You have 29 coins, the total value of which is $1.77. The number of quarter is eight less than the number of pennies and the rest are nickels. How many of each coin do you have?
Q=P-8
N=29-Q-P
25Q+P+5N=177
can you solve this by substution?
To solve this problem, we can use a system of linear equations. Let's assign variables to represent the number of each type of coin.
Let:
P = Number of pennies
Q = Number of quarters
N = Number of nickels
From the given information, we can establish the following equations:
1) P + Q + N = 29 (The total number of coins is 29)
2) 0.01P + 0.25Q + 0.05N = 1.77 (The total value of the coins is $1.77)
Now let's find the relationship between P, Q, and N based on the given information:
- The number of quarters is eight less than the number of pennies: Q = P - 8
We can substitute this relationship into equation 1:
P + (P - 8) + N = 29
2P - 8 + N = 29
2P + N = 37
Now we have a system of equations:
2P + N = 37
0.01P + 0.25(P - 8) + 0.05N = 1.77
We can solve this system of equations to find the values of P, Q, and N.
First, let's simplify the second equation:
0.01P + 0.25P - 2 + 0.05N = 1.77
0.26P + 0.05N = 3.77
Now we can solve the system using these two equations:
2P + N = 37 ...(3)
0.26P + 0.05N = 3.77 ...(4)
We can solve this system by either substitution, elimination, or matrix operations. Let's use the substitution method:
From equation (3), we can express N in terms of P:
N = 37 - 2P
Substitute this into equation (4):
0.26P + 0.05(37 - 2P) = 3.77
0.26P + 1.85 - 0.1P = 3.77
0.16P = 3.77 - 1.85
0.16P = 1.92
P = 1.92 / 0.16
P = 12
Substitute P = 12 back into equation (3):
2(12) + N = 37
24 + N = 37
N = 37 - 24
N = 13
Now that we have the values of P and N, we can find Q:
Q = P - 8
Q = 12 - 8
Q = 4
Therefore, you have 12 pennies, 4 quarters, and 13 nickels.