A jar contains only pennies, nickels, and dimes. The number of dimes is one more than the
number of nickels, and the number of pennies is six more than the number of nickels.
How many of each denomination can be found in the jar if the total value is $4.80?
Both conditions are referenced in terms of nickels, so
let the number of nickels be x
number of dimes = x+1
number of pennies = x+6
Value equation:
5x + 10(x+1) + x+6 = 480
continue ....
Let's solve this step by step.
Let's assume the number of nickels in the jar is "x".
According to the given information:
- The number of dimes is one more than the number of nickels. So, the number of dimes is "x + 1".
- The number of pennies is six more than the number of nickels. So, the number of pennies is "x + 6".
Now, let's calculate the value of each denomination in terms of nickels:
- Value of nickels = 5 cents * x
- Value of dimes = 10 cents * (x + 1)
- Value of pennies = 1 cent * (x + 6)
The total value of all the coins is $4.80, so we can write the equation:
5x + 10(x + 1) + 1(x + 6) = 480 cents
Now, let's solve the equation to find the value of "x" (the number of nickels).
5x + 10x + 10 + x + 6 = 480
16x + 16 = 480
16x = 480 - 16
16x = 464
x = 464 / 16
x = 29
So, the number of nickels in the jar is 29.
The number of dimes is x + 1 = 29 + 1 = 30.
The number of pennies is x + 6 = 29 + 6 = 35.
Therefore, there are 29 nickels, 30 dimes, and 35 pennies in the jar.
To solve this problem, we need to set up a system of equations based on the given information.
Let's use the variables:
P = number of pennies
N = number of nickels
D = number of dimes
Based on the given information, we can form the following equations:
1) The number of dimes is one more than the number of nickels:
D = N + 1
2) The number of pennies is six more than the number of nickels:
P = N + 6
3) The total value of the coins is $4.80:
The value of pennies (in cents): P * 1
The value of nickels (in cents): N * 5
The value of dimes (in cents): D * 10
Using the given values, we can form the equation:
P * 1 + N * 5 + D * 10 = 480 cents
Now, substituting the values from equations 1 and 2 into equation 3, we get:
(N + 6) * 1 + N * 5 + (N + 1) * 10 = 480
Simplifying the equation:
N + 6 + 5N + 10N + 10 = 480
16N + 16 = 480
16N = 464
N = 464 / 16
N = 29
Now, we can substitute the value of N back into equations 1 and 2 to find the values of P and D.
From equation 2:
P = N + 6
P = 29 + 6
P = 35
From equation 1:
D = N + 1
D = 29 + 1
D = 30
Therefore, there are 35 pennies, 29 nickels, and 30 dimes in the jar.