Need to find value of X in the below example.

Question

A payroll service has given us the total amount to pay for holidays. For the given shift, we have to pay $656.25, but this total is broken down into 2 segments, regular time, and holiday time. The total hours for this shift is 11.50. The regular time hours are 5. The holiday time hours are 6.50. The holiday time is paid at 1.25% of the regular time. The answer to the value of X in this case is $50, but I need to know the algebraic formula for this equation. See the calculation example below.

50*5=250

50*1.25*6.5=406.25

Total=$656.25

Answer 1

Interpreting your calculation I see that the first line seems to be the cost of 5 hours at $50 per hour regular time, giving $250. The second line doesn't quite seem to make sense, because you quoted holiday time as being paid at a rate of 1.25%. You haven't converted that to a fraction or decimal so my guess is that it should have been 125%, which is 1.25 as a fraction. That means that holiday time is paid at "time and a quarter", compared to regular time. In other words 1 hour of holiday time is worth 1 hour 15 minutes of regular time. When interpreted like this the second line makes sense, because it's the cost of 6.5 hours paid at time and a quarter compared to regular time. The third line simply computes the total for 11.5 hours.

OK. Now the algebra. Let's call the number of hours of regular time R and the holiday hours H. Let's call the regular rate of pay $X per hour. Let's call the enhanced rate for the holiday pay r%, meaning the additional amount to be added to the regular rate. In the example, r would be 25%. Let's call the total amount paid for the 11-hour shift T. Now we can write:

T = RX+ HX(1+r/100) or X(R+H(1+r/100))