a) To find a formula for the sum registered ($C) in terms of the number of notes (n) counted, we can use the given information to set up a system of equations.
From the given information, we have two data points:
When 120 notes were counted, the registered sum was $775.
When 160 notes were counted, the registered sum was $975.
Let's denote the formula connecting $C and n as C = an + b, where a and b are constants to be determined.
Using the first data point, we can set up the equation:
775 = a * 120 + b
Using the second data point, we can set up the equation:
975 = a * 160 + b
We now have a system of two equations in two variables. To solve for a and b, we can subtract the first equation from the second equation:
975 - 775 = (a * 160 + b) - (a * 120 + b)
200 = a * (160 - 120)
Simplifying the equation further:
200 = a * 40
a = 200/40
a = 5
Substituting the value of a back into the first equation, we can solve for b:
775 = 5 * 120 + b
775 = 600 + b
b = 775 - 600
b = 175
Therefore, the formula for the sum registered ($C) in terms of the number of notes (n) counted is:
C = 5n + 175.
b) To determine if there was a sum already on the register when counting began, we can substitute n = 0 into the formula C = 5n + 175:
C = 5 * 0 + 175
C = 0 + 175
C = 175
The sum already on the register when counting began was $175.