A family has a coin jar that is now full. The children count the change and calculate the total value to be $35.37. Let Q represent the number of quarters and use the information below to find the number of each coin.
There are:
112 more dimes than quarters
7 times as many nickels as quarters
17 more than 10 times as many pennies as quarters
d = Q + 112
n = 7 Q
p = 10 Q + 17
p + 5 n + 10 d + 25 Q = 3537 ... substitute a Q equivalent for d, n, and p
solve for Q , then substitute back to find the other values
To find the number of each coin, we need to set up a system of equations based on the given information.
Let's use variables to represent the number of each coin:
Q = number of quarters
D = number of dimes
N = number of nickels
P = number of pennies
According to the information given:
1. "There are 112 more dimes than quarters."
This can be represented as: D = Q + 112
2. "7 times as many nickels as quarters."
This can be represented as: N = 7Q
3. "17 more than 10 times as many pennies as quarters."
This can be represented as: P = 10Q + 17
Now, let's use the value of each coin and set up an equation for the total value of the coins:
Total value = (value of a quarter * number of quarters) + (value of a dime * number of dimes) + (value of a nickel * number of nickels) + (value of a penny * number of pennies)
The value of a quarter is $0.25, a dime is $0.10, a nickel is $0.05, and a penny is $0.01.
The equation for the total value of the coins is:
$35.37 = (0.25 * Q) + (0.10 * D) + (0.05 * N) + (0.01 * P)
Now, substitute the values from the previous equations into the total value equation:
$35.37 = (0.25 * Q) + (0.10 * (Q + 112)) + (0.05 * 7Q) + (0.01 * (10Q + 17))
Simplify and solve this equation to find the value of Q (number of quarters). Once you have the value of Q, you can substitute it back into the other equations to find the number of each coin.