Two teachers are grading math homework. one teacher can grade a stack of papers in 50 minutes and the other can complete the job in 100 minutes. how long does it take them to grade all the papers if the first teacher grades alone for 23 minutes, then they work together to finish?

If the amount of papers of homework are P, then the time needed to correct them is: T = (P - 23/50)*(100/3) + 23 minutes.

Step-by-step explanation:

First, we need to calculate the amount of work done per minute by each teacher.

We know that the first teacher, T1, can complete one work in 50 minutes, so per minute he does 1/50 of the work

The second teacher, T2, can complete one work in 100 minutes, so he completes 1/100 of the work per minute.

Now, if we define t as the time in minutes, the equation for the amount of work corrected by minute is:

W(t) = (1/50)*23 + (1/50)*t + (1/100)*t = (23/50) + (1/50 + 1/100)*t

W(t) = 23/50 + (3/100)*t

where the 23/50 apears because the first teacher started 23 minutes earlier.

Now if P is the total amount of papers to correct, P a whole number, the equation we need to solve is:

W(t) = P = 23/50 + (3/100)*t

We need to isolate t in this equation.

t = (P - 23/50)*(100/3)

but remember that we also need to take into account the 23 minutes that the first teacher worked alone, so the time that the teachers would need to correct P papers is:

T = (P - 23/50)*(100/3) + 23 minutes.

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