20 points) Lawyers in a law firm prepare documents and spend their time
mentoring law. For a lawyer to prepare any page of paperwork, they need
to receive 15m of law mentoring from another lawyer. Each hour of men-
torship, no matter who receives it, needs to be recorded by filling out half
a page of paperwork.
The law firm’s main business is to file paperwork on behalf of its clients.
In the past year, the law firm filed 48 pages of paperwork on behalf of
its clients. For community outreach, the law firm also mentored students
from a local high school for 79 hours.
a) (4 points) Express the law firm’s activity in the previous year as a
system of equations.
b) (4 points) How many hours of mentoring were internal?

### Part (a) - Expressing the firm's activities as a system of equations

Let's define the variables:
- Let \( P \) be the number of pages of paperwork filed by the firm.
- Let \( M_i \) be the number of hours of internal mentoring between lawyers.
- Let \( M_s \) be the number of hours of mentoring given to students (external mentoring).

We know the following:
1. **Paperwork and mentoring relationship:** For every page of paperwork filed, a lawyer must have received 15 minutes (i.e., \( \frac{1}{4} \) hours) of internal mentoring from another lawyer. Hence, for \( P \) pages of paperwork, the internal mentoring received by all lawyers must be \( \frac{1}{4} P \) hours.
   
   So, \( M_i = \frac{1}{4} P \).

2. **Mentorship records:** Each hour of mentoring requires filling out \( \frac{1}{2} \) page of paperwork. Therefore, the number of pages of paperwork required for internal mentoring \( M_i \) and external mentoring \( M_s \) combined is \( \frac{1}{2}(M_i + M_s) \).

In the past year, we also know the following:
- The firm filed 48 pages of paperwork, so \( P = 48 \).
- The firm mentored students for 79 hours, so \( M_s = 79 \).

Thus, the system of equations is:

\[
M_i = \frac{1}{4} P
\]
\[
P = \frac{1}{2}(M_i + M_s) + 48
\]

### Part (b) - Finding internal mentoring hours

To find how many hours of internal mentoring occurred, substitute \( P = 48 \) and \( M_s = 79 \) into the system of

Lawyers in a law firm prepare documents and spend their time mentoring law. For a lawyer to prepare any page of paperwork, they need to receive 15m of law mentoring from another lawyer. Each hour of mentorship, no matter who receives it, needs to be recorded by filling out half a page of paperwork. The law firm’s main business is to file paperwork on behalf of its clients. In the past year, the law firm filed 48 pages of paperwork on behalf of its clients. For community outreach, the law firm also mentored students from a local high school for 79 hours. a) Express the law firm’s activity in the previous year as a system of equations. b) How many hours of mentoring were internal?

Explore how a law firm's internal mentoring and paperwork filing system can be expressed as a system of equations. Learn how to calculate internal mentoring hours and paperwork requirements. Suitable for high school students learning algebra and systems of equations.

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