) = 80.00 - 36.00
8.00d - 8.00d + 10.00q + 1.00q = 44.00
11.00q = 44.00
Divide both sides of the equation by 11:
11.00q/11 = 44.00/11
Simplifying:
q = 4
Now that we know the value of 'q', we can substitute it back into Equation 4 to find 'd':
3d + q = 19
3d + 4 = 19
3d = 19 - 4
3d = 15
Divide both sides by 3:
d = 5
Finally, let's substitute the values of 'q' and 'd' into Equation 3 to find 'n':
n = 2d
n = 2 * 5
n = 10
Therefore, the number of nickels (n), dimes (d), and quarters (q) in the bank are 10, 5, and 4 respectively.